On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains: clustering concentration layers
aa r X i v : . [ m a t h . A P ] F e b ON AMBROSETTI-MALCHIODI-NI CONJECTURE ON TWO-DIMENSIONALSMOOTH BOUNDED DOMAINS: CLUSTERING CONCENTRATION LAYERS
SUTING WEI ‡ AND JUN YANG § Abstract.
We consider the problem ε div (cid:0) ∇ a ( y ) u (cid:1) − V ( y ) u + u p = 0 , u > , ∇ a ( y ) u · ν = 0 on ∂ Ω , where Ω is a bounded domain in R with smooth boundary, the exponent p is greater than 1, ε > V is a uniformly positive smooth potential on ¯Ω, and ν denotes the outward normal of ∂ Ω.For two positive smooth functions a ( y ) , a ( y ) on ¯Ω, the operator ∇ a ( y ) is given by ∇ a ( y ) u = a ( y ) ∂u∂y , a ( y ) ∂u∂y ! . (1). Let Γ ⊂ ¯Ω be a smooth curve intersecting orthogonally with ∂ Ω at exactly two points and dividingΩ into two parts. Moreover, Γ is a non-degenerate geodesic embedded in the Riemannian manifold R withmetric V σ ( y ) (cid:2) a ( y )d y + a ( y )d y (cid:3) , where σ = p +1 p − − . By assuming some additional constraints onthe functions a ( y ), V ( y ) and the curves Γ, ∂ Ω, we prove that there exists a sequence of ε such that theproblem has solutions u ε with clustering concentration layers directed along Γ, exponentially small in ε atany positive distance from it.(2). If ˜Γ is a simple closed smooth curve in Ω (not touching the boundary ∂ Ω), which is also a non-degenerate geodesic embedded in the Riemannian manifold R with metric V σ ( y ) (cid:2) a ( y )d y + a ( y )d y (cid:3) ,then a similar result of concentrated solutions is still true. Keywords:
Ambrosetti-Malchiodi-Ni conjecture, Clustering concentration layers, Toda-Jacobi system,Resonance phenomena
MSC 2020: Introduction
We consider the following problem for the existence of solutions with concentration phenomena ε div (cid:0) ∇ a ( y ) u (cid:1) − V ( y ) u + u p = 0 , u > , ∇ a ( y ) u · ν = 0 on ∂ Ω , (1.1)where Ω is a bounded domain in R d with smooth boundary, ε > V is a uniformlypositive, smooth potential on ¯Ω, and ν denotes the outward normal of ∂ Ω, the exponent p >
1. For a ( y ) = (cid:0) a ( y ) , · · · , a d ( y ) (cid:1) , the operator is defined in the form ∇ a ( y ) u = (cid:0) a ( y ) u y , · · · , a d ( y ) u y d (cid:1) , where a ( y ) , · · · , a d ( y ) are positive smooth functions on ¯Ω. • In the case a ≡ V ≡
1, problem (1.1) takes the form ε ∆ u − u + u p = 0 , u > , ∇ u · ν = 0 on ∂ Ω , (1.2)which is known as the stationary equation of Koeller-Segel system in chemotaxis [42]. It can also be viewedas a limiting stationary equation of Gierer-Meinhardt system in biological pattern formation [34].In the pioneering papers [42, 51, 52], under the condition that p is a subcritical Sobolev exponent, C.-S.Lin, W.-M. Ni and I. Takagi established, for ε sufficiently small, the existence of a least-energy solution U ε of (1.2) with only one local maximum point locating at the most curved point of ∂ Ω. Such a solution iscalled a spike-layer, which has concentration phenomena at interior or boundary points. For the existence of ‡ Department of Mathematics, South China Agricultural University, Guangzhou, 510642, P. R. China. Email:[email protected]. § School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, P. R. China. Email:[email protected] author: Jun Yang, [email protected]. ‡ AND JUN YANG § interior spikes, we refer the reader to the articles [10, 14, 21, 35, 36, 56] and the references therein. On theother hand, boundary spikes related to the mean curvature of ∂ Ω can be found in [11, 13, 20, 38, 41, 55, 58],and the references therein. The coexistence of interior and boundary spikes was due to C. Gui and J. Wei[37]. A good review of the subject up to 2004 can be found in [50].There is a conjecture on higher-dimensional concentration by W.-M. Ni [49] (see also [50]):
Conjecture 1.
For any integer ≤ k ≤ d − , there exists p k ∈ (1 , ∞ ) such that for all < p < p k , problem(1.2) has a solution U ε which concentrates on a k -dimensional subset of ¯Ω , provided that ε is sufficientlysmall. (cid:3) We here mention some results for the existence of higher dimensional boundary concentration phenomenain the papers [43, 45, 46, 47, 48]. The papers [59, 60] set up the existence of concentration on an interiorline, which connects the boundary ∂ Ω and is non-degenerate in the sense of variation of arc-length. Thereare also some other results [5, 6, 28, 29, 30] to exhibit concentration phenomena on interior line segmentsconnecting the boundary of Ω. For higher dimensional extension, the reader can refer to [7, 15, 26, 39, 40].The reader can also refer to the survey paper by J. Wei [57]. • In the case a ≡ V constant, problem (1.1) on the whole space corresponds to the following problem ε ∆ u − V ( y ) u + u p = 0 , u > R d , (1.3)where ε > p >
1, and V is a smooth function withinf y ∈ R d V ( y ) > . Started by [33], solutions exhibiting concentration around one or more points of space under various assump-tions on the potential and the nonlinearity were given by many authors [1, 4, 12, 16, 17, 18, 19, 32, 54]. Onthe other hand, radially symmetric solutions with concentration on sphere of radius r can be constructedin [2], whenever V is a radial function and r > M ( r ) = r d − V σ ( r ) , where σ = p + 1 p − − . Based on heuristic arguments, in 2003, A. Ambrosetti, A. Malchiodi and W.-M. Ni raised the followingconjecture (p.465, [2]):
Conjecture 2.
Let K be a non-degenerate k -dimensional stationary manifold of the following functional Z K V p +1 p − − ( d − k ) , where the exponent p is subcritical w.r.t. R d − k . Then there exists a solution to (1.3) concentrating near K ,at least along a subsequence ε j → . (cid:3) The validity of this conjecture was confirmed in two-dimensional general case with concentration on station-ary and non-degenerate curves for ε satisfying a gap condition due to the resonance character of the problem,see [22]. More results can be found in [8, 9, 53, 44].Let us go back to the problem on smooth bounded domains with homogeneous Neumann boundarycondition, ε ∆ u − V ( y ) u + u p = 0 , u > , ∇ u · ν = 0 on ∂ Ω , (1.4)where Ω is a bounded domain in R d with smooth boundary, ε > V is a uniformlypositive, smooth potential on ¯Ω, and ν denotes the outward normal of ∂ Ω and the exponent p >
1. If Ω is aunit ball B (0), the existence of radial solutions to (1.4) was shown in [3], where the concentration lying onspheres in B (0) will approach the boundary with speed O ( ε | log ε | ) as ε →
0. For the interior concentrationphenomena connecting the boundary ∂ Ω, there is a conjecture by A. Ambrosetti, A. Malchiodi and W.-M.Ni in 2004 (p. 327, [3]), which can be stated as:
Conjecture 3.
Let K be a k -dimensional manifold intersecting ∂ Ω perpendicularly, which is also stationaryand non-degenerate with respect to the following functional Z K V p +1 p − − ( d − k ) , MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 3 where the exponent p is subcritical w.r.t. R d − k . Then there exists a solution to (1.4) concentrating near K ,at least along a subsequence ε j → . (cid:3) In [62], S. Wei, B. Xu and J. Yang considered (1.4) and provided an affirmative answer to the mentionedconjecture only in the case: d = 2 and k = 1 for the existence of solutions with single concentration layerconnecting the boundary ∂ Ω. • In present paper, we will consider a little bit more general case, i.e., problem (1.1), and investigate firstthe existence of clustering phenomena of multiple concentration layers, which connect the boundary ∂ Ω. Asthe descriptions in Concluding remarks of [24], the difficulties arise from the multiple resonance phenomena,see also Remark 1.5. On the other hand, the much more complicated situation is the balance betweenneighbouring layers plus the interaction among the interior concentration layers, the boundary of Ω and thecompetition between a and V . It is natural to introduce the new ingredient (1.9) to handle this delicatething. Whence, in the present paper we focus on the two dimensional case of problem (1.1). We make thefollowing assumptions: (A1). Let Ω be a smooth and bounded domain in R , Γ be a curve intersecting ∂ Ω at exactly two points,saying P , P , and, at these points Γ ⊥ ∂ Ω . In the small neighborhoods of P , P , the boundary ∂ Ω are two curves, say C and C , which can be represented by the graphs of two functions respectively: y = ϕ ( y ) with (0 , ϕ (0)) = P ,y = ϕ ( y ) with (0 , ϕ (0)) = P . Without loss of generality, we can assume that Γ has length , and then denote k , k the signedcurvatures of C and C respectively, also k the curvature of Γ . (A2). Γ separates the domain Ω into two disjoint components Ω and Ω . (A3). The functions a ( y ) and a ( y ) satisfy the condition a = a at the points P and P . (1.5) The curve Γ is a non-degenerate geodesic embedded in the Riemannian manifold R with the followingmetric V σ ( y ) (cid:2) a ( y )d y + a ( y )d y (cid:3) , with σ = p + 1 p − − . (1.6) This will be clarified in the next section (see (2.99) and (2.120)). (cid:3)
Let w denote the unique positive solution of the problem w ′′ − w + w p = 0 , w > R , w ′ (0) = 0 , w ( ±∞ ) = 0 . (1.7)We can formulate the first result. Theorem 1.1.
Let d = 2 , p > and recall the assumptions in (A1) - (A3) as well as the modified Fermicoordinates ( t, θ ) in (2.13) . Moreover, we assume that τ ( θ ) ≡ H ′ ( θ ) − H ( θ ) + 2 | β ′ ( θ ) | β ( θ ) H ( θ ) − β ′′ ( θ ) β ( θ ) H ( θ ) − β ′ ( θ ) β ( θ ) H ′ ( θ ) > , (1.8) and also the validity of the admissibility conditions b b + β ′ (0) β (0) = 0 , b b + β ′ (1) β (1) = 0 , (1.9) where the function β > is defined in (3.24), the functions H , H and H are given in (2.111) - (2.113) ,and the constants b , b , b , b are given in (2.77) and (2.80) . Then for each N , there exists a sequenceof ε , say { ε l } , such that problem (1.1) has a positive solution u ε l with exactly N concentration layers atmutual distances O ( ε l | ln ε l | ) . In addition, the center of mass for N concentration layers collapses to Γ atspeed O ( ε µl ) for some small positive constant µ . More precisely, u ε l has the form u ε l ( y , y ) ≈ N X j =1 V (0 , θ ) p − w vuut V (0 , θ ) (cid:16) a (0 , θ ) | n ( θ ) | + a (0 , θ ) | n ( θ ) | (cid:17) | a (0 , θ ) | + | a (0 , θ ) | t − ε l f j ( θ ) ε l , (1.10) SUTING WEI ‡ AND JUN YANG § where n ( θ ) = ( n ( θ ) , n ( θ )) is the unit normal to Γ . The functions f j ’s satisfy k f j k ∞ ≤ C | ln ε l | , N X j =1 f j = O ( ε µl ) with µ > , (1.11)min ≤ j ≤ N − ( f j +1 − f j ) ≈ | ln ε l | vuut | a (0 , θ ) | + | a (0 , θ ) | V (0 , θ ) (cid:16) a (0 , θ ) | n ( θ ) | + a (0 , θ ) | n ( θ ) | (cid:17) , (1.12) and solve the Jacobi-Toda system, for j = 1 , · · · , N , ε l ς h H f ′′ j + H ′ f ′ j + (cid:0) H ′ − H (cid:1) f j i − e − β ( f j − f j − ) + e − β ( f j +1 − f j ) ≈ in (0 , , (1.13) with boundary conditions b f ′ j (1) − b f j (1) ≈ , b f ′ j (0) − b f j (0) ≈ , ∀ j = 1 , · · · , N, (1.14) where the function ς > is defined in (5.10) with the conventions f = −∞ , f N +1 = ∞ . (cid:3) Here is the second result for the existence of interior clustering concentration layers, which do not touchthe boundary ∂ Ω. Theorem 1.2.
Let d = 2 and p > . Suppose that (ˆ t, ˆ θ ) are the modified Fermi coordinates given in (2.121) .Assume that ˆΓ is a simple closed smooth curve with unit length in Ω , which is also a non-degenerate geodesicembedded in the Riemannian manifold R with the following metric V σ ( y ) (cid:2) a ( y )d y + a ( y )d y (cid:3) , with σ = p + 1 p − − , (1.15) see (2.124) and (2.128). Moreover, we assume that ˆ τ (ˆ θ ) ≡ b H ′ (ˆ θ ) − b H (ˆ θ ) + 2 | β ′ (ˆ θ ) | β (ˆ θ ) b H (ˆ θ ) − β ′′ (ˆ θ ) β (ˆ θ ) b H (ˆ θ ) − β ′ (ˆ θ ) β (ˆ θ ) b H ′ (ˆ θ ) > , (1.16) where the functions β > , b H , b H and b H are given in (3.24) , (2.125) - (2.127) . Then for each N , there existsa sequence of ε , say { ˆ ε l } , such that problem (1.1) has a positive solution u ˆ ε l with exactly N concentrationlayers at mutual distances O (ˆ ε l | ln ˆ ε l | ) . In addition, the center of mass for N concentration layers collapsesto ˆΓ at speed O (ˆ ε µl ) for some small positive constant µ . More precisely, u ˆ ε l has the form u ˆ ε l ( y , y ) ≈ N X j =1 V (0 , ˆ θ ) p − w vuut V (0 , ˆ θ ) (cid:16) a (0 , ˆ θ ) | ˆ n (ˆ θ ) | + a (0 , ˆ θ ) | ˆ n (ˆ θ ) | (cid:17) | a (0 , ˆ θ ) | + | a (0 , ˆ θ ) | ˆ t − ˆ ε l ˆ f j (ˆ θ )ˆ ε l , (1.17) where ˆ n (ˆ θ ) = (ˆ n (ˆ θ ) , ˆ n (ˆ θ )) is the unit normal to ˆΓ . The functions ˆ f j ’s satisfy k ˆ f j k ∞ ≤ C | ln ˆ ε l | , N X j =1 ˆ f j = O (ˆ ε µl ) with µ > , (1.18)min ≤ j ≤ N − ( ˆ f j +1 − ˆ f j ) ≈ | ln ˆ ε l | vuut | a (0 , ˆ θ ) | + | a (0 , ˆ θ ) | V (0 , ˆ θ ) (cid:16) a (0 , ˆ θ ) | ˆ n (ˆ θ ) | + a (0 , ˆ θ ) | ˆ n (ˆ θ ) | (cid:17) , (1.19) and solve the Jacobi-Toda system, for j = 1 , · · · , N , ˆ ε l ς h b H ˆ f ′′ j + b H ′ ˆ f ′ j + (cid:0) b H ′ − b H (cid:1) ˆ f j i − e − β ( ˆ f j − ˆ f j − ) + e − β ( ˆ f j +1 − ˆ f j ) ≈ in (0 , , (1.20) with boundary conditions ˆ f ′ j (0) = ˆ f ′ j (1) , ˆ f j (0) = ˆ f j (1) , ∀ j = 1 , · · · , N, where ς > is defined in (5.10) with the conventions ˆ f = −∞ , ˆ f N +1 = ∞ . (cid:3) MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 5
Here are some words for further discussions. Since the solutions have exponential decaying as y leavesaway the curve ˜Γ, the proof of Theorem 1.2 is much more simpler than that of Theorem 1.1. Whence,in the present paper, we will only provide the details to show the validity of Theorem 1.1. Based on thesame reason, a same result as in Theorem 1.2 also holds for the first equation of (1.1) in the whole space R under the condition u ( y ) → | y | → ∞ . Whence, Theorems 1.1 and 1.2 for the existence of clusterof multiple concentration layers can be concerned as the extensions of the results in [62] and [22], wheresolutions with single concentration layer were constructed for partial confirmation of the two dimensionalcases of Conjectures 2 and 3.However, in addition to the interaction between neighbouring layers in the cluster of multiple concentrationlayers, the new ingredient is the role of the term a ( y ) = (cid:0) a ( y ) , a ( y ) (cid:1) . This is the reason that we shall setup the new local coordinates, see (2.8) together with (2.13) and also (2.121). In the procedure of variationalcalculus, the deformations of the curves Γ and ˜Γ are no longer directed along their normal directions, see(2.82)-(2.83) and (2.122)-(2.123). The term a will play an effect in the variational properties of the curves,see the notions of non-degenerate stationary curves in Section 2.4. Remark 1.3.
The Toda system was used first in [23] to construct the clustered interfaces for Allen-Cahnmodel in a two dimensional bounded domain. Later, M. del Pino, M. Kowalczyk, J. Wei and J. Yang [27] used the Jacobi-Toda system in the construction of clustered phase transition layers for Allen-Cahn modelon general Riemannian manifolds. The reader can refer to [24, 25, 60, 61, 63, 64, 65] for more results.For a d -dimensional smooth compact Riemannian manifold ( ˜ M , g ) , M. del Pino, M. Kowalczyk, J. Weiand J. Yang [27] considered the singularly perturbed Allen-Cahn equation ε ∆ g u + (1 − u ) u = 0 in ˜ M , where ε is a small parameter. We let in what follows K be a minimal ( d − -dimensional embedded subman-ifold of ˜ M , which divides ˜ M into two open components ˜ M ± . (The latter condition is not needed in somecases.) Assume that K is non-degenerate in the sense that it does not support non-trivial Jacobi fields, andthat |A K | + Ric g ( ν K , ν K ) > along K . (1.21) Then for each integer N ≥ , they established the existence of a sequence ε = ε j → , and solutions u ε with N -transition layers near K , with mutual distance O ( ε | ln ε | ) .As the above geometric language, we consider R as a manifold with the metric ˜ g = V σ ( y ) (cid:2) a ( y )d y + a ( y )d y (cid:3) , with σ in (1.6) and Γ as its submanifold with boundary. In the manifold ( R , ˜ g ) , Γ is a non-degenerategeodesic with endpoints on ∂ Ω . In other words, in order to construct the clustering phase transition layersconnecting the boundary ∂ Ω in Theorem 1.1, we need the condition (1.8), which is similar as (1.21) in [27] .The reader can refer to Section 6.2. (cid:3) Remark 1.4. At P and P (the intersection points of Γ and ∂ Ω ), the conditions in (1.9) set up relationsbetween the terms a , V and the geometric properties of the curves ∂ Ω and Γ . For example, by recalling theunit normal n ( θ ) = ( n ( θ ) , n ( θ )) to Γ and also the curvatures k and k of ∂ Ω at P and P , the first onein (1.9) can be exactly expressed in the following form √ a (0 , h ∂ t a (0 , − ∂ t a (0 , i n (0) n (0) + √ (cid:16) ˜ a ′ (0) | n (0) | + ˜ a ′ (0) | n (0) | (cid:17) + k = β ′ (0) β (0) , (1.22) with the conventions ˜ a ( θ ) = a (0 , θ ) p | a (0 , θ ) | + | a (0 , θ ) | , ˜ a ( θ ) = a (0 , θ ) p | a (0 , θ ) | + | a (0 , θ ) | ,β ( θ ) = s V (0 , θ ) (cid:0) a (0 , θ ) | n ( θ ) | + a (0 , θ ) | n ( θ ) | (cid:1) | a (0 , θ ) | + | a (0 , θ ) | , where we have used (2.6) , (3.24) , (E.26) , (E.27) . These conditions will be used to decompose the interactionof neighbouring layers on the boundary ∂ Ω , see Remark 6.3. On the other hand, we still have to deal withthe delicate boundary terms for the reduced equations in Section 6.2. (cid:3) SUTING WEI ‡ AND JUN YANG § Remark 1.5.
We construct solutions with multiple clustering concentration layers only for a sequence of ε due to the coexistence of two types of resonances, see also the fourth open question in ”Concluding Remarks”of [24] . The first one is due to the instability of the profile function w , see Proposition 6.1, in which we canimpose the following gap condition for ε | λ ∗ − j ε | ≥ ˜ c ε, ∀ j ∈ N , (1.23) where ˜ c is a given small positive constant. In the above, λ ∗ is a positive constant given by λ ∗ = λ ℓ π , (1.24) where λ and ℓ are the positive constants given in (3.3) and (3.40) . More details about this resonancephenomena were described in [22] . The other one comes from the Jacobi-Toda system which was concernedin [27] . In this case, we shall choose a sequence of ε from those satisfying (1.23), see Proposition 6.2. (cid:3) By the rescaling y = ε ˜ y (1.25)in R , problem (1.1) will be rewritten asdiv (cid:0) ∇ a ( ε ˜ y ) u (cid:1) − V ( ε ˜ y ) u + u p = 0 in Ω ε , ∇ a ( ε ˜ y ) u · ν ε = 0 on ∂ Ω ε , (1.26)where Ω ε = Ω /ε and Γ ε = Γ /ε , ν ε is the unit outer normal of ∂ Ω ε . The remaining part of this paper isdevoted to the proof of Theorem 1.1, which will be organized as follows:1. In Section 2, we will set up a coordinate system in a neighborhood of Γ. Next we write down thelocal form of (1.26), especially the differential operators the differential operators div (cid:0) ∇ a ( y ) u (cid:1) and ∇ a ( y ) u · ν . This local coordinate system also help us set up the stationary and non-degeneracyconditions for the curve Γ, see (2.99) and (2.120).2. We will set up an outline of the proof in Section 3, which involves the gluing procedure from [22], sothat we can transform (1.26) into a projected form, see (3.52)-(3.55).3. In Section 4, we are devoted to the constructing of a local approximate solution in such a way thatit solves the nonlinear problem locally up to order O ( ε ).4. To get a real solution, the well-known infinite dimensional reduction method [22] will be needed inSections 5-6. In fact, the reduced problem involves a Toda-Jacobi system and inherits the resonancephenomena, which will be handled by complicated Fourier analysis.2. Geometric preliminaries
In this section, we will set up a coordinate system in a neighborhood of Γ. This system is similar to themodified Fermi coordinates in [62]. However, some adaptions should be introduced due to the existence of theterm a in (1.1), which make the geometric computations much more complicated. The differential operatorsin (1.1) will be then derived in the local coordinates. The notion of a stationary and non-degenerate curveΓ will be also derived in the last part of this section.2.1. Modified Fermi coordinates.
Recall the assumptions (A1) - (A3) in Section 1 and notation therein. For basic notions of curves, suchas the signed curvature, the reader can refer to the book by do Carmo [31]. Step 1.
Let the natural parameterization of the curve Γ be as follows. γ : [0 , → Γ ⊂ ¯Ω ⊂ R . For some small positive number σ , one can make a smooth extension and define the mapping γ = ( γ , γ ) : ( − σ , σ ) → R , such that γ (˜ θ ) = γ (˜ θ ) , ∀ ˜ θ ∈ [0 , . There holds the Frenet formula γ ′′ = kn and n ′ = − kγ ′ , (2.1) MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 7 where k , n = ( n , n ) are the curvature and the normal of γ . The relations | γ ′ (˜ θ ) | + | γ ′ (˜ θ ) | = 1 , | n (˜ θ ) | + | n (˜ θ ) | = 1 , γ ′ (˜ θ ) n (˜ θ ) + γ ′ (˜ θ ) n (˜ θ ) = 0 , (2.2)will give that (cid:0) γ ′ (˜ θ ) , γ ′ (˜ θ ) (cid:1) = (cid:0) − n (˜ θ ) , n (˜ θ ) (cid:1) , (2.3)and − γ ′ (˜ θ ) n (˜ θ ) + γ ′ (˜ θ ) n (˜ θ ) = 1 . (2.4)Choosing δ > S ≡ ( − δ , δ ) × ( − σ , σ ) , we construct the following mappingˆ H : S → ˆ H ( S ) ≡ ˆΩ δ ,σ with ˆ H (˜ t, ˜ θ ) = γ (˜ θ ) + ˜ t n (˜ θ ) . (2.5)Note that ˆ H is a diffeomorphism (locally) and ˆ H (0 , ˜ θ ) = γ (˜ θ ). By this, we will write the functions a , a inthe forms a (˜ t, ˜ θ ) and a (˜ t, ˜ θ ) and then set˜ a (˜ θ ) = a (0 , ˜ θ ) q | a (0 , ˜ θ ) | + | a (0 , ˜ θ ) | , ˜ a (˜ θ ) = a (0 , ˜ θ ) q | a (0 , ˜ θ ) | + | a (0 , ˜ θ ) | . (2.6)Note that ˜ a (0) = ˜ a (0) = ˜ a (1) = ˜ a (1) = 1 √ , (2.7)due to the assumptions in (1.5). After that, we construct another mapping H : S → H ( S ) ≡ Ω δ ,σ with H (˜ t, ˜ θ ) = γ (˜ θ ) + ˜ t (cid:0) ˜ a (˜ θ ) n (˜ θ ) , ˜ a (˜ θ ) n (˜ θ ) (cid:1) , (2.8)in such a way that it is a local diffeomorphism and H (0 , ˜ θ ) = γ (˜ θ ). This is due to the fact that a and a are positive functions. Step 2 . Recall C , C given in the assumptions (A1) - (A3) in Section 1 and then denote the preimages˜ C ≡ H − ( C ) and ˜ C ≡ H − ( C ) , which are two smooth curves in (˜ t, ˜ θ ) coordinates of (2.8) and can be parameterized respectively by (cid:0) ˜ t, ˜ ϕ (˜ t ) (cid:1) and (cid:0) ˜ t, ˜ ϕ (˜ t ) (cid:1) for some smooth functions ˜ ϕ (˜ t ) and ˜ ϕ (˜ t ) with the properties˜ ϕ (0) = 0 , ˜ ϕ (0) = 1 . (2.9)We define a mapping ˜ H : S → S ≡ ˜ H ( S ) ⊂ R , such that t = ˜ t, θ = ˜ θ − ˜ ϕ (˜ t )˜ ϕ (˜ t ) − ˜ ϕ (˜ t ) . This transformation will straighten up the curves ˜ C and ˜ C . It is obvious that˜ H − (0 , θ ) = (0 , θ ) , θ ∈ [0 , . Moreover, we have
Lemma 2.1.
There hold ˜ ϕ ′ (0) = 0 , ˜ ϕ ′ (0) = 0 , ˜ ϕ ′′ (0) = ˜ k , ˜ ϕ ′′ (0) = ˜ k , (2.10) where ˜ k = (cid:0) | ˜ a (0) n (0) | + | ˜ a (0) n (0) | (cid:1) / ˜ a (0) | n (0) | + ˜ a (0) | n (0) | k = 12 k , (2.11)˜ k = (cid:0) | ˜ a (1) n (1) | + | ˜ a (1) n (1) | (cid:1) / ˜ a (1) | n (1) | + ˜ a (1) | n (1) | k = 12 k . (2.12) SUTING WEI ‡ AND JUN YANG § Proof.
In fact, the curves can be expressed in the following forms C : H (cid:0) ˜ t, ˜ ϕ (˜ t ) (cid:1) = γ (cid:0) ˜ ϕ (˜ t ) (cid:1) + ˜ t (cid:16) ˜ a (cid:0) ˜ ϕ (˜ t ) (cid:1) n (cid:0) ˜ ϕ (˜ t ) (cid:1) , ˜ a (cid:0) ˜ ϕ (˜ t ) (cid:1) n (cid:0) ˜ ϕ (˜ t ) (cid:1)(cid:17) , C : H (cid:0) ˜ t, ˜ ϕ (˜ t ) (cid:1) = γ (cid:0) ˜ ϕ (˜ t ) (cid:1) + ˜ t (cid:16) ˜ a (cid:0) ˜ ϕ (˜ t ) (cid:1) n (cid:0) ˜ ϕ (˜ t ) (cid:1) , ˜ a (cid:0) ˜ ϕ (˜ t ) (cid:1) n (cid:0) ˜ ϕ (˜ t ) (cid:1)(cid:17) , Γ : H (˜ θ ) = γ (˜ θ ) . It follows that the tangent vectors of C at P can be written asd C d˜ t (cid:12)(cid:12)(cid:12) ˜ t =0 = ∂γ∂ ˜ θ (cid:12)(cid:12)(cid:12) ˜ θ = ˜ ϕ (0) · d ˜ ϕ d˜ t (cid:12)(cid:12)(cid:12) ˜ t =0 + (cid:16) ˜ a (cid:0) ˜ ϕ (˜ t ) (cid:1) n (cid:0) ˜ ϕ (˜ t ) (cid:1) , ˜ a (cid:0) ˜ ϕ (˜ t ) (cid:1) n (cid:0) ˜ ϕ (˜ t ) (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) ˜ t =0 = γ ′ (0) ˜ ϕ ′ (0) + (cid:0) ˜ a (0) n (0) , ˜ a (0) n (0) (cid:1) , and the tangent vector of Γ at P is γ ′ (0). According to the condition: Γ ⊥ ∂ Ω at P , we have that (cid:28) d C d˜ t (cid:12)(cid:12)(cid:12) ˜ t =0 , γ ′ (0) (cid:29) = 0 . By (2.2) and (2.7), we have ˜ a (0) n (0) γ ′ (0) + ˜ a (0) n (0) γ ′ (0) = 0 , and then drive from the above to get ˜ ϕ ′ (0) = 0 . Similarly, we can show ˜ ϕ ′ (0) = 0.The curve C can be expressed in the following form C : H (cid:0) ˜ t, ˜ ϕ (˜ t ) (cid:1) = (cid:16) γ (cid:0) ˜ ϕ (˜ t ) (cid:1) + ˜ t ˜ a (cid:0) ˜ ϕ (˜ t ) (cid:1) n (cid:0) ˜ ϕ (˜ t ) (cid:1) , γ (cid:0) ˜ ϕ (˜ t ) (cid:1) + ˜ t ˜ a (cid:0) ˜ ϕ (˜ t ) (cid:1) n (cid:0) ˜ ϕ (˜ t ) (cid:1)(cid:17) ≡ (cid:0) y (˜ t ) , y (˜ t ) (cid:1) . The calculations y ′ (˜ t ) = γ ′ ( ˜ ϕ ) · d ˜ ϕ d˜ t + ˜ a ( ˜ ϕ ) n ( ˜ ϕ ) + ˜ t · ˜ a ( ˜ ϕ ) · n ′ ( ˜ ϕ ) · d ˜ ϕ d˜ t + ˜ t · ˜ a ′ ( ˜ ϕ ) · n ( ˜ ϕ ) · d ˜ ϕ d˜ t ,y ′ (˜ t ) = γ ′ ( ˜ ϕ ) · d ˜ ϕ d˜ t + ˜ a ( ˜ ϕ ) n ( ˜ ϕ ) + ˜ t · ˜ a ( ˜ ϕ ) · n ′ ( ˜ ϕ ) · d ˜ ϕ d˜ t + ˜ t · ˜ a ′ ( ˜ ϕ ) · n ( ˜ ϕ ) · d ˜ ϕ d˜ t , and y ′′ (˜ t ) = γ ′′ ( ˜ ϕ ) · (cid:16) d ˜ ϕ d˜ t (cid:17) + γ ′ ( ˜ ϕ ) · d ˜ ϕ d˜ t + 2 ˜ a ′ ( ˜ ϕ ) · n ( ˜ ϕ ) · d ˜ ϕ d˜ t + 2 ˜ a ( ˜ ϕ ) · n ′ ( ˜ ϕ ) · d ˜ ϕ d˜ t + 2˜ t ˜ a ′ ( ˜ ϕ ) n ′ ( ˜ ϕ ) · (cid:16) d ˜ ϕ d˜ t (cid:17) + ˜ t ˜ a ( ˜ ϕ ) n ′′ ( ˜ ϕ ) · (cid:16) d ˜ ϕ d˜ t (cid:17) + ˜ t ˜ a ( ˜ ϕ ) n ′ ( ˜ ϕ ) · d ˜ ϕ d˜ t + ˜ t ˜ a ′′ ( ˜ ϕ ) n ( ˜ ϕ ) · (cid:16) d ˜ ϕ d˜ t (cid:17) + ˜ t ˜ a ′ ( ˜ ϕ ) n ( ˜ ϕ ) · d ˜ ϕ d˜ t ,y ′′ (˜ t ) = γ ′′ ( ˜ ϕ ) · (cid:16) d ˜ ϕ d˜ t (cid:17) + γ ′ ( ˜ ϕ ) · d ˜ ϕ d˜ t + 2 ˜ a ′ ( ˜ ϕ ) · n ( ˜ ϕ ) · d ˜ ϕ d˜ t + 2 ˜ a ( ˜ ϕ ) · n ′ ( ˜ ϕ ) · d ˜ ϕ d˜ t + 2˜ t ˜ a ′ ( ˜ ϕ ) n ′ ( ˜ ϕ ) · (cid:16) d ˜ ϕ d˜ t (cid:17) + ˜ t ˜ a ( ˜ ϕ ) n ′′ ( ˜ ϕ ) · (cid:16) d ˜ ϕ d˜ t (cid:17) + ˜ t ˜ a ( ˜ ϕ ) n ′ ( ˜ ϕ ) · d ˜ ϕ d˜ t + ˜ t ˜ a ′′ ( ˜ ϕ ) n ( ˜ ϕ ) · (cid:16) d ˜ ϕ d˜ t (cid:17) + ˜ t ˜ a ′ ( ˜ ϕ ) n ( ˜ ϕ ) · d ˜ ϕ d˜ t , imply that | y ′ (0) | + | y ′ (0) | = | ˜ a (0) n (0) | + | ˜ a (0) n (0) | , and y ′ (0) y ′′ (0) − y ′′ (0) y ′ (0) = (cid:0) ˜ a (0) n (0) γ ′ (0) − ˜ a (0) n (0) γ ′ (0) (cid:1) ˜ ϕ ′′ (0) MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 9 = (cid:0) ˜ a (0) | n (0) | + ˜ a (0) | n (0) | (cid:1) ˜ ϕ ′′ (0) . Therefore, the signed curvature of the curve C at the point P is k = y ′ (0) y ′′ (0) − y ′′ (0) y ′ (0) (cid:2) | y ′ (0) | + | y ′ (0) | (cid:3) = ˜ a (0) | n (0) | + ˜ a (0) | n (0) | (cid:0) | ˜ a (0) n (0) | + | ˜ a (0) n (0) | (cid:1) / ˜ ϕ ′′ (0) = 2 ˜ ϕ ′′ (0) . Similarly, we can show k = ˜ a (1) | n (1) | + ˜ a (1) | n (1) | (cid:0) | ˜ a (1) n (1) | + | ˜ a (1) n (1) | (cid:1) / ˜ ϕ ′′ (0) = 2 ˜ ϕ ′′ (0) . Step 3 . We define the modified Fermi coordinates ( y , y ) = F ( t, θ ) = H ◦ ˜ H − ( t, θ ) : ( − δ , δ ) × ( − σ , σ ) → R (2.13)for given small positive constants σ and δ . More precisely, we write F ( t, θ ) = (cid:0) F ( t, θ ) , F ( t, θ ) (cid:1) with F i ( t, θ ) = γ i (cid:0) Θ( t, θ ) (cid:1) + t ˜ a i (cid:0) Θ( t, θ ) (cid:1) n i (cid:0) Θ( t, θ ) (cid:1) , i = 1 , , (2.14)where Θ( t, θ ) ≡ (cid:0) ˜ ϕ ( t ) − ˜ ϕ ( t ) (cid:1) θ + ˜ ϕ ( t ) . (2.15)From (2.9)-(2.12), we haveΘ(0 , θ ) = θ, Θ t (0 , θ ) = 0 , Θ θ (0 , θ ) = 1 , Θ θt (0 , θ ) = 0 , (2.16)Θ tt (0 , θ ) = (cid:0) ˜ k − ˜ k (cid:1) θ + ˜ k , Θ ttθ (0 , θ ) = ˜ k − ˜ k . (2.17)These quantities will play an important role in the further settings.We now derive the asymptotical behaviors of the coordinates. For given i = 1 or i = 2, consider thederivative of first order ∂F i ∂t = γ ′ i (Θ) · Θ t + ˜ a i (Θ) n i (Θ) + t ˜ a ′ i (Θ) n i (Θ) · Θ t + t ˜ a i (Θ) n ′ i (Θ) · Θ t , (2.18)and also the derivative of second order ∂ F i ∂t = γ ′′ i (Θ) · | Θ t | + γ ′ i (Θ) · Θ tt + 2˜ a ′ i (Θ) n i (Θ) · Θ t + 2˜ a i (Θ) n ′ i (Θ) · Θ t + 2 t ˜ a ′ i (Θ) n ′ i (Θ) · | Θ t | + t ˜ a ′′ i (Θ) n i (Θ) · | Θ t | + t ˜ a ′ i (Θ) n i (Θ) · Θ tt + t ˜ a i (Θ) n ′′ i (Θ) · | Θ t | + t ˜ a i (Θ) n ′ i (Θ) · Θ tt . (2.19)These imply that ∂F i ∂t (0 , θ ) = ˜ a i ( θ ) n i ( θ ) , (2.20)and ∂ F i ∂t (0 , θ ) = γ ′ i ( θ ) · Θ tt (0 , θ ) = (cid:0) ˜ k − ˜ k (cid:1) θ + ˜ k ≡ q i ( θ ) . (2.21)Hence Θ tt (0 , θ ) = q γ ′ + q γ ′ = − q n + q n . (2.22)Here comes the derivative of third order ∂ F i ∂t (0 , θ ) = (cid:20) γ ′′′ i (Θ) · (Θ t ) + 3 γ ′′ i (Θ) · Θ t · Θ tt + γ ′ i (Θ) · Θ ttt + 3˜ a ′′ i (Θ) n i (Θ) · | Θ t | + 8˜ a ′ i (Θ) n ′ i (Θ) · | Θ t | + 3˜ a ′ i (Θ) n i (Θ) · Θ tt + 3˜ a i (Θ) n ′′ i (Θ) · | Θ t | + 3˜ a i (Θ) n ′ i (Θ) · Θ tt (cid:21)(cid:12)(cid:12)(cid:12) ( t,θ )=(0 ,θ ) = γ ′ i ( θ ) · Θ ttt (0 , θ ) + 3˜ a ′ i ( θ ) n i ( θ ) · Θ tt (0 , θ ) + 3˜ a i ( θ ) n ′ i ( θ ) · Θ tt (0 , θ ) ≡ m i ( θ ) . (2.23) ‡ AND JUN YANG § These results will be collected in the following way.
Lemma 2.2.
The mapping F has the following properties: (1). F (0 , θ ) = γ ( θ ) , ∂F∂t (0 , θ ) = (cid:0) ˜ a ( θ ) n ( θ ) , ˜ a ( θ ) n ( θ ) (cid:1) , (2). ∂ F∂t (0 , θ ) = q ( θ ) with q ( θ ) = (cid:0) q ( θ ) , q ( θ ) (cid:1) ⊥ n ( θ ) , (3). ∂ F∂t (0 , θ ) = m ( θ ) with m ( θ ) = (cid:0) m ( θ ) , m ( θ ) (cid:1) .Here m i ’s and q i ’s are given in (2.21) - (2.23) . (cid:3) As a conclusion, as t is small enough, there holds the expansion F ( t, θ ) = γ ( θ ) + t (cid:0) ˜ a ( θ ) n ( θ ) , ˜ a ( θ ) n ( θ ) (cid:1) + t q ( θ )+ t m ( θ ) + O ( t ) , ∀ θ ∈ [0 , , t ∈ ( − δ , δ ) , (2.24)where δ > ∂F∂t ( t, θ ) = (cid:0) ˜ a ( θ ) n ( θ ) , ˜ a ( θ ) n ( θ ) (cid:1) + tq ( θ ) + t m ( θ ) + O ( t ) , (2.25) ∂F ∂θ ( t, θ ) = γ ′ ( θ ) + t ˜ a ′ ( θ ) n ( θ ) − tk ˜ a ( θ ) γ ′ ( θ ) + t q ′ ( θ ) + O ( t )= − n ( θ ) + t ˜ a ′ ( θ ) n ( θ ) + tk ˜ a ( θ ) n ( θ ) + t q ′ ( θ ) + O ( t ) , (2.26) ∂F ∂θ ( t, θ ) = γ ′ ( θ ) + t ˜ a ′ ( θ ) n ( θ ) − tk ˜ a ( θ ) γ ′ ( θ ) + t q ′ ( θ ) + O ( t )= n ( θ ) + t ˜ a ′ ( θ ) n ( θ ) − tk ˜ a ( θ ) n ( θ ) + t q ′ ( θ ) + O ( t ) , (2.27)where we have used (2.3) and the Frenet formula (2.1). Moreover, there hold q ′ i ( θ ) = γ ′′ i ( θ ) · Θ tt (0 , θ ) + γ ′ i ( θ )Θ ttθ (0 , θ ) , (2.28)and specially, q ′ (0) = γ ′′ (0)˜ k + γ ′ (0)(˜ k − ˜ k ) , q ′ (1) = γ ′′ (1)˜ k + γ ′ (1)(˜ k − ˜ k ) . The metric.
In the local coordinates ( t, θ ) in (2.13), here are the preparing computations for the metricmatrix: ∂F i ∂t ∂F i ∂t = (˜ a i ) | n i | + 2 t ˜ a i n i q i + t ˜ a i n i m i + t | q i | + O ( t ) , (2.29) ∂F ∂t ∂F ∂θ = − ˜ a n n + t ˜ a ˜ a ′ | n | + tk ˜ a n n − tq n + t a q ′ n + t ˜ a ′ q n + t k ˜ a q n − t m n + O ( t ) , (2.30) ∂F ∂t ∂F ∂θ = ˜ a n n + t ˜ a ˜ a ′ | n | − tk ˜ a n n + tq n + t a q ′ n + t ˜ a ′ q n − t k ˜ a q n + t m n + O ( t ) , (2.31) ∂F ∂θ ∂F ∂θ = | n | − t ˜ a ′ n n − tk ˜ a | n | − t q ′ n + t (˜ a ′ ) | n | + 2 t k ˜ a ˜ a ′ n n + t k (˜ a ) | n | + O ( t ) , (2.32)and ∂F ∂θ ∂F ∂θ = | n | + 2 t ˜ a ′ n n − tk ˜ a | n | + t q ′ n + t (˜ a ′ ) | n | − t k ˜ a ˜ a ′ n n + t k (˜ a ) | n | + O ( t ) . (2.33)The elements of the metric matrix are: g = ∂F ∂t ∂F ∂t + ∂F ∂t ∂F ∂t = (cid:2) ˜ a | n | + ˜ a | n | (cid:3) + 2 t (cid:2) ˜ a n q + ˜ a n q (cid:3) + t (cid:2) ˜ a n m + ˜ a n m (cid:3) + t (cid:2) | q | + | q | (cid:3) + O ( t ) ,g = ∂F ∂t ∂F ∂θ + ∂F ∂t ∂F ∂θ = (cid:2) − ˜ a n n + ˜ a n n (cid:3) + t (cid:2) ˜ a ˜ a ′ | n | + ˜ a ˜ a ′ | n | (cid:3) − tk (cid:2) − ˜ a n n + ˜ a n n (cid:3) + t (cid:2) − q n + q n (cid:3) + t (cid:2) ˜ a q ′ n + ˜ a q ′ n (cid:3) + t (cid:2) ˜ a ′ q n + ˜ a ′ q n (cid:3) − t k (cid:2) − ˜ a q n + ˜ a q n (cid:3) + t (cid:2) − m n + m n (cid:3) + O ( t ) , (2.34)and g = ∂F ∂θ ∂F ∂θ + ∂F ∂θ ∂F ∂θ =1 + 2 t (cid:2) − ˜ a ′ n n + ˜ a ′ n n (cid:3) − tk (cid:2) ˜ a | n | + ˜ a | n | (cid:3) + t (cid:2) − q ′ n + q ′ n (cid:3) + t (cid:2) (˜ a ′ ) | n | + (˜ a ′ ) | n | (cid:3) − t k (cid:2) − ˜ a ˜ a ′ n n + ˜ a ˜ a ′ n n (cid:3) + t k (cid:2) ˜ a | n | + ˜ a | n | (cid:3) + O ( t ) . (2.35)So the determinant of the metric matrix is g = g g − g g where g g = (cid:2) ˜ a | n | + ˜ a | n | (cid:3) + 2 t (cid:2) ˜ a n q + ˜ a n q (cid:3) + 2 t (cid:2) ˜ a | n | + ˜ a | n | (cid:3)(cid:2) − ˜ a ′ n n + ˜ a ′ n n (cid:3) − tk (cid:2) ˜ a | n | + ˜ a | n | (cid:3)(cid:2) ˜ a | n | + ˜ a | n | (cid:3) + t (cid:2) ˜ a n m + ˜ a n m (cid:3) + t (cid:2) | q | + | q | (cid:3) + 4 t (cid:2) ˜ a n q + ˜ a n q (cid:3)(cid:2) − ˜ a ′ + ˜ a ′ (cid:3) n n + t (cid:2) ˜ a | n | + ˜ a | n | (cid:3)(cid:2) − q ′ n + q ′ n (cid:3) + t (cid:2) ˜ a | n | + ˜ a | n | (cid:3)(cid:2) (˜ a ′ ) | n | + (˜ a ′ ) | n | (cid:3) − t k (cid:2) ˜ a | n | + ˜ a | n | (cid:3)(cid:2) ˜ a n q + ˜ a n q (cid:3) − t k (cid:2) ˜ a | n | + ˜ a | n | (cid:3)(cid:2) − ˜ a ˜ a ′ n n + ˜ a ˜ a ′ n n (cid:3) + t k (cid:2) ˜ a | n | + ˜ a | n | (cid:3)(cid:2) ˜ a | n | + ˜ a | n | (cid:3) + O ( t ) , and g g = (cid:2) − ˜ a n n + ˜ a n n (cid:3) + 2 t (cid:2) − ˜ a n n + ˜ a n n (cid:3)(cid:2) ˜ a ˜ a ′ | n | + ˜ a ˜ a ′ | n | (cid:3) + 2 t (cid:2) − ˜ a n n + ˜ a n n (cid:3)(cid:2) − q n + q n (cid:3) − tk (cid:2) − ˜ a n n + ˜ a n n (cid:3)(cid:2) − ˜ a n n + ˜ a n n (cid:3) + t (cid:2) ˜ a ˜ a ′ | n | + ˜ a ˜ a ′ | n | (cid:3) + 2 t (cid:2) ˜ a ˜ a ′ | n | + ˜ a ˜ a ′ | n | (cid:3)(cid:2) − q n + q n (cid:3) + t (cid:2) − q n + q n (cid:3)
22 SUTING WEI ‡ AND JUN YANG § + t (cid:2) ˜ a q ′ n + ˜ a q ′ n (cid:3)(cid:2) − ˜ a n n + ˜ a n n (cid:3) + 2 t (cid:2) ˜ a ′ q n + ˜ a ′ q n (cid:3)(cid:2) − ˜ a n n + ˜ a n n (cid:3) + t (cid:2) − m n + m n (cid:3)(cid:2) − ˜ a n n + ˜ a n n (cid:3) − t k (cid:2) ˜ a ˜ a ′ | n | + ˜ a ˜ a ′ | n | (cid:3)(cid:2) − ˜ a n n + ˜ a n n (cid:3) − t k (cid:2) − ˜ a n n + ˜ a n n (cid:3)(cid:2) − q n + q n (cid:3) − t k (cid:2) − ˜ a q n + ˜ a q n (cid:3)(cid:2) − ˜ a n n + ˜ a n n (cid:3) + t k (cid:2) − ˜ a n n + ˜ a n n (cid:3) + O ( t ) . We now make a rearrangement of the terms in g and then consider the following terms. ♣ Term 1: (cid:2) ˜ a | n | + ˜ a | n | (cid:3) − (cid:2) − ˜ a n n + ˜ a n n (cid:3) = (cid:2) ˜ a | n | + ˜ a | n | (cid:3) . ♣ Term 2: 2 t (cid:2) ˜ a n q + ˜ a n q (cid:3) − t (cid:2) − ˜ a n n + ˜ a n n (cid:3)(cid:2) − q n + q n (cid:3) = 2 t (cid:2) ˜ a n q (cid:0) − | n | (cid:1) + ˜ a n q (cid:0) − | n | (cid:1) + ˜ a q n | n | + ˜ a q | n | n (cid:3) = 2 t (cid:2) ˜ a n q | n | + ˜ a n q | n | + ˜ a q n | n | + ˜ a q | n | n (cid:3) = 2 t (cid:2) ˜ a | n | (cid:0) q n + q n (cid:1) − ˜ a | n | (cid:0) − q n − q n (cid:1)(cid:3) = 0 , due to the fact that q ⊥ n given in Lemma 2.2. ♣ Term 3:2 t (cid:2) − ˜ a ′ n n + ˜ a ′ n n (cid:3)(cid:2) ˜ a | n | + ˜ a | n | (cid:3) − t (cid:2) − ˜ a n n + ˜ a n n (cid:3)(cid:2) ˜ a ˜ a ′ | n | + ˜ a ˜ a ′ | n | (cid:3) = 2 t (cid:2) ˜ a ′ ˜ a n | n | − ˜ a ′ ˜ a n | n | − ˜ a ˜ a ˜ a ′ n | n | + ˜ a ˜ a ˜ a ′ n | n | (cid:3) = 2 t (cid:0) ˜ a ′ ˜ a − ˜ a ′ ˜ a (cid:1)(cid:0) ˜ a | n | + ˜ a | n | (cid:1) n n . ♣ Term 4: − tk (cid:2) ˜ a | n | + ˜ a | n | (cid:3)(cid:2) ˜ a | n | + ˜ a | n | (cid:3) + 2 tk (cid:2) − ˜ a n n + ˜ a n n (cid:3)(cid:2) − ˜ a n n + ˜ a n n (cid:3) = − tk h ˜ a ˜ a | n | | n | + ˜ a ˜ a | n | | n | + ˜ a ˜ a | n | | n | + ˜ a ˜ a | n | | n | i = − tk ˜ a ˜ a (cid:2) ˜ a | n | + ˜ a | n | (cid:3) . ♣ Term 5: t (cid:2) ˜ a n m + ˜ a n m (cid:3) − t (cid:2) − m n + m n (cid:3)(cid:2) − ˜ a n n + ˜ a n n (cid:3) = t (cid:2) ˜ a n m (cid:0) − | n | (cid:1) + ˜ a n m (cid:0) − | n | (cid:1) + ˜ a m n n + ˜ a m n n (cid:3) = t (cid:0) m n + m n (cid:1)(cid:2) ˜ a | n | + ˜ a | n | (cid:3) . ♣ Term 6: t (cid:2) | q | + | q | (cid:3) − t (cid:2) − q n + q n (cid:3) = t (cid:2) | q | + | q | (cid:3) − t (cid:0) q | n | − q q n n + q | n | (cid:1) = t (cid:2) | q | (cid:0) − | n | (cid:1) + | q | (cid:0) − | n | (cid:1) + 2 q q n n (cid:3) = t (cid:2) q n + q n (cid:3) = 0 , due to the fact that q ⊥ n given in Lemma 2.2. ♣ Term 7:4 t (cid:2) − ˜ a ′ n n + ˜ a ′ n n (cid:3)(cid:2) ˜ a n q + ˜ a n q (cid:3) − t (cid:2) ˜ a ′ q n + ˜ a ′ q n (cid:3)(cid:2) − ˜ a n n + ˜ a n n (cid:3) − t (cid:2) ˜ a ˜ a ′ | n | + ˜ a ˜ a ′ | n | (cid:3)(cid:2) − q n + q n (cid:3) = t (cid:2) a ′ ˜ a n n (cid:0) q n + q n (cid:1) − a ′ ˜ a n n (cid:0) q n + q n (cid:1) MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 13 + 2˜ a ′ q n ˜ a | n | + 2˜ a ′ q n ˜ a | n | − a ′ q n ˜ a | n | − a ′ q n ˜ a | n | (cid:3) = 2 t (cid:2) ˜ a ′ q n − ˜ a ′ q n (cid:3)(cid:2) ˜ a | n | + ˜ a | n | (cid:3) , where we have used the fact that q ⊥ n given in Lemma 2.2. ♣ Term 8: t (cid:2) ˜ a | n | + ˜ a | n | (cid:3)(cid:2) − q ′ n + q ′ n (cid:3) − t (cid:2) ˜ a q ′ n + ˜ a q ′ n (cid:3)(cid:2) − ˜ a n n + ˜ a n n (cid:3) = t (cid:2) − ˜ a | n | q ′ + ˜ a | n | q ′ + ˜ a ˜ a q ′ n n − ˜ a ˜ a q ′ n n (cid:3) = t (cid:2) ˜ a q ′ n − ˜ a q ′ n (cid:3)(cid:2) ˜ a | n | + ˜ a | n | (cid:3) . ♣ Term 9: t (cid:2) ˜ a | n | + ˜ a | n | (cid:3)(cid:2) (˜ a ′ ) | n | + (˜ a ′ ) | n | (cid:3) − t (cid:2) ˜ a ˜ a ′ | n | + ˜ a ˜ a ′ | n | (cid:3) = t (cid:2) ˜ a (˜ a ′ ) | n | | n | + ˜ a (˜ a ′ ) | n | | n | − a ˜ a ′ ˜ a ˜ a ′ | n | | n | (cid:3) = t (cid:2) ˜ a ˜ a ′ − ˜ a ′ ˜ a (cid:3) | n | | n | . ♣ Term 10: − t k (cid:2) ˜ a | n | + ˜ a | n | (cid:3)(cid:2) ˜ a n q + ˜ a n q (cid:3) + 2 t k (cid:2) − ˜ a n n + ˜ a n n (cid:3)(cid:2) − q n + q n (cid:3) + 2 t k (cid:2) − ˜ a q n + ˜ a q n (cid:3)(cid:2) − ˜ a n n + ˜ a n n (cid:3) = − t k h a ˜ a q n (cid:0) | n | + | n | (cid:1) + 2˜ a ˜ a q n (cid:0) | n | + | n | (cid:1) + 2˜ a q n (cid:0) ˜ a | n | + ˜ a | n | (cid:1) + 2˜ a q n (cid:0) ˜ a | n | + ˜ a | n | (cid:1)i = − t k h a ˜ a (cid:0) q n + q n (cid:1) + 2 (cid:0) ˜ a q n + ˜ a q n (cid:1)(cid:0) ˜ a | n | + ˜ a | n | (cid:1)i = − t k (cid:0) ˜ a q n + ˜ a q n (cid:1)(cid:0) ˜ a | n | + ˜ a | n | (cid:1) , where we have used the fact that q ⊥ n given in Lemma 2.2. ♣ Term 11: − t k (cid:2) ˜ a | n | + ˜ a | n | (cid:3)(cid:2) − ˜ a ˜ a ′ n n + ˜ a ˜ a ′ n n (cid:3) + 2 t k (cid:2) ˜ a ˜ a ′ | n | + ˜ a ˜ a ′ | n | (cid:3)(cid:2) − ˜ a n n + ˜ a n n (cid:3) = − t k (cid:2) − ˜ a ˜ a ˜ a ′ n | n | + ˜ a ˜ a ˜ a ′ | n | n + ˜ a ˜ a ′ ˜ a n | n | − ˜ a ˜ a ′ ˜ a | n | n (cid:3) = 2 t k (cid:2) ˜ a ˜ a ˜ a ′ − ˜ a ˜ a ˜ a ′ (cid:3) n n . ♣ Term 12: t k (cid:2) ˜ a | n | + ˜ a | n | (cid:3)(cid:2) ˜ a | n | + ˜ a | n | (cid:3) − t k (cid:2) − ˜ a n n + ˜ a n n (cid:3) = t k (cid:2) ˜ a ˜ a | n | | n | + ˜ a ˜ a | n | | n | + 2˜ a ˜ a n n (cid:3) = t k ˜ a ˜ a (cid:0) n + n (cid:1) = t k ˜ a ˜ a . Therefore, we obtain that g = det( g ij ) = g g − g g = (cid:2) ˜ a | n | + ˜ a | n | (cid:3) + t n (cid:0) ˜ a ′ ˜ a − ˜ a ′ ˜ a (cid:1)(cid:2) ˜ a | n | + ˜ a | n | (cid:3) n n − k ˜ a ˜ a (cid:2) ˜ a n + ˜ a n (cid:3)o + t n (cid:2) ˜ a ′ q n − ˜ a ′ q n (cid:3)(cid:2) ˜ a | n | + ˜ a | n | (cid:3) + (cid:2) ˜ a q ′ n − ˜ a q ′ n (cid:3)(cid:2) ˜ a | n | + ˜ a | n | (cid:3) + (cid:0) m n + m n (cid:1)(cid:2) ˜ a | n | + ˜ a | n | (cid:3) + (cid:2) ˜ a ˜ a ′ − ˜ a ′ ˜ a (cid:3) | n | | n |
24 SUTING WEI ‡ AND JUN YANG § + 2 k (cid:2) ˜ a ˜ a ˜ a ′ − ˜ a ˜ a ˜ a ′ (cid:3) n n − k (cid:0) ˜ a q n + ˜ a q n (cid:1)(cid:2) ˜ a | n | + ˜ a | n | (cid:3) + k ˜ a ˜ a o + O ( t ) ≡ h ( θ ) + t h ( θ ) + t h ( θ ) + O ( t ) . (2.36)We now compute the inverse of the metric matrix. By the formula (cid:2) at + bt + O ( t ) (cid:3) − = 1 − at + ( a − b ) t + O ( t ) , we have 1 g = 1 h − h h t + h h h − h h i t + O ( t ) ≡ h + g t + g t + O ( t ) . (2.37)Note that if the function has the following asymptotic expansion f ( s ) = 1 + as + bs + O ( s ) , f (0) = 1 , then for s close to zero p f ( s ) = 1 + a s + 12 (cid:18) b − a (cid:19) s + O ( s ) . We can get the following formulas √ g = p h + 12 h √ h t + h h √ h t − h ( √ h ) i + O ( t ) , (2.38)and also 1 √ g = 1 √ h − h ( √ h ) t + h h ( √ h ) − h ( √ h ) i t + O ( t ) ≡ √ h + r t + r t + O ( t ) . (2.39)Whence g = − g g = 1 h (cid:2) ˜ a − ˜ a (cid:3) n n + t n − h (cid:2) ˜ a ˜ a ′ | n | + ˜ a ˜ a ′ | n | (cid:3) + k h (cid:2) − ˜ a + ˜ a (cid:3) n n − h (cid:2) − q n + q n (cid:3) + g (cid:2) ˜ a − ˜ a (cid:3) n n o + t n −
12 1 h (cid:2) ˜ a q ′ n + ˜ a q ′ n (cid:3) − h (cid:2) ˜ a ′ q n + ˜ a ′ q n (cid:3) + k h (cid:2) − ˜ a q n + ˜ a q n (cid:3) + 12 1 h (cid:2) − m n + m n (cid:3) + g (cid:2) ˜ a ˜ a ′ | n | + ˜ a ˜ a ′ | n | (cid:3) + k g (cid:2) − ˜ a + ˜ a (cid:3) n n − g (cid:2) − q n + q n (cid:3) + g (cid:2) ˜ a − ˜ a (cid:3) n n o + O ( t ) ≡ g + g t + g t + O ( t ) , (2.40)and g = g g = 1 h (cid:2) ˜ a | n | + ˜ a | n | (cid:3) + t n h (cid:2) ˜ a n q + ˜ a n q (cid:3) + g (cid:2) ˜ a | n | + ˜ a | n | (cid:3)o + t n h (cid:2) ˜ a n m + ˜ a n m (cid:3) + 1 h (cid:2) | q | + | q | (cid:3) MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 15 + 2 g (cid:2) ˜ a n q + ˜ a n q (cid:3) + g (cid:2) ˜ a | n | + ˜ a | n | (cid:3)o + O ( t ) ≡ g + g t + g t + O ( t ) . (2.41)Similar asymptotic expression holds for the term g = g /g .2.3. Local forms of the differential operators in (1.1) . In this section, we are devoted to presenting the expressions of the differential operators div (cid:0) ∇ a ( y ) u (cid:1) and ∇ a ( y ) u · ν in problem (1.1). Part 1: the operator div (cid:0) ∇ a ( y ) u (cid:1) We recall the relation of (2.13) and then haved y = ∂F ∂t d t + ∂F ∂θ d θ, d y = ∂F ∂t d t + ∂F ∂θ d θ. This implies that d t = 1 ∂F ∂t ∂F ∂θ − ∂F ∂t ∂F ∂θ (cid:20) ∂F ∂θ d y − ∂F ∂θ d y (cid:21) , (2.42)d θ = 1 ∂F ∂t ∂F ∂θ − ∂F ∂t ∂F ∂θ (cid:20) ∂F ∂t d y − ∂F ∂t d y (cid:21) . (2.43)On the other hand, there hold ∂u∂t = ∂u∂y ∂F ∂t + ∂u∂y ∂F ∂t , ∂u∂θ = ∂u∂y ∂F ∂θ + ∂u∂y ∂F ∂θ , (2.44)which give that ∂u∂y = 1 ∂F ∂t ∂F ∂θ − ∂F ∂t ∂F ∂θ (cid:20) ∂u∂t ∂F ∂θ − ∂u∂θ ∂F ∂t (cid:21) , (2.45) ∂u∂y = 1 ∂F ∂t ∂F ∂θ − ∂F ∂t ∂F ∂θ (cid:20) ∂u∂θ ∂F ∂t − ∂u∂t ∂F ∂θ (cid:21) . (2.46)We now compute ∇ a ( y ) u = a ∂u∂y ∂∂y + a ∂u∂y ∂∂y = a ∂u∂y (cid:20) ∂t∂y ∂∂t + ∂θ∂y ∂∂θ (cid:21) + a ∂u∂y (cid:20) ∂t∂y ∂∂t + ∂θ∂y ∂∂θ (cid:21) = (cid:20) a ∂t∂y ∂u∂y + a ∂t∂y ∂u∂y (cid:21) ∂∂t + (cid:20) a ∂θ∂y ∂u∂y + a ∂θ∂y ∂u∂y (cid:21) ∂∂θ . (2.47)By substituting (2.42), (2.43), (2.45) and (2.46) in (2.47), we obtain ∇ a ( y ) u = 1 g (cid:20) a ∂F ∂θ ∂F ∂θ + a ∂F ∂θ ∂F ∂θ (cid:21) ∂u∂t ∂∂t − g (cid:20) a ∂F ∂θ ∂F ∂t + a ∂F ∂θ ∂F ∂t (cid:21) ∂u∂θ ∂∂t − g (cid:20) a ∂F ∂t ∂F ∂θ + a ∂F ∂t ∂F ∂θ (cid:21) ∂u∂t ∂∂θ + 1 g (cid:20) a ∂F ∂t ∂F ∂t + a ∂F ∂t ∂F ∂t (cid:21) ∂u∂θ ∂∂θ . (2.48)Furthermore, by setting ˜ g = a ∂F ∂θ ∂F ∂θ + a ∂F ∂θ ∂F ∂θ , (2.49)˜ g = ˜ g = a ∂F ∂θ ∂F ∂t + a ∂F ∂θ ∂F ∂t , (2.50)˜ g = a ∂F ∂t ∂F ∂t + a ∂F ∂t ∂F ∂t , (2.51) ‡ AND JUN YANG § the definition of div operator will give thatdiv (cid:0) ∇ a ( y ) u (cid:1) = 1 √ g ∂∂t (cid:20) √ g ˜ g ∂u∂t (cid:21) − √ g ∂∂t (cid:20) √ g ˜ g ∂u∂θ (cid:21) − √ g ∂∂θ (cid:20) √ g ˜ g ∂u∂t (cid:21) + 1 √ g ∂∂θ (cid:20) √ g ˜ g ∂u∂θ (cid:21) . (2.52)Here are the computations of all coefficients in (2.52). By using the Taylor expansion a i ( t, θ ) = a i (0 , θ ) + t∂ t a i (0 , θ ) + t ∂ tt a i (0 , θ ) + O ( t ) , ∀ θ ∈ [0 , , t ∈ ( − δ , δ ) , (2.53)and recalling (2.29)-(2.32), we obtain˜ g = a ∂F ∂θ ∂F ∂θ + a ∂F ∂θ ∂F ∂θ = a h | n | + 2 t ˜ a ′ n n − tk ˜ a | n | + t q ′ n + t (˜ a ′ ) | n | − t k ˜ a ˜ a ′ n n + t k ˜ a | n | i + a h | n | − t ˜ a ′ n n − tk ˜ a | n | − t q ′ n + t (˜ a ′ ) | n | + 2 t k ˜ a ˜ a ′ n n + t k ˜ a | n | i + O ( t )= h a (0 , θ ) | n | + a (0 , θ ) | n | i + t n (cid:2) a (0 , θ )˜ a ′ n n − a (0 , θ )˜ a ′ n n (cid:3) − k (cid:2) a (0 , θ )˜ a | n | + a (0 , θ )˜ a | n | (cid:3) + (cid:2) ∂ t a (0 , θ ) | n | + ∂ t a (0 , θ ) | n | (cid:3)o + t n(cid:2) a (0 , θ ) n q ′ − a (0 , θ ) n q ′ (cid:3) + (cid:2) a (0 , θ )(˜ a ′ ) | n | + a (0 , θ )(˜ a ′ ) | n | (cid:3) − k (cid:2) a (0 , θ )˜ a ˜ a ′ n n − a (0 , θ )˜ a ˜ a ′ n n (cid:3) + k (cid:2) a (0 , θ )˜ a | n | + a (0 , θ )˜ a | n | (cid:3) + 2 (cid:2) ∂ t a (0 , θ )˜ a ′ n n − ∂ t a (0 , θ )˜ a ′ n n (cid:3) − k (cid:2) ∂ t a (0 , θ )˜ a | n | + ∂ t a (0 , θ )˜ a | n | (cid:3) + 12 (cid:2) ∂ tt a (0 , θ ) | n | + ∂ tt a (0 , θ ) | n | (cid:3)o + O ( t ) ≡ f + t f + t f + O ( t ) . (2.54)Similarly, there hold˜ g = a ∂F ∂θ ∂F ∂t + a ∂F ∂θ ∂F ∂t = a h ˜ a n n + t ˜ a ˜ a ′ | n | − tk ˜ a n n + tq n + t a q ′ n i + a h − ˜ a n n + t ˜ a ˜ a ′ | n | + tk ˜ a n n − tq n i + O ( t )= t n(cid:2) a (0 , θ )˜ a ˜ a ′ | n | + a (0 , θ )˜ a ˜ a ′ | n | (cid:3) − k (cid:2) a (0 , θ )˜ a n n − a (0 , θ )˜ a n n (cid:3) + (cid:2) a (0 , θ ) n q − a (0 , θ ) n q (cid:3) + (cid:2) ∂ t a (0 , θ )˜ a n n − ∂ t a (0 , θ )˜ a n n (cid:3)o + O ( t ) ≡ t l + O ( t ) , (2.55)and ˜ g = a ∂F ∂t ∂F ∂t + a ∂F ∂t ∂F ∂t MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 17 = a h ˜ a | n | + 2 t ˜ a n q + t ˜ a n m + t | q | i + a h ˜ a | n | + 2 t ˜ a n q + t ˜ a n m + t | q | i + O ( t )= h a (0 , θ )˜ a | n | + a (0 , θ )˜ a | n | i + t n a (0 , θ )˜ a n q + 2 a (0 , θ )˜ a n q + ∂ t a (0 , θ )˜ a | n | + ∂ t a (0 , θ )˜ a | n | o + O ( t ) ≡ w + t w + O ( t ) . (2.56)By recalling (2.37)-(2.39) and (2.54)-(2.56), we can obtain that˜ g g = h h + g t + g t + O ( t ) i × h f + t f + t f + O ( t ) i = 1 h f + t h f + t g f + t h f + t g f + t g f + O ( t ) , ˜ g √ g = h √ h + r t + r t + O ( t ) i × h f + t f + t f + O ( t ) i = 1 √ h f + t √ h f + t r f + t √ h f + t r f + t r f + O ( t ) , ˜ g g = h h + g t + g t + O ( t ) i × h w + t w + O ( t ) i = 1 h w + t h w + t g w + O ( t ) , ˜ g √ g = h √ h + r t + r t + O ( t ) i × (cid:2) w + t w + t w + O ( t ) (cid:3) = 1 √ h w + t √ h w + t r w + O ( t ) , ˜ g g = h h + g t + g t + O ( t ) i × (cid:2) t l + O ( t ) (cid:3) = t h l + O ( t ) , ˜ g √ g = h √ h + r t + r t + O ( t ) i × h t l + O ( t ) i = t √ h l + O ( t ) . Then, there hold ∂ t h √ g ˜ g i = 1 √ h l + O ( t ) , ∂ θ h √ g ˜ g i = t h √ h l i ′ + O ( t ) , and ∂ t h √ g ˜ g i = 1 √ h f + r f + 2 t √ h f + 2 t r f + 2 t r f + O ( t ) ,∂ θ h √ g ˜ g i = h √ h w i ′ + t h √ h w i ′ + t (cid:2) r w (cid:3) ′ + O ( t ) . Combining the expression of ( √ g ) − as in (2.39), we can obtain that1 √ g ∂ t h √ g ˜ g i = h √ h + r t + r t + O ( t ) i × h √ h l + O ( t ) i = 1 h l + O ( t ) , ‡ AND JUN YANG § √ g ∂ θ h √ g ˜ g i = h √ h + r t + r t + O ( t ) i × h t (cid:16) √ h l (cid:17) ′ + O ( t ) i = t √ h h √ h l i ′ + O ( t ) , and 1 √ g ∂ t h √ g ˜ g i = 1 h f + r √ h f + 2 t h f + 3 t √ h r f + 2 t √ h r f + t r r f + O ( t ) , √ g ∂ θ h √ g ˜ g i = 1 √ h h √ h w i ′ + t √ h h √ h w i ′ + t √ h (cid:2) r w (cid:3) ′ + t r h √ h w i ′ + O ( t ) . Notation 1:
By collecting all the computations in the above, we set the following conventions. h ( θ ) = f ( θ ) h ( θ ) = | a (0 , θ ) | + | a (0 , θ ) | a (0 , θ ) | n ( θ ) | + a (0 , θ ) | n ( θ ) | , (2.57) h ( θ ) = w ( θ ) h ( θ ) = a (0 , θ ) a (0 , θ ) a (0 , θ ) | n ( θ ) | + a (0 , θ ) | n ( θ ) | , (2.58) h ( θ ) = 1 h f + r √ h f = 1 h f − h h f , h ( θ ) = 1 √ h h √ h w i ′ − h l , (2.59) h ( θ ) = 2 1 h f + 3 r √ h f + 2 r √ h f + r r f − √ h h √ h l i ′ = 2 1 h f − h h f + h h f − h h f − √ h h √ h l i ′ , (2.60) h ( θ ) = − h l , h ( θ ) = 1 h f + g f + g f = 1 h f − h h f + h h h − h h i f , (2.61) h ( θ ) = 1 h f + g f = 1 h f − h h f . (2.62) (cid:3) Hence, the term div (cid:0) ∇ a ( y ) u (cid:1) in (2.52) has the following form in the modified Fermi coordinate systemdiv (cid:0) ∇ a ( y ) u (cid:1) = ˜ g g u tt + 1 √ g ∂ t h √ g ˜ g i u t − g g u θt − √ g ∂ t h √ g ˜ g i u θ − √ g ∂ θ h √ g ˜ g i u t + ˜ g g u θθ + 1 √ g ∂ θ h √ g ˜ g i u θ = h ( θ ) u tt + h ( θ ) u θθ + h ( θ ) u t + h ( θ ) u θ + ¯ B ( u ) + ¯ B ( u ) , (2.63)where ¯ B ( u ) = h ( θ ) tu t + h ( θ ) tu tθ + h ( θ ) t u tt + h ( θ ) tu tt , (2.64)and ¯ B ( u ) = h ( t, θ ) tu θθ + h ( t, θ ) t u θθ + h ( t, θ ) t u θt + h ( t, θ ) tu θ . (2.65)Here, h , · · · , h are smooth functions. Part 2: the operator ∇ a ( y ) u · ν We finally show the local expression of ∇ a ( y ) u · ν in (1.1). Suppose that, in the local coordinates ( t, θ ) of(2.13), the unit outer normal of ∂ Ω is expressed in the form ν = σ ∂F∂t + σ ∂F∂θ . If θ = 0 or θ = 1, the expression of F ( t, θ ) in (2.13) gives the curves C or C . We then have h ∂F /∂t, ν i = 0 at θ = 0 , . MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 19
For the convenience of notation, in the following lines of this part, we will always take θ = 0 or θ = 1 withoutany further announcement. Hence σ = 0 , ∂F∂θ = 0 , and σ g + σ g = 0 . On the other hand, h ν, ν i = 1, that is (cid:28) σ ∂F∂t + σ ∂F∂θ , σ ∂F∂t + σ ∂F∂θ (cid:29) = 1 , which implies that σ g + σ g + 2 σ σ g = 1 . Combining above two equations, one can get σ = ± g p g , σ = ± p g . By choosing the sign ” + ” and using (2.40)-(2.41), it is easy to check that σ = h g + g t + g t + O ( t ) i × n g − g g t + h g g − g g i t + O ( t ) o = g g + t h g g − g g g i + t h g g − g g g + g g i + O ( t ) ≡ y ( θ ) + t y ( θ ) + t y ( θ ) + O ( t ) , (2.66)and σ = p g = √ g + 12 g √ g t + 12 h g √ g − g ( √ g ) i t + O ( t ) ≡ y ( θ ) + t y ( θ ) + t y ( θ ) + O ( t ) . (2.67)In the modified Fermi coordinates ( t, θ ) in (2.13), the normal derivative ∇ a ( y ) u · ν has a local form asfollows ∇ a ( y ) u · ν = (cid:16) a ( y ) ∂u∂y , a ( y ) ∂u∂y (cid:17)(cid:16) σ ∂F ∂t + σ ∂F ∂θ , σ ∂F ∂t + σ ∂F ∂θ (cid:17) = a ( y ) ∂u∂y (cid:16) σ ∂F ∂t + σ ∂F ∂θ (cid:17) + a ( y ) ∂u∂y (cid:16) σ ∂F ∂t + σ ∂F ∂θ (cid:17) = a ( y ) 1 √ g (cid:20) ∂u∂t ∂F ∂θ − ∂u∂θ ∂F ∂t (cid:21) (cid:16) σ ∂F ∂t + σ ∂F ∂θ (cid:17) + a ( y ) 1 √ g (cid:20) ∂u∂θ ∂F ∂t − ∂u∂t ∂F ∂θ (cid:21) (cid:16) σ ∂F ∂t + σ ∂F ∂θ (cid:17) = σ √ g (cid:20) a ( y ) ∂F ∂t ∂F ∂θ − a ( y ) ∂F ∂θ ∂F ∂t (cid:21) ∂u∂t + σ √ g (cid:20) a ( y ) ∂F ∂θ ∂F ∂θ − a ( y ) ∂F ∂θ ∂F ∂θ (cid:21) ∂u∂t + σ √ g (cid:20) a ( y ) ∂F ∂t ∂F ∂t − a ( y ) ∂F ∂t ∂F ∂t (cid:21) ∂u∂θ + σ √ g (cid:20) a ( y ) ∂F ∂t ∂F ∂θ − a ( y ) ∂F ∂θ ∂F ∂t (cid:21) ∂u∂θ . According to the expressions of ∂F i ∂t and ∂F i ∂θ as in (2.25)-(2.27), it is easy to derive that ∂F ∂t ∂F ∂t = ˜ a ˜ a n n + t (cid:0) ˜ a q n + ˜ a q n (cid:1) + t q q + t (cid:0) ˜ a m n + ˜ a m n (cid:1) + O ( t ) , (2.68) ∂F ∂θ ∂F ∂θ = − n n + t (cid:0) − ˜ a ′ | n | + ˜ a ′ | n | (cid:1) + tk (cid:0) ˜ a + ˜ a (cid:1) n n + t (cid:0) − q ′ n + q ′ n (cid:1) + t ˜ a ′ ˜ a ′ n n − t k (cid:0) ˜ a ′ ˜ a | n | − ˜ a ˜ a ′ | n | (cid:1) − t k ˜ a ˜ a n n + O ( t ) , (2.69) ‡ AND JUN YANG § and ∂F ∂t ∂F ∂θ = ˜ a | n | + t ˜ a ˜ a ′ n n − tk ˜ a ˜ a | n | + tq n + t q ˜ a ′ n − t k ˜ a q n + t a n q ′ + t m n + O ( t ) , (2.70) ∂F ∂θ ∂F ∂t = − ˜ a | n | + t ˜ a ′ ˜ a n n + tk ˜ a ˜ a | n | − tq n + t ˜ a ′ n q + t k ˜ a q n + t a n q ′ − t m n + O ( t ) . (2.71)By using the Taylor expansion of a i ( y ) as in (2.53) and (2.68)-(2.71), it is easy to derive that a ( y ) ∂F ∂t ∂F ∂θ − a ( y ) ∂F ∂θ ∂F ∂t = a ( y ) (cid:2) ˜ a | n | + t ˜ a ˜ a ′ n n − tk ˜ a ˜ a | n | + tq n + O ( t ) (cid:3) − a ( y ) (cid:2) − ˜ a | n | + t ˜ a ′ ˜ a n n + tk ˜ a ˜ a | n | − tq n + O ( t ) (cid:3) = (cid:2) a (0 , θ )˜ a | n | + a (0 , θ )˜ a | n | (cid:3) + t n(cid:2) a (0 , θ )˜ a ˜ a ′ n n − a (0 , θ )˜ a ′ ˜ a n n (cid:3) − k (cid:2) a (0 , θ )˜ a ˜ a | n | − a (0 , θ )˜ a ˜ a | n | (cid:3) + (cid:2) a (0 , θ ) q n + a (0 , θ ) q n (cid:3) + (cid:2) ∂ t a (0 , θ )˜ a | n | + ∂ t a (0 , θ )˜ a | n | (cid:3)o + O ( t ) ≡ p ( θ ) + t p ( θ ) + O ( t ) , (2.72) a ( y ) ∂F ∂θ ∂F ∂θ − a ( y ) ∂F ∂θ ∂F ∂θ = h a ( y ) − a ( y ) i × h − n n + t (cid:0) ˜ a ′ | n | − ˜ a ′ | n | (cid:1) + tk (cid:0) ˜ a + ˜ a (cid:1) n n + t (cid:0) q ′ n − q ′ n (cid:1) + t ˜ a ′ ˜ a ′ n n − t k (cid:0) ˜ a ′ ˜ a | n | − ˜ a ˜ a ′ | n | (cid:1) − t k ˜ a ˜ a n n + O ( t ) i = − (cid:2) a (0 , θ ) − a (0 , θ ) (cid:3) n n + t n(cid:2) a (0 , θ ) − a (0 , θ ) (cid:3)(cid:0) − ˜ a ′ | n | + ˜ a ′ | n | (cid:1) + k (cid:2) a (0 , θ ) − a (0 , θ ) (cid:3)(cid:0) ˜ a + ˜ a (cid:1) n n − (cid:2) ∂ t a (0 , θ ) − ∂ t a (0 , θ ) (cid:3) n n o + t n (cid:2) a (0 , θ ) − a (0 , θ ) (cid:3)(cid:0) − q ′ n + q ′ n (cid:1) + (cid:2) a (0 , θ ) − a (0 , θ ) (cid:3) ˜ a ′ ˜ a ′ n n − k (cid:2) a (0 , θ ) − a (0 , θ ) (cid:3)(cid:0) ˜ a ′ ˜ a | n | − ˜ a ˜ a ′ | n | (cid:1) − k (cid:2) a (0 , θ ) − a (0 , θ ) (cid:3) ˜ a ˜ a n n + (cid:2) ∂ t a (0 , θ ) − ∂ t a (0 , θ ) (cid:3)(cid:0) − ˜ a ′ | n | + ˜ a ′ | n | (cid:1) + k (cid:2) ∂ t a (0 , θ ) − ∂ t a (0 , θ ) (cid:3)(cid:0) ˜ a + ˜ a (cid:1) n n o + O ( t ) ≡ p ( θ ) + t p ( θ ) + t p ( θ ) + O ( t ) , (2.73) a ( y ) ∂F ∂t ∂F ∂t − a ( y ) ∂F ∂t ∂F ∂t = (cid:2) a (0 , θ ) − a (0 , θ ) (cid:3) ˜ a ˜ a n n + t n(cid:2) a (0 , θ ) − a (0 , θ ) (cid:3)(cid:0) ˜ a q n + ˜ a q n (cid:1) + (cid:2) ∂ t a (0 , θ ) − ∂ t a (0 , θ ) (cid:3) ˜ a ˜ a n n o + O ( t ) ≡ p ( θ ) + t p ( θ ) + O ( t ) , (2.74)and a ( y ) ∂F ∂t ∂F ∂θ − a ( y ) ∂F ∂θ ∂F ∂t = a ( y ) (cid:2) ˜ a | n | + t ˜ a ˜ a ′ n n − tk ˜ a ˜ a | n | + tq n + t ˜ a ′ n q − t k ˜ a q n + t a n q ′ + t m n + O ( t ) (cid:3) − a ( y ) (cid:2) − ˜ a | n | + t ˜ a ′ ˜ a n n + tk ˜ a ˜ a | n | − tq n + t ˜ a ′ n q + t k ˜ a q n + t a n q ′ − t m n + O ( t ) (cid:3) = (cid:2) a (0 , θ )˜ a | n | + a (0 , θ )˜ a | n | (cid:3) + t n(cid:2) ∂ t a (0 , θ )˜ a | n | + ∂ t a (0 , θ )˜ a | n | (cid:3) + (cid:2) a (0 , θ )˜ a ˜ a ′ − a (0 , θ )˜ a ′ ˜ a (cid:3) n n − k (cid:2) a (0 , θ )˜ a ˜ a | n | + a (0 , θ )˜ a ˜ a | n | (cid:3) + (cid:2) a (0 , θ ) q n + a (0 , θ ) q n (cid:3)o + t n(cid:2) ∂ t a (0 , θ )˜ a ˜ a ′ − ∂ t a (0 , θ )˜ a ′ ˜ a (cid:3) n n − k (cid:2) ∂ t a (0 , θ )˜ a ˜ a | n | + ∂ t a (0 , θ )˜ a ˜ a | n | (cid:3) + (cid:2) ∂ t a (0 , θ ) q n + ∂ t a (0 , θ ) q n (cid:3) + (cid:2) a (0 , θ )˜ a ′ n q − a (0 , θ )˜ a ′ n q (cid:3) − k (cid:2) a (0 , θ )˜ a q n − a (0 , θ )˜ a q n (cid:3) + 12 (cid:2) a (0 , θ )˜ a n q ′ − a (0 , θ )˜ a n q ′ (cid:3) + 12 (cid:2) a (0 , θ ) m n + a (0 , θ ) m n (cid:3)o + O ( t ) ≡ p ( θ ) + t p ( θ ) + t p ( θ ) + O ( t ) . (2.75)On the other hand, by recalling (2.39), (2.66) and (2.67), we can obtain that1 √ g σ = h √ h + r t + r t + O ( t ) i × (cid:2) y ( θ ) + t y ( θ ) + t y ( θ ) + O ( t ) (cid:3) = 1 √ h y ( θ ) + t r y ( θ ) + t √ h y ( θ ) + t h r y ( θ ) + 1 √ h y ( θ ) + c y ( θ ) i + O ( t ) ≡ √ h y ( θ ) + t r y ( θ ) + t √ h y ( θ ) + t k + O ( t ) , √ g σ = h √ h + r t + r t + O ( t ) (cid:3) × (cid:2) y ( θ ) + t y ( θ ) + t y ( θ ) + O ( t ) i = 1 √ h y ( θ ) + t r y ( θ ) + t √ h y ( θ ) + t h r y ( θ ) + y ( θ ) r + 1 √ h y ( θ ) i + O ( t ) ≡ √ h y ( θ ) + t r y ( θ ) + t √ h y ( θ ) + t k + O ( t ) . ‡ AND JUN YANG § Therefore, we obtain that1 √ g σ (cid:20) a ( y ) ∂F ∂t ∂F ∂θ − a ( y ) ∂F ∂θ ∂F ∂t (cid:21) u t + 1 √ g σ (cid:20) a ( y ) ∂F ∂θ ∂F ∂θ − a ( y ) ∂F ∂θ ∂F ∂θ (cid:21) u t = h √ h y ( θ ) + t r y ( θ ) + t √ h y ( θ ) + t k + O ( t ) i × h p ( θ ) + t p ( θ ) + O ( t ) i u t + h √ h y ( θ ) + t r y ( θ ) + t √ h y ( θ ) + t k + O ( t ) i × h p ( θ ) + t p ( θ ) + t p ( θ ) + O ( t ) i u t ≡ √ h y ( θ ) p ( θ ) u t + 1 √ h y ( θ ) p ( θ ) u t + h r y ( θ ) p ( θ ) + 1 √ h y ( θ ) p ( θ ) + 1 √ h y ( θ ) p ( θ )+ 1 √ h y ( θ ) p ( θ ) + r y ( θ ) p ( θ ) + 1 √ h y ( θ ) p ( θ ) i tu t + b ( θ ) t u t + O ( t ) , and 1 √ g σ (cid:20) a ( y ) ∂F ∂t ∂F ∂t − a ( y ) ∂F ∂t ∂F ∂t (cid:21) u θ + 1 √ g σ (cid:20) a ( y ) ∂F ∂t ∂F ∂θ − a ( y ) ∂F ∂θ ∂F ∂t (cid:21) u θ = h √ h y ( θ ) + t r y ( θ ) + t √ h y ( θ ) + t k + O ( t ) i × h p ( θ ) + t p ( θ ) + O ( t ) i u θ + 1 √ h y ( θ ) + t r y ( θ ) + t √ h y ( θ ) + t k + O ( t ) i × h p ( θ ) + t p ( θ ) + t p ( θ ) + O ( t ) i u θ ≡ √ h y ( θ ) p ( θ ) u θ + 1 √ h y ( θ ) p ( θ ) u θ + b ( θ ) tu θ + b ( θ ) t u θ + O ( t ) . Whence, the terms in ∇ a ( y ) u · ν will be rearranged in the form ∇ a ( y ) u · ν = 1 √ h y ( θ ) p ( θ ) u t + 1 √ h y ( θ ) p ( θ ) u t + 1 √ h y ( θ ) p ( θ ) u θ + 1 √ h y ( θ ) p ( θ ) u θ + h r y ( θ ) p ( θ ) + 1 √ h y ( θ ) p ( θ ) + 1 √ h y ( θ ) p ( θ )+ 1 √ h y ( θ ) p ( θ ) + r y ( θ ) p ( θ ) + 1 √ h y ( θ ) p ( θ ) i tu t + b ( θ ) tu θ + b ( θ ) t u t + b ( θ ) t u θ + O ( t ) . More precisely, we shall derive the boundary operator in the following way. According to (2.7), we can obtainthat y (0) = y (1) = 0 , p (0) = p (1) = 0 . For θ = 0, the boundary operator becomes b u θ + b tu t + b t u t + b tu θ + b t u θ + ¯ D ( u ) , (2.76)where b = 1 p h (0) y (0) p (0) , b = 1 p h (0) (cid:2) y (0) p (0) + y (0) p (0) (cid:3) , (2.77) b = b (0) , b = b (0) , b = b (0) , ¯ D ( u ) = σ ( t ) u t + σ ( t ) u θ . (2.78) MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 23
On the other hand, for θ = 1, it has the form b u θ + b tu t + b t u t + b tu θ + b t u θ + ¯ D ( u ) , (2.79)with the notation b = 1 p h (1) y (1) p (1) , b = 1 p h (1) (cid:2) y (1) p (1) + y (1) p (1) (cid:3) , (2.80) b = b (1) , b = b (1) , b = b (1) , ¯ D ( u ) = σ ( t ) u t + σ ( t ) u θ . (2.81)In the above, the functions σ , · · · , σ are smooth functions of t with the properties | σ i ( t ) | ≤ C | t | , i = 3 , , , . Stationary and non-degenerate curves.
In the following, for a simple smooth curve Γ connecting the boundary ∂ Ω, we will make precisely thenotion of a non-degenerate geodesic with respect to the metric d s = V σ ( y ) (cid:2) a ( y )d y + a ( y )d y (cid:3) . Considerthe deformation of Γ in the formΓ t ( θ ) = (cid:0) Γ t ( θ ) , Γ t ( θ ) (cid:1) : γ (cid:0) Θ( t ( θ ) , θ ) (cid:1) + t ( θ ) (cid:16) ˜ a ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) , ˜ a ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1)(cid:17) , (2.82)where θ ∈ [0 ,
1] and t is a smooth function of θ with small L ∞ -norm. Note that the end points of Γ t stayon ∂ Ω. We denote a ( F ( t, θ )) , a ( F ( t, θ )) and V ( F ( t, θ )) as a ( t, θ ), a ( t, θ ), V ( t, θ ). The weighted length ofthe curve Γ t is given by the functional of t J ( t ) ≡ Z V σ (cid:0) Γ t ( θ ) (cid:1)q a (Γ t ( θ )) (cid:12)(cid:12) Γ ′ t ( θ ) (cid:12)(cid:12) + a (Γ t ( θ )) (cid:12)(cid:12) Γ ′ t ( θ ) (cid:12)(cid:12) d θ = Z V σ (cid:0) Γ t ( θ ) (cid:1)q W (cid:0) t ( θ ) (cid:1) d θ. (2.83)where we have denoted W (cid:0) t ( θ ) (cid:1) = (cid:16) Θ t ( t ( θ ) , θ ) t ′ ( θ ) + Θ θ ( t ( θ ) , θ ) (cid:17) × n (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1)(cid:12)(cid:12) + (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1)(cid:12)(cid:12) o + | t ′ ( θ ) | (cid:20) a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) ˜ a ( θ ) (cid:12)(cid:12) (cid:12)(cid:12) n (cid:0) Θ( t ( θ ) , θ ) (cid:1)(cid:12)(cid:12) + a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) ˜ a ( θ ) (cid:12)(cid:12) (cid:12)(cid:12) n (cid:0) Θ( t ( θ ) , θ ) (cid:1)(cid:12)(cid:12) (cid:21) + 2 t ′ ( θ ) t ( θ ) (cid:20) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ )˜ a ′ ( θ ) | n (cid:0) Θ( t ( θ ) , θ ) (cid:1) | + a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ )˜ a ′ ( θ ) | n (cid:0) Θ( t ( θ ) , θ ) (cid:1) | (cid:21) + | t ( θ ) | (cid:20) a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) ˜ a ′ ( θ ) (cid:12)(cid:12) | n (cid:0) Θ( t ( θ ) , θ ) (cid:1) | + a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) ˜ a ′ ( θ ) (cid:12)(cid:12) | n (cid:0) Θ( t ( θ ) , θ ) (cid:1) | (cid:21) + 2 t ′ ( θ ) (cid:16) Θ t ( t ( θ ) , θ ) t ′ ( θ ) + Θ θ ( t ( θ ) , θ ) (cid:17) × n(cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) + (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1)o + 2 t ( θ ) (cid:16) Θ t ( t ( θ ) , θ ) t ′ ( θ ) + Θ θ ( t ( θ ) , θ ) (cid:17) × n(cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ′ ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) ‡ AND JUN YANG § + (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ′ ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1)(cid:3)o . (2.84) Step 1.
The first variation of J at t along the direction h is given by J ′ ( t )[ h ] = dd s J ( t + sh ) (cid:12)(cid:12)(cid:12) s =0 = Z σV σ − (cid:0) Γ t ( θ ) (cid:1) V t (cid:0) Γ t ( θ ) (cid:1) h q W (cid:0) t ( θ ) (cid:1) d θ + 12 Z V σ (cid:0) Γ t ( θ ) (cid:1)q W (cid:0) t ( θ ) (cid:1) dd s W ( t + sh ) (cid:12)(cid:12)(cid:12) s =0 d θ, (2.85)where dd s W ( t + sh ) (cid:12)(cid:12)(cid:12) s =0 = W (cid:0) t ( θ ) (cid:1) [ h ] + W (cid:0) t ( θ ) (cid:1) [ h ] + W (cid:0) t ( θ ) (cid:1) [ h ]+ W (cid:0) t ( θ ) (cid:1) [ h ] + W (cid:0) t ( θ ) (cid:1) [ h ] + W (cid:0) t ( θ ) (cid:1) [ h ] , (2.86)in which W i (cid:0) t ( θ ) (cid:1) [ h ] , ( i = 1 , · · · ,
6) are given by W (cid:0) t ( θ ) (cid:1) [ h ] = (cid:16) Θ t ( t ( θ ) , θ ) t ′ ( θ ) + Θ θ ( t ( θ ) , θ ) (cid:17) × n (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) ( − k ˜ a ) a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1)(cid:12)(cid:12) + 2 (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) ( − k ˜ a ) a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1)(cid:12)(cid:12) o h + 2 (cid:16) Θ t ( t ( θ ) , θ ) t ′ ( θ ) + Θ θ ( t ( θ ) , θ ) (cid:17)(cid:16) Θ tt ( t ( θ ) , θ ) t ′ ( θ ) h + Θ t ( t ( θ ) , θ ) h ′ + Θ θt ( t ( θ ) , θ ) h (cid:17) × n (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1)(cid:12)(cid:12) + (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1)(cid:12)(cid:12) o + (cid:16) Θ t ( t ( θ ) , θ ) t ′ ( θ ) + Θ θ ( t ( θ ) , θ ) (cid:17) × n (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) ∂ t a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1)(cid:12)(cid:12) + (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) Θ t ( t ( θ ) , θ )+ (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) ∂ t a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1)(cid:12)(cid:12) + (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) Θ t ( t ( θ ) , θ ) o h, (2.87) W (cid:0) t ( θ ) (cid:1) [ h ] = 2 t ′ ( θ ) (cid:20) a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) ˜ a ( θ ) (cid:12)(cid:12) · (cid:12)(cid:12) n (cid:0) Θ( t ( θ ) , θ ) (cid:1)(cid:12)(cid:12) + a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) ˜ a ( θ ) (cid:12)(cid:12) · (cid:12)(cid:12) n (cid:0) Θ( t ( θ ) , θ ) (cid:1)(cid:12)(cid:12) (cid:21) h ′ + | t ′ ( θ ) | n ∂ t a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) ˜ a ( θ ) (cid:12)(cid:12) · (cid:12)(cid:12) n (cid:0) Θ( t ( θ ) , θ ) (cid:1)(cid:12)(cid:12) + ∂ t a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) ˜ a ( θ ) (cid:12)(cid:12) · (cid:12)(cid:12) n (cid:0) Θ( t ( θ ) , θ ) (cid:1)(cid:12)(cid:12) + a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) ˜ a ( θ ) (cid:12)(cid:12) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) n ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) Θ t ( t ( θ ) , θ )+ a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) ˜ a ( θ ) (cid:12)(cid:12) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) n ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) Θ t ( t ( θ ) , θ ) o h, (2.88) W (cid:0) t ( θ ) (cid:1) [ h ] = 2 t ( θ ) h a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ )˜ a ′ ( θ ) | n (cid:0) Θ( t ( θ ) , θ ) (cid:1) | + a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ )˜ a ′ ( θ ) | n (cid:0) Θ( t ( θ ) , θ ) (cid:1) | i h ′ + 2 t ′ ( θ ) h a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ )˜ a ′ ( θ ) | n (cid:0) Θ( t ( θ ) , θ ) (cid:1) | + a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ )˜ a ′ ( θ ) | n (cid:0) Θ( t ( θ ) , θ ) (cid:1) | i h MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 25 + 2 t ′ ( θ ) t ( θ ) n ∂ t a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ )˜ a ′ ( θ ) | n (cid:0) Θ( t ( θ ) , θ ) (cid:1) | + ∂ t a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ )˜ a ′ ( θ ) | n (cid:0) Θ( t ( θ ) , θ ) (cid:1) | + a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ )˜ a ′ ( θ )2 n (cid:0) Θ( t ( θ ) , θ ) n ′ (cid:0) Θ( t ( θ ) , θ )Θ t ( t ( θ ))+ a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ )˜ a ′ ( θ )2 n (cid:0) Θ( t ( θ ) , θ ) (cid:1) n ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) Θ t ( t ( θ )) o h, (2.89) W (cid:0) t ( θ ) (cid:1) [ h ] = 2 t ( θ ) h a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) ˜ a ′ ( θ ) (cid:12)(cid:12) | n (cid:0) Θ( t ( θ ) , θ ) (cid:1) | + a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) ˜ a ′ ( θ ) (cid:12)(cid:12) | n (cid:0) Θ( t ( θ ) , θ ) (cid:1) | i h + | t ( θ ) | n ∂ t a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) ˜ a ′ ( θ ) (cid:12)(cid:12) | n (cid:0) Θ( t ( θ ) , θ ) (cid:1) | + ∂ t a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) ˜ a ′ ( θ ) (cid:12)(cid:12) | n (cid:0) Θ( t ( θ ) , θ ) (cid:1) | + a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) ˜ a ′ ( θ ) (cid:12)(cid:12) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) n ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) Θ t ( t ( θ ) , θ )+ a (cid:0) Γ t ( θ ) (cid:1)(cid:12)(cid:12) ˜ a ′ ( θ ) (cid:12)(cid:12) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) n ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) Θ t ( t ( θ ) , θ ) o h, (2.90) W (cid:0) t ( θ ) (cid:1) [ h ] = 2 (cid:16) Θ t ( t ( θ ) , θ ) t ′ ( θ ) + Θ θ ( t ( θ ) , θ ) (cid:17) × n(cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) + (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1)o h ′ + 2 t ′ ( θ ) (cid:16) Θ tt ( t ( θ ) , θ ) t ′ ( θ ) h + Θ t ( t ( θ ) , θ ) h ′ + Θ θt ( t ( θ ) , θ ) h (cid:17) × n(cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) + (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1)o h + 2 t ′ ( θ ) (cid:16) Θ t ( t ( θ ) , θ ) t ′ ( θ ) + Θ θ ( t ( θ ) , θ ) (cid:17) × n(cid:2) − k ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) + (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) ∂ t a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) − (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ ) n ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) Θ t ( t ( θ )) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) − (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) n ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) Θ t ( t ( θ ) , θ )+ (cid:2) − k ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) + (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) ∂ t a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) + (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ ) n ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) Θ t ( t ( θ ) , θ )+ (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) n ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) Θ( t ( θ ) , θ ) o h, (2.91)and W (cid:0) t ( θ ) (cid:1) [ h ] = 2 (cid:16) Θ t ( t ( θ ) , θ ) t ′ ( θ ) + Θ θ ( t ( θ ) , θ ) (cid:17) × n(cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ′ ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) + (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ′ ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1)o h ‡ AND JUN YANG § + 2 t ( θ ) (cid:16) Θ tt ( t ( θ ) , θ ) t ′ ( θ ) h + Θ t ( t ( θ ) , θ ) h ′ + Θ θt ( t ( θ ) , θ ) h (cid:17) × n(cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ′ ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) + (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ′ ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1)o + 2 t ( θ ) (cid:16) Θ t ( t ( θ ) , θ ) t ′ ( θ ) + Θ θ ( t ( θ ) , θ ) (cid:17) × n(cid:2) − k ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ′ ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) + (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) ∂ t a (cid:0) Γ t ( θ ) (cid:1) ˜ a ′ ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) − (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ′ ( θ ) n ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) Θ t ( t ( θ ) , θ )+ (cid:2) − k ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ′ ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) + (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) ∂ t a (cid:0) Γ t ( θ ) (cid:1) ˜ a ′ ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) γ ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) + (cid:2) − k ( θ ) t ( θ )˜ a (cid:3) a (cid:0) Γ t ( θ ) (cid:1) ˜ a ′ ( θ ) n (cid:0) Θ( t ( θ ) , θ ) (cid:1) n ′ (cid:0) Θ( t ( θ ) , θ ) (cid:1) Θ t ( t ( θ ) , θ ) o h. (2.92)By substituting the relations in (2.16)-(2.17) into (2.87)-(2.92), we get W (0) = a (0 , θ ) (cid:12)(cid:12) n (cid:12)(cid:12) + a (0 , θ ) (cid:12)(cid:12) n (cid:12)(cid:12) = f ( θ ) , (2.93) W (0)[ h ] = − k (cid:20) ˜ a a (0 , θ ) (cid:12)(cid:12) n (cid:12)(cid:12) + ˜ a a (0 , θ ) (cid:12)(cid:12) n (cid:12)(cid:12) (cid:21) h + (cid:20) ∂ t a (0 , θ ) (cid:12)(cid:12) n (cid:12)(cid:12) + ∂ t a (0 , θ ) (cid:12)(cid:12) n (cid:12)(cid:12) (cid:21) h, (2.94) W (0)[ h ] = 0 , W (0)[ h ] = 0 , W (0)[ h ] = 0 , (2.95) W (0)[ h ] = 2 h ′ (cid:2) − a (0 , θ )˜ a ( θ ) + a (0 , θ )˜ a ( θ ) (cid:3) n n = 0 , (2.96) W (0)[ h ] = 2 h (cid:2) − a (0 , θ )˜ a ′ ( θ ) + a (0 , θ )˜ a ′ ( θ ) (cid:3) n n . (2.97)From the definitions of f as in (2.54), we know that W (0)[ h ] + W (0)[ h ] + W (0)[ h ] + W (0)[ h ] + W (0)[ h ] + W (0)[ h ] = f . The above computations give that J ′ (0)[ h ] = Z ( σV t (0 , θ ) V − σ (0 , θ ) p f + 12 V σ (0 , θ ) √ f f ) h d θ. (2.98)The curve Γ is said to be stationary with respect to the weighted length in (2.83) if the first variationof J at t = 0 is equal to zero. That is, for any smooth function h ( θ ) defined at [0 ,
1] there holds J ′ (0)[ h ] = 0 . This is equivalent to the relation σV t (0 , θ ) V − σ (0 , θ ) p f + 12 f √ f V σ (0 , θ ) = 0 , ∀ θ ∈ (0 , , (2.99)where f and f are given in (2.54). Specially, if the parameter σ in (2.83) is p +1 p − − , then (2.99) has theform (cid:16) p + 1 p − − (cid:17) V t (0 , θ ) p f + 12 V (0 , θ ) √ f f = 0 , ∀ θ ∈ (0 , , (2.100) MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 27 i.e., k = a (0 , θ )˜ a ′ n n − a (0 , θ )˜ a ′ n n a (0 , θ )˜ a | n | + a (0 , θ )˜ a | n | + 12 ∂ t a (0 , θ ) | n | + ∂ t a (0 , θ ) | n | a (0 , θ )˜ a | n | + a (0 , θ )˜ a | n | + σ V t (0 , θ ) V (0 , θ ) a (0 , θ ) | n | + a (0 , θ ) | n | a (0 , θ )˜ a | n | + a (0 , θ )˜ a | n | , ∀ θ ∈ (0 , . (2.101) Step 2.
We now consider the second variation of JJ ′′ (0)[ h, f ] = dd s J ′ (0 + sf )[ h ] (cid:12)(cid:12)(cid:12) s =0 = Z " σV tt (0 , θ ) V − σ (cid:0) , θ (cid:1) + σ ( σ − (cid:12)(cid:12) V t (0 , θ ) (cid:12)(cid:12) V − σ (0 , θ ) W (0) hf d θ + σ Z V t (0 , θ ) hV − σ (cid:0) , θ (cid:1) p W (0) dd s W ( sf ) (cid:12)(cid:12)(cid:12) s =0 d θ + σ Z V t (0 , θ ) fV − σ (cid:0) , θ (cid:1) p W (0) dd s W ( sh ) (cid:12)(cid:12)(cid:12) s =0 d θ − Z V σ (cid:0) , θ (cid:1)(cid:0)p W (0) (cid:1) dd s W ( sf ) (cid:12)(cid:12)(cid:12) s =0 dd s W ( sh ) (cid:12)(cid:12)(cid:12) s =0 d θ + 12 Z V σ (cid:0) , θ (cid:1)p W (0) dd s (cid:2) X i =1 W i (cid:0) sf (cid:1) [ h ] (cid:3)(cid:12)(cid:12)(cid:12) s =0 d θ. (2.102)From the definitions of W ( f ) and f in (2.84), (2.54), we can obtain thatdd s W ( sf ) (cid:12)(cid:12)(cid:12) s =0 = − k (cid:2) a (0 , θ )˜ a | n | + a (0 , θ )˜ a | n | (cid:3) f + (cid:2) ∂ t a (0 , θ ) | n | + ∂ t a (0 , θ ) | n | (cid:3) f + 2 (cid:2) a (0 , θ )˜ a n γ ′ + a (0 , θ )˜ a n γ ′ (cid:3) f ′ + 2 (cid:2) a (0 , θ )˜ a ′ n γ ′ + a (0 , θ )˜ a ′ n γ ′ (cid:3) f = − k (cid:2) a (0 , θ )˜ a | n | + a (0 , θ )˜ a | n | (cid:3) f + (cid:2) ∂ t a (0 , θ ) | n | + ∂ t a (0 , θ ) | n | (cid:3) f + 2 (cid:2) − a (0 , θ )˜ a ′ + a (0 , θ )˜ a ′ (cid:3) n n f = f f. Moreover, dd s W ( sh ) (cid:12)(cid:12)(cid:12) s =0 = f h, dd s W ( sf ) (cid:12)(cid:12)(cid:12) s =0 dd s W ( sh ) (cid:12)(cid:12)(cid:12) s =0 = f hf. On the other hand, we use (2.87)-(2.92) together with the relations in (2.16)-(2.17) to derive the followingdd s W (cid:0) sf (cid:1) [ h ] (cid:12)(cid:12)(cid:12) s =0 = 2 k (cid:2) a (0 , θ )˜ a | n | + a (0 , θ )˜ a | n | (cid:3) f h − k (cid:2) ∂ t a (0 , θ )˜ a | n | + ∂ t a (0 , θ )˜ a | n | (cid:3) f h + 2 (cid:2) Θ tt (0 , θ ) f ′ h + Θ tt (0 , θ ) f h ′ + Θ θtt (0 , θ ) f h (cid:3)(cid:2) a (0 , θ ) | n | + a (0 , θ ) | n | (cid:3) + (cid:2) ∂ tt a (0 , θ ) | n | + ∂ tt a (0 , θ ) | n | (cid:3) f h + 2Θ tt (0 , θ ) (cid:2) a (0 , θ ) γ ′ γ ′′ + a (0 , θ ) γ ′ γ ′′ (cid:3) f h, (2.103)dd s W (cid:0) sf (cid:1) [ h ] (cid:12)(cid:12)(cid:12) s =0 = 2 f ′ h ′ (cid:2) a (0 , θ )˜ a | n | + a (0 , θ )˜ a | n | (cid:3) = 2 f ′ h ′ w , (2.104)dd s W (cid:0) sf (cid:1) [ h ] (cid:12)(cid:12)(cid:12) s =0 = 2 ( h f ′ + h ′ f ) (cid:2) a (0 , θ )˜ a ˜ a ′ | n | + a (0 , θ )˜ a ˜ a ′ | n | (cid:3) , (2.105) ‡ AND JUN YANG § dd s W (cid:0) sf (cid:1) [ h ] (cid:12)(cid:12)(cid:12) s =0 = 2 f h (cid:2) a (0 , θ ) | ˜ a ′ | | n | + a (0 , θ ) | ˜ a ′ | | n | (cid:3) , (2.106)dd s W (cid:0) sf (cid:1) [ h ] (cid:12)(cid:12)(cid:12) s =0 = − k ( h f ′ + h ′ f ) (cid:2) a (0 , θ ) | ˜ a | γ ′ n + a (0 , θ ) | ˜ a | γ ′ n (cid:3) + 2( h f ′ + h ′ f ) (cid:2) ∂ t a (0 , θ )˜ a γ ′ n + ∂ t a (0 , θ )˜ a γ ′ n (cid:3) = − k ( h f ′ + h ′ f ) (cid:2) − a (0 , θ ) | ˜ a | + a (0 , θ ) | ˜ a | (cid:3) n n + 2( h f ′ + h ′ f ) (cid:2) − ∂ t a (0 , θ )˜ a + ∂ t a (0 , θ )˜ a (cid:3) n n , (2.107)and dd s W (cid:0) sf (cid:1) [ h ] (cid:12)(cid:12)(cid:12) s =0 = − k (cid:2) a (0 , θ )˜ a ˜ a ′ γ ′ n + a (0 , θ )˜ a ˜ a ′ γ ′ n (cid:3) f h + 4 (cid:2) ∂ t a (0 , θ )˜ a ′ γ ′ n + ∂ t a (0 , θ )˜ a ′ γ ′ n (cid:3) f h = − k (cid:2) − a (0 , θ )˜ a ˜ a ′ + a (0 , θ )˜ a ˜ a ′ (cid:3) n n f h + 4 (cid:2) − ∂ t a (0 , θ )˜ a ′ + ∂ t a (0 , θ )˜ a ′ (cid:3) n n f h, (2.108)where w is given in (2.56). Hence, we obtaindd s (cid:2) X i =1 W i (cid:0) sf (cid:1) [ h ] (cid:3)(cid:12)(cid:12)(cid:12) s =0 = n − k (cid:2) − a (0 , θ )˜ a ˜ a ′ + a (0 , θ )˜ a ˜ a ′ (cid:3) n n + 4 (cid:2) − ∂ t a (0 , θ )˜ a ′ + ∂ t a (0 , θ )˜ a ′ (cid:3) n n + 2 (cid:2) a (0 , θ ) | ˜ a ′ | | n | + a (0 , θ ) | ˜ a ′ | | n | (cid:3) + 2 k (cid:2) a (0 , θ )˜ a | n | + a (0 , θ )˜ a | n | (cid:3) − k (cid:2) ∂ t a (0 , θ )˜ a | n | + ∂ t a (0 , θ )˜ a | n | (cid:3) + 2Θ θtt (0 , θ ) (cid:2) a (0 , θ ) | n | + a (0 , θ ) | n | (cid:3) + (cid:2) ∂ tt a (0 , θ ) | n | + ∂ tt a (0 , θ ) | n | (cid:3) + 2Θ tt (0 , θ ) (cid:2) a (0 , θ ) γ ′ γ ′′ + a (0 , θ ) γ ′ γ ′′ (cid:3)o f h + n (cid:2) a (0 , θ ) | n | + a (0 , θ ) | n | (cid:3) Θ tt (0 , θ ) + 2 k (cid:2) a (0 , θ )˜ a ˜ a ′ | n | + a (0 , θ )˜ a ˜ a ′ | n | (cid:3) − k (cid:2) − a (0 , θ ) | ˜ a ′ | + a (0 , θ ) | ˜ a ′ | (cid:3) n n + 2 (cid:2) − ∂ t a (0 , θ )˜ a + ∂ t a (0 , θ )˜ a (cid:3) n n o ( f ′ h + f h ′ ) + 2 f ′ h ′ w ≡ f f h + f ( f ′ h + f h ′ ) + 2 f ′ h ′ w . (2.109)The combinations of the above computations together with (2.93)-(2.97) will give that J ′′ (0)[ h, f ] = Z " σV tt (0 , θ ) V − σ (cid:0) , θ (cid:1) + σ ( σ − (cid:12)(cid:12) V t (0 , θ ) (cid:12)(cid:12) V − σ (0 , θ ) f hf d θ + σ Z V t (0 , θ ) V − σ (cid:0) , θ (cid:1) √ f f hf d θ − Z V σ (cid:0) , θ (cid:1)(cid:0) √ f (cid:1) f hf d θ + 12 Z V σ (cid:0) , θ (cid:1) √ f (cid:2) f f h + f ( f ′ h + f h ′ ) + 2 f ′ h ′ w (cid:3)(cid:12)(cid:12)(cid:12) s =0 d θ. (2.110) Notation 2:
By recalling w , l and f , f , f in (2.56), (2.55) and (2.54), we introduce following functions: MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 29 H ( θ ) = V σ (cid:0) , θ (cid:1) √ f w , (2.111) H ( θ ) = V σ (cid:0) , θ (cid:1) √ f n (cid:2) a (0 , θ ) | n | + a (0 , θ ) | n | (cid:3) Θ tt (0 , θ ) + 2 (cid:2) a (0 , θ )˜ a ˜ a ′ | n | + a (0 , θ )˜ a ˜ a ′ | n | (cid:3) − k (cid:2) − a (0 , θ ) | ˜ a ′ | + a (0 , θ ) | ˜ a ′ | (cid:3) n n + 2 (cid:2) − ∂ t a (0 , θ )˜ a + ∂ t a (0 , θ )˜ a (cid:3) n n o = V σ (cid:0) , θ (cid:1) √ f n l − (cid:2) a (0 , θ ) n q + a (0 , θ ) n q (cid:3) + (cid:2) a (0 , θ ) | n | + a (0 , θ ) | n | (cid:3) Θ tt (0 , θ ) o = V σ (cid:0) , θ (cid:1) √ f l , (2.112)and H ( θ ) = V σ (cid:0) , θ (cid:1) √ f n − k (cid:2) a (0 , θ )˜ a ˜ a ′ − a (0 , θ )˜ a ˜ a ′ (cid:3) n n + 4 (cid:2) ∂ t a (0 , θ )˜ a ′ − ∂ t a (0 , θ )˜ a ′ (cid:3) n n + 2 (cid:2) a (0 , θ ) | ˜ a ′ | | n | + a (0 , θ ) | ˜ a ′ | | n | (cid:3) + 2 k (cid:2) a (0 , θ )˜ a | n | + a (0 , θ )˜ a | n | (cid:3) − k (cid:2) ∂ t a (0 , θ )˜ a | n | + ∂ t a (0 , θ )˜ a | n | (cid:3) + (cid:2) ∂ tt a (0 , θ ) | n | + ∂ tt a (0 , θ ) | n | (cid:3) + 2Θ θtt (0 , θ ) (cid:2) a (0 , θ ) | n | + a (0 , θ ) | n | (cid:3) + 2Θ tt (0 , θ ) (cid:2) a (0 , θ ) γ ′ γ ′′ + a (0 , θ ) γ ′ γ ′′ (cid:3)o + " σV tt (0 , θ ) V − σ (cid:0) , θ (cid:1) + σ ( σ − (cid:12)(cid:12) V t (0 , θ ) (cid:12)(cid:12) V − σ (0 , θ ) f + σ V t (0 , θ ) V − σ (cid:0) , θ (cid:1) √ f f − V σ (cid:0) , θ (cid:1)(cid:0) √ f (cid:1) f = V σ (cid:0) , θ (cid:1) √ f n f − (cid:2) a (0 , θ ) n q ′ − a (0 , θ ) n q ′ (cid:3) + 2Θ θtt (0 , θ ) (cid:2) a (0 , θ ) | n | + a (0 , θ ) | n | (cid:3) + 2Θ tt (0 , θ ) (cid:2) a (0 , θ ) γ ′ γ ′′ + a (0 , θ ) γ ′ γ ′′ (cid:3)o + " σV tt (0 , θ ) V − σ (cid:0) , θ (cid:1) + σ ( σ − (cid:12)(cid:12) V t (0 , θ ) (cid:12)(cid:12) V − σ (0 , θ ) f + σ V t (0 , θ ) V − σ (cid:0) , θ (cid:1) √ f f − V σ (cid:0) , θ (cid:1)(cid:0) √ f (cid:1) f = V σ (cid:0) , θ (cid:1) √ f f + " σV tt (0 , θ ) V − σ (cid:0) , θ (cid:1) + σ ( σ − (cid:12)(cid:12) V t (0 , θ ) (cid:12)(cid:12) V − σ (0 , θ ) f + σ V t (0 , θ ) V − σ (cid:0) , θ (cid:1) √ f f − V σ (cid:0) , θ (cid:1)(cid:0) √ f (cid:1) f . (2.113)In the above, we have used the facts (2.21) and (2.28).The terms in (2.110) can be rearranged in the following way J ′′ (0)[ h, f ] = Z (cid:16) H f ′ + H f (cid:17) h ′ d θ + Z (cid:16) H f ′ + H f (cid:17) h d θ = (cid:16) H f ′ + H f (cid:17) h (cid:12)(cid:12)(cid:12) θ =0 − Z (cid:2) H f ′′ + H ′ f ′ + (cid:0) H ′ − H (cid:1) f (cid:3) h d θ. (2.114) ‡ AND JUN YANG § Note that H (1) f ′ (1) + H (1) f (1) = V σ (cid:0) , (cid:1)p f (1) (cid:2) w (1) f ′ (1) + l (1) f (1) (cid:3) (2.115)= V σ (cid:0) , (cid:1) p f (1) (cid:2) b f ′ (1) − b f (1) (cid:3) , (2.116)and H (0) f ′ (0) + H (0) f (0) = V σ (cid:0) , (cid:1)p f (0) (cid:2) w (0) f ′ (0) + l (0) f (0) (cid:3) (2.117)= V σ (cid:0) , (cid:1) p f (0) (cid:2) b f ′ (0) − b f (0) (cid:3) . (2.118)For more details of the derivation of the equalities (2.116) and (2.118), the reader can refer to the computa-tions in Appendix E.For a stationary curve Γ, we say that it is non-degenerate in the sense that if J ′′ (0)[ h, f ] = 0 , ∀ h ∈ H (0 , , (2.119)then f ≡
0. It is equivalent to that the boundary problem H f ′′ + H ′ f ′ + (cid:0) H ′ − H (cid:1) f = 0 in (0 , , b f ′ (0) − b f (0) = 0 , b f ′ (1) − b f (1) = 0 , (2.120)has only the trivial solution.Paralleling with the above arguments, we finally give the following Remark. Remark 2.3.
For a simple closed curve ˆΓ in R with unit length, we construct another modified Fermicoordinates y = ˆ γ (ˆ θ ) + ˆ t (cid:0) ˜ a (ˆ θ )ˆ n (ˆ θ ) , ˜ a (ˆ θ )ˆ n (ˆ θ ) (cid:1) , (2.121) where ˆ γ (ˆ θ ) = (ˆ γ (ˆ θ ) , ˆ γ (ˆ θ )) is a natural parametrization of ˆΓ in R , ˆ n (ˆ θ ) = (ˆ n (ˆ θ ) , ˆ n (ˆ θ )) is the unit normalof ˆΓ , and the functions ˜ a and ˜ a have similar expressions as in (2.6) for ˆ θ ∈ [0 , . Consider the deformationof ˆΓ in the form ˆΓ t (ˆ θ ) = (cid:0) ˆΓ t (ˆ θ ) , ˆΓ t (ˆ θ ) (cid:1) : ˆ γ (ˆ θ ) + t (ˆ θ ) (cid:16) ˜ a (ˆ θ )ˆ n (ˆ θ ) , ˜ a (ˆ θ )ˆ n (ˆ θ ) (cid:17) , (2.122) where t is a smooth function of ˆ θ with small L ∞ -norm, and then the length functional ˆ J ( t ) ≡ Z V σ (cid:0) ˆΓ t (ˆ θ ) (cid:1)q a (cid:0) ˆΓ t (ˆ θ ) (cid:1)(cid:12)(cid:12) ˆΓ ′ t (ˆ θ ) (cid:12)(cid:12) + a (cid:0) ˆΓ t (ˆ θ ) (cid:1)(cid:12)(cid:12) ˆΓ ′ t (ˆ θ ) (cid:12)(cid:12) dˆ θ. (2.123) We can do the same variational calculations to the functional ˆ J and then derive the following notions. Ifthe curvature ˆ k of ˆΓ satisfies ˆ k = a (0 , ˆ θ )˜ a ′ ˆ n ˆ n − a (0 , ˆ θ )˜ a ′ ˆ n ˆ n a (0 , ˆ θ )˜ a | ˆ n | + a (0 , ˆ θ )˜ a | ˆ n | + 12 ∂ ˆ t a (0 , ˆ θ ) | ˆ n | + ∂ ˆ t a (0 , ˆ θ ) | ˆ n | (cid:2) a (0 , ˆ θ )˜ a | ˆ n | + a (0 , ˆ θ )˜ a | ˆ n | (cid:3) + σ V ˆ t (0 , ˆ θ ) V (0 , ˆ θ ) a (0 , ˆ θ ) | ˆ n | + a (0 , ˆ θ ) | ˆ n | (cid:2) a (0 , ˆ θ )˜ a | ˆ n | + a (0 , ˆ θ )˜ a | ˆ n | (cid:3) , ∀ ˆ θ ∈ (0 , , (2.124) then the curve ˆΓ is said to be stationary . Set the notation b H = V σ (cid:0) , ˆ θ (cid:1)q ˆ f (cid:2) a (0 , ˆ θ )˜ a | ˆ n | + a (0 , ˆ θ )˜ a | ˆ n | (cid:3) (2.125) MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 31 b H = V σ (cid:0) , ˆ θ (cid:1) q ˆ f n (cid:2) a (0 , ˆ θ )˜ a ˜ a ′ | ˆ n | + a (0 , ˆ θ )˜ a ˜ a ′ | ˆ n | (cid:3) − k (cid:2) a (0 , ˆ θ ) | ˜ a ′ | − a (0 , ˆ θ ) | ˜ a ′ | (cid:3) ˆ n ˆ n + 2 (cid:2) − ∂ ˆ t a (0 , ˆ θ )˜ a + ∂ ˆ t a (0 , ˆ θ )˜ a (cid:3) ˆ n ˆ n o (2.126) b H = V σ (cid:0) , ˆ θ (cid:1) q ˆ f n − k (cid:2) a (0 , ˆ θ )˜ a ˜ a ′ − a (0 , ˆ θ )˜ a ˜ a ′ (cid:3) ˆ n ˆ n + 4 (cid:2) ∂ ˆ t a (0 , ˆ θ )˜ a ′ − ∂ ˆ t a (0 , ˆ θ )˜ a ′ (cid:3) ˆ n ˆ n + 2 (cid:2) a (0 , ˆ θ ) | ˜ a ′ | | ˆ n | + a (0 , ˆ θ ) | ˜ a ′ | | ˆ n | (cid:3) + 2ˆ k (cid:2) a (cid:0) , ˆ θ (cid:1) ˜ a | ˆ n | + a (cid:0) , ˆ θ (cid:1) ˜ a | ˆ n | (cid:3) − k (cid:2) ∂ ˆ t a (cid:0) , ˆ θ (cid:1) ˜ a | ˆ n | + ∂ ˆ t a (cid:0) , ˆ θ (cid:1) ˜ a | ˆ n | (cid:3) + (cid:2) ∂ ˆ t ˆ t a (cid:0) , ˆ θ (cid:1) | ˆ n | + ∂ ˆ t ˆ t a (cid:0) , ˆ θ (cid:1) | ˆ n | (cid:3)o + " σV ˆ t ˆ t (0 , ˆ θ ) V − σ (cid:0) , ˆ θ (cid:1) + σ ( σ − (cid:12)(cid:12) V ˆ t (0 , ˆ θ ) (cid:12)(cid:12) V − σ (0 , ˆ θ ) ˆ f + σ V ˆ t (0 , ˆ θ ) V − σ (cid:0) , ˆ θ (cid:1) q ˆ f ˆ f − V σ (cid:0) , ˆ θ (cid:1)(cid:0)q ˆ f (cid:1) ˆ f , (2.127) where ˆ f (ˆ θ ) = a (0 , ˆ θ ) | ˆ n (ˆ θ ) | + a (0 , ˆ θ ) | ˆ n (ˆ θ ) | , ˆ f (ˆ θ ) = 2 (cid:20) a (0 , ˆ θ )˜ a ′ (ˆ θ )ˆ n (ˆ θ )ˆ n (ˆ θ ) − a (0 , ˆ θ )˜ a ′ (ˆ θ )ˆ n (ˆ θ )ˆ n (ˆ θ ) (cid:21) − k (ˆ θ ) (cid:20) a (0 , ˆ θ )˜ a (ˆ θ ) | ˆ n (ˆ θ ) | + a (0 , ˆ θ )˜ a (ˆ θ ) | ˆ n (ˆ θ ) | (cid:21) + (cid:20) ∂ ˆ t a (0 , ˆ θ ) | ˆ n (ˆ θ ) | + ∂ ˆ t a (0 , ˆ θ ) | ˆ n (ˆ θ ) | (cid:21) . For a simple closed curve ˆΓ satisfying the stationary condition, if the boundary problem b H f ′′ + b H ′ f ′ + (cid:0) b H ′ − b H (cid:1) f = 0 in (0 , ,f ′ (1) = f ′ (0) , f (1) = f (0) , (2.128) has only the trivial solution, we call ˆΓ a non-degenerate stationary curve. (cid:3) Outline of the proof
Recall that w is the solution to (1.7). In fact, w is an even function defined in the form w ( x ) = C p (cid:2) e ( p − x + e − ( p − x (cid:3) − p − , ∀ x ∈ R . (3.1)We consider the associated linearized eigenvalue problem, h ′′ − h + pw p − h = λh in R , h ( ±∞ ) = 0 . (3.2)It is well known that this equation possesses a unique positive eigenvalue λ , with associated eigenfunction Z (even and positive) in H ( R ), which can be normalized so that R R Z = 1. In fact, a simple computationshows that λ = 14 ( p − p + 3) , Z = 1 qR R w p +1 w p +12 . (3.3)In this section, the strategy to prove Theorem 1.1 will be provided step by step. ‡ AND JUN YANG § The gluing procedure.
Recall that δ > η ε δ ( s ) = η δ ( ε | s | ) where η δ ( t ) is also a smooth cut-off function defined as η δ ( t ) = 1 , ∀ ≤ t ≤ δ and η δ ( t ) = 0 , ∀ t > δ, for a fixed number δ < δ / w (to be chosen later, cf. (4.59)) anda perturbation term ˜ φ (˜ y ) = η ε δ ( s ) ˇ φ (˜ y ) + ˇ ψ (˜ y ) on Ω ε , the function u (˜ y ) = w (˜ y ) + ˜ φ (˜ y ) satisfies (1.26) if( ˇ φ, ˇ ψ ) satisfies the following coupled system: η ε δ L ( ˇ φ ) = η εδ (cid:2) − N ( η ε δ ˇ φ + ˇ ψ ) − E − p w p − ˇ ψ (cid:3) in Ω ε , (3.4) η ε δ ∇ a ( ε ˜ y ) ˇ φ · ν ε + η εδ ∇ a ( ε ˜ y ) w · ν ε = 0 on ∂ Ω ε , (3.5)and X i =1 ∂ ˜ y i (cid:0) a i ( ε ˜ y ) ˇ ψ ˜ y i (cid:1) − V ( ε ˜ y ) ˇ ψ + (1 − η εδ ) p w p − ˇ ψ = − (1 − η εδ ) E − X i =1 a i ( ε ˜ y ) ˇ φ ˜ y i ∂ ˜ y i (cid:0) η ε δ ( s ) (cid:1) − X i =1 ∂ ˜ y i h a i ( ε ˜ y ) ˇ φ∂ ˜ y i (cid:0) η ε δ ( s ) (cid:1)i − (1 − η εδ ) N ( η ε δ ˇ φ + ˇ ψ ) in Ω ε , (3.6) ∇ a ( ε ˜ y ) ˇ ψ · ν ε + (cid:0) − η εδ (cid:1) ∇ a ( ε ˜ y ) w · ν ε + ε ∇ a ( ε ˜ y ) η ε δ · ν ε ˇ φ = 0 on ∂ Ω ε , (3.7)where E = X i =1 ∂ ˜ y i (cid:0) a i ( ε ˜ y ) w ˜ y i (cid:1) − V ( ε ˜ y ) w + w p , (3.8) L ( ˇ φ ) = X i =1 ∂ ˜ y i (cid:0) a i ( ε ˜ y ) ˇ φ ˜ y i (cid:1) − V ( ε ˜ y ) ˇ φ + p w p − ˇ φ, N ( ˜ φ ) = ( w + ˜ φ ) p − w p − p w p − ˜ φ. (3.9)Assume now that ˇ φ satisfies the following decay property (cid:12)(cid:12) ∇ ˇ φ (˜ y ) (cid:12)(cid:12) + (cid:12)(cid:12) ˇ φ (˜ y ) (cid:12)(cid:12) ≤ Ce − ρ/ε if dist(˜ y, Γ ε ) > δ/ε, (3.10)for a certain constant ρ >
0, and also that w (˜ y ) is exponentially small if dist(˜ y, Γ ε ) > δ/ε. (3.11)Since N is power-like with power greater than one, a direct application of Contraction Mapping Principleyields that (3.6)-(3.7) has a unique (small) solution ˇ ψ = ˇ ψ ( ˇ φ ) with k ˇ ψ ( ˇ φ ) k L ∞ ≤ Cε (cid:2) k ˇ φ k L ∞ ( | s | >δ/ε ) + k∇ ˇ φ k L ∞ ( | s | >δ/ε ) + e − δ/ε (cid:3) , (3.12)where | s | > δ/ε denotes the complement in Ω ε of δ/ε -neighborhood of Γ ε . Moreover, the nonlinear operatorˇ ψ satisfies a Lipschitz condition of the form k ˇ ψ ( ˇ φ ) − ˇ ψ ( ˇ φ ) k L ∞ ≤ Cε (cid:2) k ˇ φ − ˇ φ k L ∞ ( | s | >δ/ε ) + k∇ ˇ φ − ∇ ˇ φ k L ∞ ( | s | >δ/ε ) (cid:3) . (3.13)The key observation is that, after solving (3.6)-(3.7), we can concern (3.4)-(3.5) as a local nonlinearproblem involving ˇ ψ = ˇ ψ ( ˇ φ ), which can be solved in local coordinates in the sense that we can decomposethe interaction among the boundary, the concentration set and the terms a , V , and then construct a goodapproximate solution and also derive the resolution theory of the nonlinear problem by delicate analysis.This whole procedure is called a gluing technique in [22].3.2. Local formulation of the problem.
As described in the above, the next step is to consider (3.4)-(3.5)in the neighbourhood of Γ ε so that by the relation ˜ y = y/ε in (1.25) close to Γ ε , the variables y can berepresented by the modified Fermi coordinates, say ( t, θ ) in (2.13), which have been set up in Section 2. MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 33
Local forms of the problem.
By recalling the local coordinates ( t, θ ) in (2.13), we can also define thelocal rescaling, ( t, θ ) = ε ( s, z ) , (3.14)and then use the results in (2.63), (2.76) and (2.79) to give local expressions of the problem. The equationin (3.4) can be locally recast in ( s, z ) coordinate system as follows η ε δ ˇ L ( ˇ φ ) = η εδ (cid:2) − N ( η ε δ ˇ φ + ˇ ψ ) − E − p w p − ˇ ψ (cid:3) , ∀ ( s, z ) ∈ ( − δ /ε, δ /ε ) × (0 , /ε ) . (3.15) • The linear operator is ˇ L ( ˇ φ ) = h ( εz ) ˇ φ ss + h ( εz ) ˇ φ zz + εh ( εz ) ˇ φ s + εh ( εz ) ˇ φ z + ˆ B ( ˇ φ ) + ˆ B ( ˇ φ ) − V ( εs, εz ) ˇ φ + p w p − ˇ φ, (3.16)where ˆ B ( · ) = ε sh ( εz ) ∂ s + εsh ( εz ) ∂ sz + ε s h ( εz ) ∂ ss + εsh ( εz ) ∂ ss , and ˆ B ( · ) = εsh ( εs, εz ) ∂ zz + ε s h ( εs, εz ) ∂ zz + ε s h ( εs, εz ) ∂ sz + ε sh ( εs, εz ) ∂ z . (3.17) • The error is then expressed in the form E = h ( εz ) w ss + h ( εz ) w zz + εh ( εz ) w s + εh ( εz ) w z + ˆ B ( w ) + ˆ B ( w ) − V ( εs, εz ) w + w p . (3.18)On the other hand, the boundary condition in (3.5) can also be expressed precisely in local coordinates.If z = 0, η ε δ D ( ˇ φ ) = − η εδ G with G = D ( w ) . (3.19)And, at z = 1 /ε there holds η ε δ D ( ˇ φ ) = − η εδ G with G = D ( w ) . (3.20)The operators on the boundary are D = b ∂ z + b ε s ∂ s + b ε s ∂ s + b εs∂ z + b ε s ∂ z + ˆ D ( · ) , (3.21)and D = b ∂ z + b εs ∂ s + b ε s ∂ s + b εs ∂ z + b ε s ∂ z + ˆ D ( · ) , (3.22)where ˆ D (cid:0) · ( s, z ) (cid:1) = ε ¯ D (cid:0) · ( t, θ ) (cid:1) and ˆ D (cid:0) · ( s, z ) (cid:1) = ε ¯ D (cid:0) · ( t, θ ) (cid:1) .3.2.2. Further changing of variables.
A further change of variables in equation (3.15) will be chosen in theforms ˇ φ ( s, z ) = α ( εz ) φ ( x, z ) , with x = β ( εz ) s, (3.23)where α ( θ ) = V (0 , θ ) p − , β ( θ ) = s V (0 , θ ) h ( θ ) = s V (0 , θ ) (cid:0) a (0 , θ ) | n ( θ ) | + a (0 , θ ) | n ( θ ) | (cid:1) | a (0 , θ ) | + | a (0 , θ ) | . (3.24)It is also convenient to expand V ( εs, εz ) = V (0 , εz ) + V t (0 , εz ) · εs + 12 V tt (0 , εz ) · ε s + a ( εs, εz ) ε s , (3.25)for a smooth function a ( t, θ ). In order to express (3.16) and (3.19)-(3.20) in terms of these new coordinates,the following identities will be preparedˇ φ s = αβφ x , ˇ φ ss = αβ φ xx , ˇ φ z = εα ′ φ + εα β ′ β xφ x + αφ z , ˇ φ sz = εα ′ βφ x + εαβ ′ φ x + εαβ ′ xφ xx + αβφ xz , and ˇ φ zz = ε α ′′ φ + 2 ε α ′ β ′ β xφ x + ε α β ′′ β xφ x ‡ AND JUN YANG § + ε α (cid:16) β ′ β (cid:17) x φ xx + 2 εα β ′ β xφ xz + 2 εα ′ φ z + αφ zz . We can deduce that1 αβ ˇ L ( ˇ φ ) = h β φ zz + h φ xx − h φ + β − p w p − φ + B ( φ ) + B ( φ ) ≡ L ( φ ) , (3.26)where B ( φ ) is a linear differential operator defined by B ( φ ) = ε h β φ x + ε h αβ (cid:2) εα ′ φ + εα β ′ β xφ x + αφ z (cid:3) + h β (cid:20) ε (cid:12)(cid:12)(cid:12) β ′ β (cid:12)(cid:12)(cid:12) x φ xx + 2 ε β ′ β xφ xz + ε β ′′ β xφ x (cid:21) + ε h α ′′ αβ φ + 2 ε h α ′ αβ β ′ β xφ x + 2 ε h α ′ αβ φ z + ε h β xφ x + ε h αβ xβ (cid:2) εα ′ βφ x + εαβ ′ φ x + εαβ ′ xφ xx + αβφ xz (cid:3) + ε h (cid:16) xβ (cid:17) φ xx + εh xβ φ xx − (cid:20) ε V t (0 , εz ) β xβ + ε V tt (0 , εz ) β − (cid:16) xβ (cid:17) (cid:21) φ. (3.27)Here B ( φ ) = 1 αβ ˆ B ( ˇ φ ) + 1 αβ a ( εs, εz ) ε s α φ, (3.28)and ˆ B ( ˇ φ ) is the operator in (3.17) where derivatives are expressed in terms of s and z through (3.23), a isgiven by (3.25), and s is replaced by β − x .In the coordinates ( x, z ), the boundary conditions in (3.19)-(3.20) can be recast in the sequel. For z = 0, η ε δ (cid:2) D ( φ ) + b φ z + D ( φ ) (cid:3) = − α η εδ G , (3.29)where D ( φ ) = ε h b + b β ′ β i xφ x + ε b α ′ α φ + ε b xβ φ z + ε h b (cid:16) xβ (cid:17) β + b (cid:16) xβ (cid:17)(cid:16) β ′ β x (cid:17)i φ x + ε b α ′ α (cid:16) xβ (cid:17) φ + ε b (cid:16) xβ (cid:17) φ z , (3.30)and D ( φ ) = 1 α ˆ D ( ˇ φ ) + ε b (cid:16) xβ (cid:17) α ′ α φ + ε b β ′ (cid:16) xβ (cid:17) φ x . (3.31)Similarly, for z = 1 /ε , we have η ε δ (cid:2) D ( φ ) + b φ z + D ( φ ) (cid:3) = − α η εδ G , (3.32)where D ( φ ) = ε h b + b β ′ β i xφ x + ε b α ′ α φ + ε b xβ φ z + ε h b (cid:16) xβ (cid:17) β + b (cid:16) xβ (cid:17)(cid:16) β ′ β x (cid:17)i φ x + ε b α ′ α (cid:16) xβ (cid:17) φ + ε b (cid:16) xβ (cid:17) φ z , (3.33)and D ( φ ) = 1 α ˆ D ( ˇ φ ) + ε b (cid:16) xβ (cid:17) α ′ α φ + ε b β ′ (cid:16) xβ (cid:17) φ x . (3.34)As a conclusion, in local coordinates ( x, z ), (3.4)-(3.5) become η ε δ L ( φ ) = ( αβ − ) − η εδ h − N ( η ε δ ˇ φ + ˇ ψ ) − E − p w p − ˇ ψ i , (3.35) MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 35 η ε δ h D ( φ ) + b φ z + D ( φ ) i = − α η εδ G , (3.36) η ε δ h D ( φ ) + b φ z + D ( φ ) i = − α η εδ G . (3.37)3.3. The projected problem.
For the convenience of presentation, we pause here to give some notation.
Notation 3:
Observe that all functions involved in (3.35)-(3.37) are expressed in ( x, z ) -variables, and thenatural domain for those variables can be extended to the infinite strip S = n ( x, z ) : −∞ < x < ∞ , < z < /ε o ,∂ S = n ( x, z ) : −∞ < x < ∞ , z = 0 o , ∂ S = n ( x, z ) : −∞ < x < ∞ , z = 1 /ε o . (3.38) Accordingly, we define ˆ S = n ( x, ˜ z ) : −∞ < x < ∞ , < ˜ z < ℓ/ε o ,∂ ˆ S = n ( x, ˜ z ) : −∞ < x < ∞ , ˜ z = 0 o , ∂ ˆ S = n ( x, ˜ z ) : −∞ < x < ∞ , ˜ z = ℓ/ε o , (3.39) where ℓ is a constant defined as ℓ ≡ Z Q ( θ )d θ, with Q ( θ ) = s V (0 , θ ) h ( θ ) = vuut V (0 , θ ) (cid:16) a (0 , θ ) | n ( θ ) | + a (0 , θ ) | n ( θ ) | (cid:17) a (0 , θ ) a (0 , θ ) . (3.40) In all what follows, we will introduce some parameters { f j } Nj =1 and { e j } Nj =1 and assume the validity ofthe following constraints, for j = 1 , · · · , N , k f j k H (0 , < C | ln ε | , f j +1 ( θ ) − f j ( θ ) > | ln ε | − | ln ε | , (3.41) k e j k ∗∗ ≡ k e j k L ∞ (0 , + ε k e ′ j k L (0 , + ε k e ′′ j k L (0 , ≤ ε , (3.42) where we have used the convention f = −∞ and f N +1 = ∞ . Set f = ( f , · · · , f N ) , f j ’s satisfy bounds (3 . , e = ( e , · · · , e N ) , e j ’s satisfy bounds (3 . , c = ( c , · · · , c N ) , d = ( d , · · · , d N ) , c j ’s and d j ’s are in L (0 , , l = ( l , , · · · , l ,N ) , l = ( l , , · · · , l ,N ) , l ,j ’s and l ,j ’s are constants , m = ( m , , · · · , m ,N ) , m = ( m , , · · · , m ,N ) , m ,j ’s and m ,j ’s are constants , and F = (cid:8) ( f , e ) : { f j } Nj =1 and { e j } Nj =1 satisfy (3 . and (3 . respectively (cid:9) . (3.43) (cid:3) One of the left job is to find the local forms of the approximate solution w with the constraint (3.11)and also of the error E . We recall the transformation in (3.23)-(3.24), and then define the local form of theapproximate solution w by the relation η ε δ ( s ) w = η ε δ ( s ) α ( εz ) v ( x, z ) with x = β ( εz ) s. (3.44)The error E can be locally recast in ( x, z ) coordinate system by the relation1 αβ η εδ ( s ) E = η εδ ( s ) E , (3.45)where E = S ( v ) with S ( v ) = h β v zz + h (cid:2) v xx − v + v p (cid:3) + B ( v ) + B ( v ) , (3.46) ‡ AND JUN YANG § with the operator B and B defined in (3.27)-(3.28). In the coordinates ( x, z ), the boundary errors can berecast as follows. For z = 0,1 α η εδ ( s ) G = η εδ ( s ) g with g = D ( v ) + b v z + D ( v ) , (3.47)and also for z = 1 /ε , 1 α η εδ ( s ) G = η εδ ( s ) g with g = D ( v ) + b v z + D ( v ) . (3.48)It is of importance that (3.45), (3.47) and (3.48) hold only in a small neighbourhood of Γ ε . Hence we willconsider v , S ( v ) as functions of the variables x and z on S , and also g , g on ∂ S and ∂ S in the sequel.We will find v = v in (4.58) step by step in Section 4, so that w will be given in (4.59) with the propertyin (3.11). In fact, to deal with the resonance, in addition to the parameters f and h , we shall add one moreparameter, say e , in the approximate solution v . The exact forms of the error terms S ( v ), g and g willbe given in (4.62) and (4.63).To make suitable extension of (3.35)-(3.37), we define an operator on the whole strip S in the form L ( φ ) ≡ h β φ zz + h (cid:2) φ xx − φ + pw p − φ (cid:3) + χ ( ε | x | ) B ( φ ) − h (1 − η ε δ ) pv p − φ in S , (3.49)and also the operators D ( φ ) = χ ( ε | x | ) D ( φ ) + b φ z + χ ( ε | x | ) D ( φ ) on ∂ S , (3.50) D ( φ ) = χ ( ε | x | ) D ( φ ) + b φ z + χ ( ε | x | ) D ( φ ) on ∂ S , (3.51)where χ ( r ) is a smooth cut-off function which equals 1 for 0 ≤ r < δ that vanishes identically for r > δ .Rather than solving problem (3.35)-(3.37) directly, we deal with the following projected problem: for eachpair of parameters f and e in F , finding functions φ ∈ H ( S ) , c , d ∈ L (0 ,
1) and constants l , l , m , m such that L ( φ ) = η εδ ( s ) (cid:2) − E − N ( φ ) (cid:3) + N X j =1 c j ( εz ) χ ( ε | x | ) w j,x + N X j =1 d j ( εz ) χ ( ε | x | ) Z j in S , (3.52) D ( φ ) = η εδ ( s ) g + N X j =1 l ,j χ ( ε | x | ) w j,x + N X j =1 m ,j χ ( ε | x | ) Z j on ∂ S , (3.53) D ( φ ) = η εδ ( s ) g + N X j =1 l ,j χ ( ε | x | ) w j,x + N X j =1 m ,j χ ( ε | x | ) Z j on ∂ S , (3.54) Z R φ ( x, z ) w j,x d x = Z R φ ( x, z ) Z j d x = 0 , < z < ε , ∀ j = 1 , · · · , N, (3.55)where we have denoted N ( φ ) = (cid:2) v + φ + ψ ( φ ) (cid:3) p − v p − pv p − φ. (3.56)The functions w j , Z j , j = 1 , · · · , N are given in (4.2). This problem has a unique solution φ such that ˇ φ satisfies (3.10). Proposition 3.1.
There is a number ˜ τ > such that for all ε small enough and all parameters ( f , e ) in F ,problem (3.52)-(3.55) has a unique solution φ = φ ( f , e ) which satisfies k φ k H ( S ) ≤ ˜ τ ε / | ln ε | q , (cid:13)(cid:13) φ (cid:13)(cid:13) L ∞ ( | x | >δ/ε ) + (cid:13)(cid:13) ∇ φ (cid:13)(cid:13) L ∞ ( | x | >δ/ε ) ≤ e − ρδ/ε . Moreover, φ depends Lipschitz-continuously on the parameters f and e in the sense of the estimate k φ ( f , e ) − φ ( f , e ) k H ( S ) ≤ Cε / | ln ε | q (cid:2) k f − f k H (0 , + k e − e k ∗∗ (cid:3) . (3.57) MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 37
Proof.
The proof is similar as that for Proposition 5.1 in [22], which will be omitted here. (cid:3)
We conclude this section by stating the following announcements: • As we have said in the above, we shall construct the approximate solution in Section 4. • To find a real solution to (1.26), the reduction procedure will be carried out in Sections 5 and 6 to kill theLangrange multipliers in (3.52)-(3.55). This can be done by suitable choice of the parameters f = ( f , · · · , f N )and e = ( e , · · · , e N ). We will first derive the equations involving the parameters f , · · · , f N and e , · · · , e N in Section 5, and then solve the coupled system involving the equations in Section 6.4. The local approximate solutions
The main objective of this section is to construct the local form v of the approximation w (cf. (4.59))and then evaluate its error E , g , g in the coordinate system ( x, z ).4.1. The first approximate solution.
Recall the notation in Section 3.3. For a fixed integer
N >
1, weassume that the locations of N concentration layers are characterized by sets (cid:8) ( x, z ) : x = β ( εz ) f j ( εz ) + β ( εz ) h ( εz ) (cid:9) , j = 1 , · · · , N, in the coordinates ( x, z ). The function h satisfies b h ′ (0) − b h (0) = 0 , b h ′ (1) − b h (1) = 0 . (4.1)In fact, h will be chosen by solving (4.57) and f j ’s can be determined in the reduction procedure.By recalling w given in (1.7) and Z in (3.3), we set w j ( x ) = w ( x j ) , Z j ( x ) = Z ( x j ) , (4.2)with x j = x − β ( εz ) f j ( εz ) − β ( εz ) h ( εz ) , and then define the first approximate solution by v ( x, z ) = N X j =1 w j ( x ) . (4.3)For every fixed n with 1 ≤ n ≤ N , we consider the following set A n = ( ( x, z ) ∈ S : βf n − ( εz ) + βf n ( εz )2 ≤ x − βh ≤ βf n ( εz ) + βf n +1 ( εz )2 ) . (4.4)For ( x, z ) ∈ A n , we expand S ( v ) by gathering terms of ε and those of order ε : S ( v ) = N X j =1 ε (cid:2) h ( f j + h ) w j,xx − V t (0 , εz ) β ( f j + h ) w j (cid:3) + N X j =1 εβ (cid:2) h w j,x − V t (0 , εz ) β x j w j + h x j w j,xx (cid:3) − N X j =1 ε "(cid:16) − h β f j + h f ′′ j β + h β ′ β f ′ j + h α ′ αβ f ′ j (cid:17) w j,x + h β ′ β f ′ j x j w j,xx − h β f j x j w j,xx + V tt (0 , εz ) β f j x j w j + N X j =1 ε h(cid:16) h f ′ j + 2 h f ′ j h ′ − h f j f ′ j − h f ′ j h + h f j (cid:17) w j,xx − β − V tt (0 , εz ) f j w j i + N X j =1 ε αβ h h (cid:16) − αβf ′ j + αβ ′ f j (cid:17) x j w j,xx + h (cid:16) α ′ βf j + αβ ′ f j (cid:17) w j,x − h αβf ′ j w j,x i ‡ AND JUN YANG § + N X j =1 ε β h(cid:16) h + h β ′′ β + h α ′ β ′ αβ (cid:17) x j w j,x + h | β ′ | β x j w j,xx + h β h ′ w j,xx + h α ′′ α w j − β V tt (0 , εz ) x j w j − V tt (0 , εz )(2 f j h + h ) w j + h β (2 f j h + h ) + h x w j,xx i − N X j =1 ε h(cid:16) − h β h + h h ′′ β + h β ′ β h ′ + h α ′ αβ h ′ (cid:17) w j,x + h β ′ β h ′ x j w j,xx − h β hx j w j,xx + V tt (0 , εz ) β hx j w j i + N X j =1 ε αβ h h (cid:16) α ′ + αβ ′ β (cid:17) x j w j,x + h αβ ′ β x j w j,xx − h αβ ( f j h ′ + hh ′ ) w j,xx + h α ′ w j i + N X j =1 ε αβ h h (cid:16) − αβh ′ + αβ ′ h (cid:17) x j w j,xx + h (cid:16) α ′ βh + αβ ′ h (cid:17) w j,x − h αβh ′ w j,x i + B ( v ) ≡ ε N X j =1 S ,j + ε N X j =1 S ,j + ε N X j =1 S ,j + ε N X j =1 S ,j + ε N X j =1 S ,j + ε N X j =1 S ,j + ε N X j =1 S ,j + ε N X j =1 S ,j + ε N X j =1 S ,j + B ( v ) , (4.5)where B ( v ) = 1 αβ (cid:2) ˆ B ( v ) + a ( εs, εz ) ε s v (cid:3) + h h pw p − n ( v − w n ) − X j = n w jp + 12 p ( p − w p − n ( v − w n ) i + max j = n O ( e − | βf j − x | ) . (4.6)Here B ( v ) turns out to be of size ( ε + ε | ln ε | q ). Let us observe that the quantities S ,j , S ,j , S ,j , S ,j and S ,j are odd functions of x j , while S ,j , S ,j , S ,j and S ,j are even functions of x j .Using the assumptions of h in (4.1), the boundary errors can be formulated as follows. For z = 0, theerror terms have the expressions ε N X j =1 β h b f j − b f ′ j i w j,x + ε h b + b β ′ β i N X j =1 x j w j,x + ε b α ′ α N X j =1 w j + ε b N X j =1 (cid:0) − β ′ f j − β f ′ j − β ′ h − β h ′ (cid:1)(cid:16) x j β + f j + h (cid:17) w j,x + ε N X j =1 h b (cid:16) x j β + f j + h (cid:17) β + b (cid:16) x j β + f j + h (cid:17) β ′ i w j,x + ε b α ′ α N X j =1 (cid:16) x j β + f j + h (cid:17) w j + ε b N X j =1 (cid:0) − β ′ f j − β f ′ j − β ′ h − β h ′ (cid:1)(cid:16) x j β + f j + h (cid:17) w j,x + D ( v ) . (4.7) MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 39
Similarly, for z = 1 /ε , we have the terms ε N X j =1 β h b f j − b f ′ j i w j,x + ε h b + b β ′ β i N X j =1 x j w j,x + ε b α ′ α N X j =1 w j + ε b N X j =1 (cid:0) − β ′ f j − β f ′ j − β ′ h − β h ′ (cid:1)(cid:16) x j β + f j + h (cid:17) w j,x + ε N X j =1 h b (cid:16) x j β + f j + h (cid:17) β + b (cid:16) x j β + f j + h (cid:17) β ′ i w j,x + ε b α ′ α N X j =1 (cid:16) x j β + f j + h (cid:17) w j + ε b N X j =1 (cid:0) − β ′ f j − β f ′ j − β ′ h − β h ′ (cid:1)(cid:16) x j β + f j + h (cid:17) w j,x + D ( v ) . (4.8)4.2. Interior correction layers.
We now want to construct correction terms and establish a further ap-proximation to a real solution that eliminates the terms of order ε in the errors. Inspired by the method inSection 2 of [22], for fixed z , we need a solution of − φ ,xx + φ − pw p − φ = N X j =1 S ,j + N X j =1 S ,j , φ ( ±∞ ) = 0 . (4.9)As it is well known, this problem is solvable provided that Z R ( S ,j + S ,j ) w j,x d x = 0 . (4.10)Furthermore, the solution is unique under the constrain Z R φ w j,x d x = 0 . (4.11)Since S ,j is odd in the variable x j , we have Z R ( S ,j + S ,j ) w j,x d x = Z R S ,j w j,x d x = 1 β h h Z R w j,x d x − V t (0 , εz ) β − Z R x j w j w j,x d x + h Z R x j w j,xx w j,x d x i = 1 β h h + σV t (0 , εz ) β − − h i Z R w j,x d x, where we have used the fact − Z R xww x d x = Z R w d x = 2 σ Z R w x d x, Z R w x d x = − Z R xw x w xx d x. (4.12)Thanks to the stationary condition (2.100), then we obtain h = 12 h − σ V t (0 , εz ) β . (4.13)For more details, the reader can refer the computations in Appendix A. Therefore, we have verified thecondition (4.10).Then the solution εφ can be written in the form ε φ = ε N X j =1 φ ,j = ε N X j =1 ( φ ,j + φ ,j + φ ,j + φ ,j ) , (4.14) ‡ AND JUN YANG § where φ ,j ( x, z ) = a ( εz ) ω ,j ( x ) = a ( εz ) ω ( x j ) , (4.15) φ ,j ( x, z ) = a ( εz ) ω ,j ( x ) = a ( εz ) ω ( x j ) , (4.16) φ ,j ( x, z ) = (cid:2) f j ( εz ) + h ( εz ) (cid:3) a ( εz ) ω ,j ( x ) = (cid:2) f j ( εz ) + h ( εz ) (cid:3) a ( εz ) ω ( x j ) , (4.17) φ ,j ( x, z ) = (cid:2) f j ( εz ) + h ( εz ) (cid:3) a ( εz ) ω ,j ( x ) = (cid:2) f j ( εz ) + h ( εz ) (cid:3) a ( εz ) ω ( x j ) , (4.18)with a ( θ ) = h ( θ ) β ( θ ) h ( θ ) , a ( θ ) = h ( θ ) β ( θ ) h ( θ ) , (4.19) a ( θ ) = − V t (0 , θ ) β ( θ ) h ( θ ) , a ( θ ) = h ( θ ) h ( θ ) . (4.20)The functions ω , ω are respectively the unique odd solutions to − ω ,xx + ω − pw p − ω = w x + σ − xw, Z R ω w x d x = 0 , (4.21) − ω ,xx + ω − pw p − ω = − σ xw + xw xx , Z R ω w x d x = 0 , (4.22)and ω , ω are respectively the unique even functions satisfying − ω ,xx + ω − pw p − ω = w, Z R ω w x d x = 0 , (4.23) − ω ,xx + ω − pw p − ω = w xx , Z R ω w x d x = 0 . (4.24)Observe that ε φ is of size O ( ε ).4.3. Boundary corrections.
In the following, we want to cancel the boundary error terms of first orderin ε given in (4.7) and (4.8), i.e., ε h b + b β ′ (0) β (0) i x j w j,x + ε b α ′ (0) α (0) w j , and ε h b + b β ′ (1) β (1) i x j w j,x + ε b α ′ (1) α (1) w j . On the other hand, the boundary terms ε h b f j (1) − b f ′ j (1) i w j,x and ε h b f j (0) − b f ′ j (0) i w j,x will be dealt with by the standard reduction procedure in Sections 5-6.This can be done by the methods in Section 2.2 of [59]. By defining two constants c = h b + b β ′ (1) β (1) i Z R x w x Z d x + b α ′ (1) α (1) Z R w Z d x, c = h b + b β ′ (0) β (0) i Z R x w x Z d x + b α ′ (0) α (0) Z R w Z d x, and also a function A (˜ θ ) = c cos (cid:0) √ λ ℓ/ε (cid:1) − c √ λ sin (cid:0) √ λ ℓ/ε (cid:1) cos (cid:0) ε − λ ˜ θ (cid:1) + c √ λ sin (cid:0) ε − λ ˜ θ (cid:1) , (4.25)where the constant ℓ is given in (3.40), we choose φ ,j ( x, z ) = A (cid:0) ˜ d ( εz ) (cid:1) Z j = A (cid:0) ˜ d ( εz ) (cid:1) Z ( x j ) , with ˜ d ( θ ) = Z θ Q ( r ) d r, (4.26) MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 41 where Q is the function given in (3.40). On the other hand, by Corollary 2.4 in [59], we then find a uniquesolution φ ∗ of the following problem:∆ φ ∗ − φ ∗ + pw p − φ ∗ = 0 in ˆ S ,∂φ ∗ ∂ ˜ z = h b + b β ′ (1) β (1) i xw x + b α ′ (1) α (1) w − c Z on ∂ ˆ S ,∂φ ∗ ∂ ˜ z = h b + b β ′ (0) β (0) i xw x + b α ′ (0) α (0) w − c Z on ∂ ˆ S , where ˆ S , ∂ ˆ S and ∂ ˆ S are defined in (3.39). Moreover, φ ∗ is even in x . By the diffeomophismΥ : [0 , /ε ] → [0 , ℓ/ε ] , Υ( z ) = ε − Z εz Q ( θ )d θ, (4.27)where Q is the function given in (3.40), we define φ ,j ( x, z ) = φ ∗ ,j = φ ∗ (cid:0) x j , Υ( z ) (cid:1) . Hence, φ ,j satisfies the following problem: h β ∂ zz φ ,j + h (cid:2) ∂ xx φ ,j − φ ,j + pw j p − φ ,j (cid:3) = ε h β Q ′ φ ∗ , ˜ z (cid:0) x j , Υ( z ) (cid:1) in S ,∂φ ,j ∂z = − Q (1) b nh b + b β ′ (1) β (1) i x j w j,x + b α ′ (1) α (1) w j − c Z j o on ∂ S , (4.28) ∂φ ,j ∂z = − Q (0) b nh b + b β ′ (0) β (0) i x j w j,x + b α ′ (0) α (0) w j − c Z j o on ∂ S , where S , ∂ S and ∂ S are defined in (3.38). We finally set the boundary correction term ε φ ( x, z ) = ε N X j =1 φ ,j ( x, z ) = ε N X j =1 ξ ( εz ) h φ ,j ( x, z ) + φ ,j ( x, z ) i , (4.29)where ξ ( θ ) ≡ χ ( θ ) Q (0) + 1 − χ ( θ ) Q (1) , and the smooth cut-off function χ is defined by χ ( θ ) = 1 if | θ | < , and χ ( θ ) = 0 if | θ | > . Note that ε φ ( x, z ) is of size O ( ε ) under the gap condition (1.23).Let v ( x, z ) = v + ε φ + ε φ be the second approximate solution. A careful computation can indicatethat the new boundary error takes the following form ε N X j =1 h b βf j − b β f ′ j i w j,x + ε h b + b β ′ β i N X j =1 (cid:16) x j β + f j + h (cid:17)h a ω ,j,x + a ω ,j,x + ( f j + h )( a ω ,j,x + a ω ,j,x )+ Q − (cid:0) A (0) Z j,x + φ ,j,x (cid:1)i + ε N X j =1 b h a ′ ω ,j + a ′ ω ,j + ( a ′ ω ,j + a ′ ω ,j )( f j + h ) + ( a ω ,j + a ω ,j )( f ′ j + h ′ ) i ‡ AND JUN YANG § − ε N X j =1 b (cid:0) β ′ f j + βf ′ j + β ′ h + βh ′ (cid:1)h a ω ,j,x + a ω ,j,x + ( f j + h )( a ω ,j,x + a ω ,j,x )+ Q − (cid:0) A (0) Z j,x + φ ,j,x (cid:1)i − ε N X j =1 b (cid:0) β ′ f j + βf ′ j + β ′ h + βh ′ (cid:1)(cid:16) x j β + f j + h (cid:17) w j,x + ε N X j =1 b α ′ α (cid:16) x j β + f j + h (cid:17) w j + ε N X j =1 h b (cid:16) x j β + f j + h (cid:17) β + b (cid:16) x j β + f j + h (cid:17)(cid:16) β ′ β x j + β ′ f j + β ′ h (cid:17)i w j,x + ε N X j =1 b α ′ α h a ω ,j + a ω ,j + ( f j + h )( a ω ,j + a ω ,j ) + Q − ( A (0) Z j + φ ,j ) i + ε N X j =1 b (cid:16) x j β + f j + h (cid:17) Q − ( εA ′ (0) Q Z j + φ ,j,z )+ D ( v + φ + φ ) + O ( ε ) on ∂ S , (4.30)where the functions are evaluated at θ = 0. Similar estimate holds on ∂ S .4.4. The second improvement.
To deal with the resonance phenomena, which were described in Section1 of [22], and improve the approximation for a solution still keeping the term of ε , we need to introduce newparameters { e j } Nj =1 . In other words, as the methods in [22] we shall set an improved approximate solutionas follows v + ε φ + ε φ + ε N X j =1 e j ( εz ) Z j ( x ) . To decompose the coupling of the parameters { f j } Nj =1 and { e j } Nj =1 on the boundary of S (in the sense ofprojection against Z in L ), by Lemma 2.2 in [59], we introduce a new term φ ∗ (even in x ) defined by thefollowing problem ∆ φ ∗ − ˜ Kφ ∗ + p w p − φ ∗ = 0 in ˆ S ,φ ∗ ˜ z = H ( x ) on ∂ ˆ S , φ ∗ ˜ z = H ( x ) on ∂ ˆ S , (4.31)where ˜ K is a large positive constant, and the function H ( x ) is given by the following H ( x ) = h b + b β ′ (0) β (0) i(cid:0) f (0) + h (0) (cid:1)n(cid:2) a (0) ω ,x + a (0) ω ,x (cid:3) + x (cid:2) a (0) ω ,x + a (0) ω ,x (cid:3) + x β (0) Q (cid:2) A (0) Z x + φ ,x (cid:3)o + 2 b (cid:2) f (0) + h (0) (cid:3) x w x + b (cid:20) β ′ (0) β (0) (cid:0) f (0) + h (0) (cid:1) + (cid:16) f ′ (0) + β ′ (0) β (0) f (0) (cid:17)(cid:21) x w x + b h β ′ (0) f (0) + β (0) f ′ (0) + β ′ (0) h (0) + β (0) h ′ (0) i h a (0) ω ,x + a (0) ω ,x i + b α ′ (0) α (0) h ( f (0) + h (0)) ( a (0) ω + a (0) ω ) + 1 β (0) (cid:16) A (0) Z + φ ( x, (cid:17)i + b (cid:2) f ′ (0) + h ′ (0) (cid:3)h a (0) ω + a (0) ω i + b (cid:2) f (0) + h (0) (cid:3)h a ′ (0) ω + a ′ (0) ω i MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 43 + b β (0) (cid:2) f (0) + h (0) (cid:3)(cid:2) εA ′ (0) β (0) Z + φ ,z ( x, (cid:3) + b (cid:2) f (0) + h (0) (cid:3) α ′ (0) α (0) w. (4.32)The function H ( x ) has a similar expression. We define a boundary correction term again ε φ ( x, z ) = ε N X j =1 φ ,j ( x, z ) = ε N X j =1 ξ ( εz ) φ ∗ ( x j , Υ( z )) . (4.33)Note that for j = 1 , · · · , N , ε φ ,j is an exponential decaying function which is of order ε and even in thevariable x j . Then define the third approximate solution to the problem near Γ ε as v ( x, z ) = v + ε φ + ε φ + ε N X j =1 e j ( εz ) Z j ( x ) + ε φ . (4.34)4.5. The third improvement.
By choosing h , we will construct a further approximation that eliminatesthe even terms (in x j ’s) in the error S ( v ). This can be fulfilled by adding a term Φ = P Nj =1 Φ j and thenconsidering the following term S ( v + Φ) = S ( v ) + L ( ε φ ) + L ( ε φ ) + L (cid:16) ε N X j =1 e j Z j (cid:17) + L ( ε φ ) + L (Φ)+ B ( ε φ ) + B ( ε φ ) + B (cid:16) ε N X j =1 e j Z j (cid:17) + B ( ε φ ) + B (Φ)+ N (cid:16) ε φ + ε φ + ε N X j =1 e j Z j + ε φ + Φ (cid:17) , (4.35)where L ( φ ) = h β φ zz + h (cid:2) φ xx − φ + pv p − φ (cid:3) , N ( φ ) = h (cid:2) ( v + φ ) p − v p − pv p − φ (cid:3) . (4.36)The details will be given in the sequel.4.5.1. Rearrangements of the error components.
The first objective of this part is to given the details tocompute the terms in formula (4.35). • It is easy to compute that L ( ε φ ) = ε h h β φ ,zz + h (cid:0) φ ,xx − φ + pv p − φ (cid:1) i = − ε N X j =1 ( S ,j + S ,j ) + ε h β φ ,zz + εh (cid:16) p v p − φ − N X j =1 p w jp − φ ,j (cid:17) . (4.37) • Recall the expression of ε φ and A (˜ θ ) defined in (4.29) and (4.25). By using the equation of φ ,j in (4.28)and the equation of Z in (3.2), we get L ( ε φ ) = ε h h β φ ,zz + h (cid:0) φ ,xx − φ + p v p − φ (cid:1)i = N X j =1 M ,j ( x, z ) + M ( x, z ) + ε h (cid:16) p v p − φ − N X j =1 p w jp − φ ,j (cid:17) , (4.38)where M ,j ( x, z ) ≡ ε β h n ξ ′ ( εz ) (cid:2) εA ′ (cid:0) ˜ d ( εz ) (cid:1) Q Z j + φ ,j,z (cid:3) + Q ′ ξ ( εz ) (cid:2) εA ′ (cid:0) ˜ d ( εz ) (cid:1) Z j + φ ∗ ,j, ˜ z (cid:3)o , (4.39)and M ( x, z ) ≡ N X j =1 ε β h ξ ′′ ( εz ) (cid:2) A ′ (cid:0) ˜ d ( εz ) (cid:1) Z j + φ ,j (cid:3) . (4.40) ‡ AND JUN YANG § • According to the equation of Z , we obtain L (cid:16) ε N X j =1 e j Z j (cid:17) = ε N X j =1 n h β ( e j Z j ) zz + h (cid:2) ( e j Z j ) xx − e j Z j + pv p − e j Z j (cid:3)o = ε N X j =1 h ε h β e ′′ j Z j + h λ e j Z j i + ε h pv p − N X j =1 e j Z j − ε h N X j =1 pw jp − e j Z j + ε a ( εs, εz ) . • Recalling the expression of ε φ defined in (4.33) and the equation of φ ∗ in (4.31), it follows that L ( ε φ ) = ε h h β φ ,zz + h (cid:0) φ ,xx − φ + p v p − φ (cid:1)i = N X j =1 ε h ( ˜ K − φ ,j + ε h h β φ ,j,zz + h (cid:0) φ ,j,xx − ˜ Kφ ,j + p v p − φ ,j (cid:1)i = N X j =1 M ,j ( x, z ) + M ( x, z ) + ε h (cid:16) p v p − φ − N X j =1 p w jp − φ ,j (cid:17) , (4.41)where M ,j ( x, z ) ≡ ε h ( ˜ K − φ ,j , (4.42)and M ( x, z ) ≡ N X j =1 h β h ε ξ ′′ ( εz ) φ ∗ ( x j , Υ( z )) + 2 ε ξ ′ ( εz ) φ ∗ ˜ z ( x j , Υ( z )) Q ( εz )+ ε ξ ( εz ) φ ∗ ˜ z ( x j , Υ( z )) Q ′ ( εz ) i . (4.43) • Recall the expression of B in (3.27), we obtain that B ( ε φ ) = ε h β N X j =1 φ ,j,x + ε h N X j =1 x j β φ ,j,xx + ε h N X j =1 ( f j + h ) φ ,j,xx − ε V t (0 , εz ) β N X j =1 x j β φ ,j − ε V t (0 , εz ) β N X j =1 ( f j + h ) φ ,j + ε a ( εs, εz )= N X j =1 M ,j ( x, z ) + M ( x, z ) , (4.44)where M ,j ( x, z ) ≡ ε ( h β (cid:0) a ω ,j,x + a ω ,j,x + a hω ,j,x + a hω ,j,x (cid:1) + h β (cid:0) a x j ω ,j,xx + a x j ω ,j,xx + a hx j ω ,j,xx + a hx j ω ,j,xx (cid:1) + h (cid:0) a f j hω ,j,xx + a f j hω ,j,xx (cid:1) + h (cid:2) ha ω ,j,xx + ha ω ,j,xx + a ( f j h + h ) ω ,j,xx + a ( f j h + h ) ω ,j,xx (cid:3) − V t (0 , εz ) β β (cid:0) a x j ω ,j + a x j ω ,j + a hx j ω ,j + a hx j ω ,j (cid:1) MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 45 − V t (0 , εz ) β (cid:2) ha ω ,j + ha ω ,j + a ( f j h + h ) ω ,j + a ( f j h + h ) ω ,j (cid:3) − V t (0 , εz ) β (cid:0) a f j hω ,j + a f j hω ,j (cid:1)) , (4.45)and M ( x, z ) ≡ N X j =1 ε ( h β f j (cid:0) a ω ,j,x + a ω ,j,x (cid:1) + h β f j (cid:0) a x j ω ,j,xx + a x j ω ,j,xx (cid:1) + h (cid:0) f j a ω ,j,xx + f j a ω ,j,xx + a f j ω ,j,xx + a f j ω ,j,xx (cid:1) − V t (0 , εz ) β f j (cid:0) a x j ω ,j + a x j ω ,j (cid:1) − V t (0 , εz ) β (cid:0) f j a ω ,j + f j a ω ,j + a f j ω ,j + a f j ω ,j (cid:1)) + ε a ( εs, εz ) . (4.46) • Moreover, we can decompose B ( ε φ ) as follows B ( ε φ ) = ε h β N X j =1 φ ,j,x + ε h β N X j =1 φ ,j,z + 2 ε h β ′ β N X j =1 (cid:16) x j β + f j + h (cid:17) φ ,j,xz + 2 ε h α ′ αβ N X j =1 φ ,j,z + ε h β N X j =1 (cid:16) x j β + f j + h (cid:17) φ ,j,xz + ε h N X j =1 (cid:16) x j β + f j + h (cid:17) φ ,j,xx − ε V t (0 , εz ) β N X j =1 (cid:16) x j β + f j + h (cid:17) φ ,j + ε a ( εs, εz )= N X j =1 M ,j ( x, z ) + M ( x, z ) , where M ,j ( x, z ) ≡ ε ξ ( εz ) β ( h (cid:2) A (cid:0) ˜ d ( εz ) (cid:1) Z j,x + φ ,j,x (cid:3) + h β (cid:2) A ′ (cid:0) ˜ d ( εz ) (cid:1) ε Q Z j + φ ,j,z (cid:3) + 2 h β ′ β (cid:16) x j β + h (cid:17)(cid:2) A ′ (cid:0) ˜ d ( εz ) (cid:1) ε Q Z j,x + φ ,j,xz (cid:3) + 2 h α ′ αβ (cid:2) A ′ (cid:0) ˜ d ( εz ) (cid:1) ε Q Z j + φ ,j,z (cid:3) + h (cid:16) x j β + h (cid:17)(cid:2) A ′ (cid:0) ˜ d ( εz ) (cid:1) ε Q Z j,x + φ ,j,xz (cid:3) + h β (cid:16) x j β + h (cid:17)(cid:2) A (cid:0) ˜ d ( εz ) (cid:1) Z j,xx + φ ,j,xx (cid:3) − V t (0 , εz ) β (cid:16) x j β + h (cid:17)(cid:2) A (cid:0) ˜ d ( εz ) (cid:1) Z j + φ ,j (cid:3)) , (4.47)and M ( x, z ) ≡ N X j =1 ε ξ ( εz ) f j β ( h β ′ β (cid:2) A ′ (cid:0) ˜ d ( εz ) (cid:1) ε Q Z j,x + φ ,j,xz (cid:3) + h (cid:2) A ′ (cid:0) ˜ d ( εz ) (cid:1) ε Q Z j,x + φ ,j,xz (cid:3) + h β (cid:2) A (cid:0) ˜ d ( εz ) (cid:1) Z j,xx + φ ,j,xx (cid:3) − V t (0 , εz ) β (cid:2) A (cid:0) ˜ d ( εz ) (cid:1) Z j + φ ,j (cid:3)) + ε a ( εs, εz ) . (4.48) ‡ AND JUN YANG § • The computations for next term is the following: B (cid:16) ε N X j =1 e j Z j (cid:17) = ε h β N X j =1 e j Z j,x + ε h N X j =1 (cid:16) x j β + f j + h (cid:17) Z j,xx − ε V t (0 , εz ) β N X j =1 (cid:16) x j β + f j + h (cid:17) e j Z j + ε a ( εs, εz )= N X j =1 M ,j ( x, z ) + M ( x, z ) , where M ,j ( x, z ) ≡ ε ( f j + h ) e j h h Z j,xx − V t (0 , εz ) β Z j i , (4.49) M ( x, z ) ≡ N X j =1 ε β e j h h Z j,x + h x j Z j,xx − V t (0 , εz ) β x j Z j i + ε a ( εs, εz ) . (4.50) • The main order of the nonlinear term N (cid:16) ε φ + ε φ + ε P Nj =1 e j Z j + ε φ + Φ (cid:17) is in the following p ( p − v p − (cid:16) ε φ + ε φ + ε N X j =1 e j Z j + ε φ + Φ (cid:17) , which can be decomposed in the form p ( p − h v p − (cid:16) ε φ + ε φ + ε N X j =1 e j Z j + ε φ + Φ (cid:17) = M ( x, z ) + M ( x, z ) + O ( ε ) , where M ( x, z ) = N X j =1 M ,j ( x, z ) ≡ N X j =1 ε h p ( p − w j p − n a ω ,j + a ω ,j + 2 a a ω ,j ω ,j + 2 h (cid:2) a a ω ,j ω ,j + a a ω ,j ω ,j (cid:3) + 2 h (cid:2) a a ω ,j ω ,j + a a ω ,j ω ,j (cid:3) + (2 f j h + h ) (cid:2) a ω ,j + a ω ,j + 2 a a ω ,j ω ,j (cid:3) + e Z j + ξ ( εz ) h A (cid:0) ˜ d ( εz ) (cid:1) Z j + φ ,j i + 2( a ω ,j + a ω ,j ) ξ ( εz ) h A (cid:0) ˜ d ( εz ) (cid:1) Z j + φ ,j i + 2( a ω ,j + a ω ,j )( f j + h ) h e j Z j + ξ ( εz ) (cid:0) A (cid:0) ˜ d ( εz ) (cid:1) Z j + φ ,j (cid:1)i + 2 ξ ( εz ) h A (cid:0) ˜ d ( εz ) (cid:1) Z j + φ ,j i e j Z j o , (4.51)and M ( x, z ) = N X j =1 M ,j ( x, z ) MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 47 ≡ N X j =1 p ( p − w jp − n ε f j (cid:0) a a ω ,j ω ,j + a a ω ,j ω ,j + a a ω ,j ω ,j + a a ω ,j ω ,j (cid:1) + ε a ω ,j f j + ε a ω ,j f j + 2 ε a ω ,j e j Z j + ε φ ( x j , z ) + Φ j + 2 ε φ ,j φ + 2 φ ,j Φ j + 2 ε φ φ + 2 φ ,j Φ j + 2 ε e j Z j ξ ( εz ) φ ∗ ( x j , z ) + 2 ε e j Z j Φ j + 2 ε φ Φ j o . (4.52)For the convenience of notation, we also denote M ( x, z ) = N (cid:16) ε φ + ε φ + ε N X j =1 e j Z j + ε φ + Φ (cid:17) − M ( x, z ) − M ( x, z ) . (4.53)Whence, according to the above rearrangements, we rewrite (4.35) in terms of S ( v + Φ) = N X j =1 h ε S ,j + ε S ,j + ε S ,j + ε S ,j + h (cid:0) Φ j,x j x j − Φ j + pw jp − Φ j (cid:1) + M ,j ( x, z )+ M ,j ( x, z ) + M ,j ( x, z ) + M ,j ( x, z ) + M ,j ( x, z ) + M ,j ( x, z ) i + N X j =1 ε (cid:16) S ,j + S ,j + S ,j (cid:17) + N X j =1 ε (cid:16) ε h β e ′′ j Z j + h λ e j Z j (cid:17) + B ( v ) + ε h β φ ,zz + h β Φ zz + M ( x, z ) + M ( x, z ) + M ( x, z ) + M ( x, z ) + M ( x, z )+ M ( x, z ) + M ( x, z ) + B ( ε φ ) + B (Φ) + εh (cid:16) pv p − φ − N X j =1 pw jp − φ ,j (cid:17) + εh (cid:16) pv p − N X j =1 e j Z j − N X j =1 pw jp − e j Z j (cid:17) + εh (cid:16) pv p − φ − N X j =1 pw jp − φ ,j (cid:17) + ε h (cid:16) pv p − φ − N X j =1 pw jp − φ ,j (cid:17) + h (cid:16) pv p − Φ − N X j =1 pw j p − Φ (cid:17) . (4.54)4.5.2. Finding new correction terms and defining the basic approximation.
In order to eliminate the termsbetween the first brackets in (4.54), for fixed z , we need a solution to the problem h ( − Φ j,xx + Φ j − pw jp − Φ j ) = ε S ,j + ε S ,j + ε S ,j + ε S ,j + M ,j ( x, z ) + M ,j ( x, z ) + M ,j ( x, z )+ M ,j ( x, z ) + M ,j ( x, z ) + M ,j ( x, z ) , ∀ x ∈ R . (4.55)It is well-known that the above problem is solvable provided that Z R (cid:2) ε S ,j + ε S ,j + ε S ,j + ε S ,j + M ,j ( x, z ) + M ,j ( x, z )+ M ,j ( x, z ) + M ,j ( x, z ) + M ,j ( x, z ) + M ,j ( x, z ) (cid:3) w j,x d x = 0 . (4.56)The computations in Appendix A give that the validity of (4.56) holds if the following problem H ( εz ) h ′′ + H ′ ( εz ) h ′ + h H ′ ( εz ) − H ( εz ) + α ( z ) ζ ( εz ) i h = 1 ζ ( εz ) (cid:2) G ( z ) + G ( z ) (cid:3) , b h ′ (0) − b h (0) = 0 , b h ′ (1) − b h (1) = 0 , (4.57) ‡ AND JUN YANG § has a solution. Here the functions H , H , H , ζ , α , G and G are given in (2.111)-(2.113), (A.19), (A.6),(A.7) and (A.9). In fact, for the solvability of problem (4.57) the reader can refer to Lemma 6.1 in [62].Moreover, h has the following estimate k h k H (0 , ≤ Cε . Now, we can find a function defined by Φ = ε φ ( x, εz ), such that the terms between the first brackets in(4.54) disappear. Finally, our basic approximate solution to the problem near the curve Γ ε is v = v + εφ + εφ + ε N X j =1 e j Z j + ε φ + ε φ . (4.58)4.6. The global approximate solution and errors.
Recall the coordinates ( s, z ) in (3.14), ( t, θ ) in(2.13), and also the local approximate solution v ( s, z ) in (4.58), which is constructed near the curve Γ ε in the coordinates ( s, z ). By the relations in (3.23), we then make an extension and simply define theapproximate solution to (1.26) in the form w (˜ y ) = η ε δ ( s ) α ( εz ) v ( x, z ) . (4.59)Note that, in the coordinates (˜ y , ˜ y ) introduced in (1.25), w is a function defined on Ω ε which is extendedglobally as 0 beyond the 6 δ/ε -neighborhood of Γ ε . The interior error can be arranged as follows E ≡ S ( v ) = ε N X j =1 (cid:0) S ,j + S ,j + S ,j (cid:1) + B ( v ) + N X j =1 h ε h β e ′′ j Z j + εh λ e j Z j i + ε h β φ ,zz + ε h β φ ,zz + M ( x, z ) + M ( x, z ) + M ( x, z )+ M ( x, z ) + M ( x, z ) + M ( x, z ) + M ( x, z ) + B ( ε φ ) + B ( ε φ )+ εh (cid:16) p v p − φ − N X j =1 pw jp − φ ,j (cid:17) + εh (cid:16) p v p − φ − N X j =1 pw j p − φ ,j (cid:17) + ε h (cid:16) p v p − φ − ε N X j =1 p w jp − φ ,j (cid:17) + ε h (cid:16) p v p − φ − ε N X j =1 p w jp − φ ,j (cid:17) , (4.60)where we have used (4.54) and the equation of φ in (4.55). The boundary error term g has the form g ( x ) = ε N X j =1 h b βf j − b β f ′ j i w j,x + ε h b + b β ′ β i N X j =1 ( f j + h ) n a ω ,j,x + a ω ,j,x + Q − (cid:0) A (0) Z j,x + φ ,j,x (cid:1)o + ε h b + b β ′ β i N X j =1 (cid:16) x j β (cid:17)(cid:2) a ω ,j,x + a ω ,j,x (cid:3) − ε N X j =1 b (cid:0) β ′ f j + βf ′ j + β ′ h + βh ′ (cid:1)n ( a ω ,j,x + a ω ,j,x )( f j + h )+ 1 β (cid:0) A (0) Z j,x + φ ,j,x ( x j , (cid:1) + e j Z j,x o − ε N X j =1 b α ′ α n ( a ω ,j + a ω ,j ) + e j Z j o + ε b α ′ α x j β w j − ε N X j =1 n b β (cid:0) x j β + f j + h + 2 f j h (cid:1) + b (cid:2) β ′ β x j + β ( f j + h ) (cid:3)o w j,x MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 49 + ε b N X j =1 h ( a ′ ω ,j + a ′ ω ,j ) + e ′ j Z j + b x j β (cid:0) εA ′ (0) β (0) Z j + φ ,j,z ( x, (cid:1)i − ε N X j =1 ( βf ′ j + β ′ f j ) h b ( f j + h ) − ε b (cid:16) x j β + f j + h (cid:17) i w j,x + O ( ε ) . (4.61)The term g has a similar expression.We decompose E = E + E , g = g + g , g = g + g , (4.62)with E = N X j =1 E ,j = N X j =1 (cid:0) ε h β e ′′ j Z j + ε h λ e j Z j (cid:1) , and E = E − E ,g = ε N X j =1 β (0) (cid:2) b f j − b f ′ j (cid:3) w j,x + N X j =1 ε b e ′ j Z j , and g = g − g , (4.63) g = ε N X j =1 β (0) (cid:2) b f j − b f ′ j (cid:3) w j,x + N X j =1 ε b e ′ j Z j , and g = g − g . For further references, it is useful to estimate the L ( S ) norm of E . From the uniform bound of e , · · · , e N in (3.42), it is easy to see that kE k L ( S ) ≤ Cε / | ln ε | q . (4.64)Since ε φ , ε φ and εe j Z j are of size O ( ε ), all terms in E carry ε in front. We claim that kE k L ( S ) ≤ Cε / | ln ε | q . (4.65)Similarly, we have the following estimate k g k L ( R ) + k g k L ( R ) ≤ Cε / | ln ε | q . (4.66)Moreover, for the Lipschitz dependence of the term of error E on the parameters f and e for the normdefined in (3.41) and (3.42), we have the validity of the estimate kE ( f , e ) − E ( f , e ) k L ( S ) ≤ Cε / | ln ε | q (cid:2) k f − f k H (0 , + k e − e k ∗∗ (cid:3) . (4.67)Similarly, we obtain k g ( f , e ) − g ( f , e ) k L ( R ) + k g ( f , e ) − g ( f , e ) k L ( R ) ≤ Cε / | ln ε | q (cid:2) k f − f k H (0 , + k e − e k ∗∗ (cid:3) . (4.68)5. Derivation of the reduced equations: Toda system
In this section, we will set up equations for the parameters f and e which are equivalent to making c ( εz ), d ( εz ), l , l , m and m are identically zero in the system (3.52)-(3.55). The equations c ( εz ) = 0 , l = 0 , l = 0 , are then equivalent to the relations, for n = 1 , · · · , N , Z R h η εδ ( s ) E + η εδ ( s ) N ( φ ) + χ ( ε | x | ) B ( φ ) + h p (cid:0) β − χ ( ε | x | ) w p − − v p − (cid:1) φ i w n,x d x = 0 , (5.1) Z R h η εδ ( s ) g − χ ( ε | x | ) D ( φ ) + χ ( ε | x | ) D ( φ ) i w n,x d x = 0 , z = 1 /ε, (5.2) Z R h η εδ ( s ) g − χ ( ε | x | ) D ( φ ) + χ ( ε | x | ) D ( φ ) i w n,x d x = 0 , z = 0 . (5.3)Similarly, d ( εz ) = 0 , m = 0 , m = 0 , ‡ AND JUN YANG § if and only if for n = 1 , · · · , N , Z R h η εδ ( s ) E + η εδ ( s ) N ( φ ) + χ ( ε | x | ) B ( φ ) + h p (cid:0) β − χ ( ε | x | ) w p − − v p − (cid:1) φ i Z n d x = 0 , (5.4) Z R h η εδ ( s ) g − χ ( ε | x | ) D ( φ ) + χ ( ε | x | ) D ( φ ) i Z n d x = 0 , z = 1 /ε, (5.5) Z R h η εδ ( s ) g − χ ( ε | x | ) D ( φ ) + χ ( ε | x | ) D ( φ ) i Z n d x = 0 , z = 0 . (5.6)5.1. Estimates for projections of the error.
For the pair ( f , e ) satisfying (3.41) and (3.42), we denote b ε and b ε , generic, uniformly bounded continuous functions of the form b lεj = b lεj (cid:0) z, f ( εz ) , e ( εz ) , f ′ ( εz ) , ε e ′ ( εz ) (cid:1) , l = 1 , , where b εj is uniformly Lipschitz in its four last arguments, and introduce the notation S = { x ∈ R : ( x, z ) ∈ S} , S n = { x ∈ R : ( x, z ) ∈ A n } n = 1 , · · · , N. (5.7)The computations in Appendix B lead to the estimate, for n = 1 , · · · , N , Z S E w n,x d x = − ̺ h ( ε ς h H f ′′ n + H ′ f ′ n + (cid:0) H ′ − H + α ( z ) (cid:1) f n i (5.8)+ e − β ( f n − f n − ) − e − β ( f n +1 − f n ) ) + P n ( εz ) , (5.9)where γ ( θ ) = ̺ β ( θ ) , ς ( θ ) = γ ( θ ) ̺ h ( θ ) , α ( z ) = α ( z ) ζ ( εz ) , (5.10)P n ( εz ) = ε γ ( θ ) (cid:2) ~ ( εz ) e n + ε ~ ( εz ) e ′′ n (cid:3) + ε µ max j = n O ( e − β | f j − f n | )+ ε N X j =1 (cid:0) b εj e ′ j + b εj f ′′ j + b εj (cid:1) + O ( ε ) N X j =1 ( f j + f ′ j + f ′′ j + f j ) , where the functions H , H , H are defined in (2.111)-(2.113), and ζ in (A.19), α ( z ) in (A.6), ̺ and ̺ in(A.2)-(B.6), ~ ( εz ) and ~ ( εz ) in (B.11)-(B.12), h in (2.57). For further references we observe that k P n k L (0 , ≤ Cε µ , for some µ > , n = 1 , · · · , N. (5.11)On the other hand, the computations in Appendix C lead to the estimate, for n = 1 , · · · , N , Z S n E Z n d x = ε h β e ′′ n + ελ h e n + ε ~ e ′ n + ̺ h (cid:2) e − β ( f n − f n − ) − e − β ( f n +1 − f n ) (cid:3) + ε ̺ ( f ′ n ) + ε ρ ( εz ) + ε ρ ( εz ) + ε ρ ( εz ) + R n ( εz ) , (5.12)where R n ( εz ) = ε f n β − ~ ( εz ) e ′′ n + 2 ε ̺ ( f ′ n h ′ + f ′ n h ) + ε β − ξ ′′ ( εz ) A ′ (cid:0) ˜ d ( εz ) (cid:1) + O ( ε ) N X j =1 (cid:0) f j + f ′ j (cid:1) + O ( ε ) N X j =1 e ′′ j + ε ˆ τ max j = n O ( e − β | f j − f n | )+ ε N X j =1 (cid:0) b εj f ′′ j + b εj e ′ j + b εj (cid:1) + O ( ε ) . Here the constants λ , ̺ , ̺ are given (3.3), (C.3) and (C.5), while the functions h , h , ~ , ~ , ρ , ρ and ρ are given in (2.57), (2.58), (C.17), (C.18), (C.7), (C.12) and (C.14). For further references we observethat k R n k L (0 , ≤ Cε µ , for some µ > , n = 1 , · · · , N. (5.13) MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 51
Projection of errors on the boundary.
In this section, we compute the projection of errors on theboundary. Without loss of generality, only the projections of the error components on ∂ S will be given.According to the expression of g as in (4.61), the main errors on the boundary integrated against w n,x and Z n in the variable x n can be computed as the following: Z R g ( x ) w n,x d x = ε N X j =1 β (0) h b f j − b f ′ j i Z R w j,x w n,x d x + ε M ,n ( f , e ) + O ( ε ) . Using the following formulas Z R Z d x = − Z R x Z x Z d x = 1 , we get the following two estimates Z R g ( x ) Z n d x = ε b (cid:20) e ′ n (0) + α ′ (0) α (0) e n (0) (cid:21) + O ( ε ) . Higher order errors can be proceeded as follows: Z R D ( φ ( x, w n,x d x = N X j =1 ε h b + b β ′ β i Z R xφ x ( x, w n,x d x − N X j =1 ε α ′ α Z R φ ( x, w n,x d x + N X j =1 ε b Z R (cid:16) x j β + f j + h (cid:17) φ z ( x, w n,x d x + N X j =1 ε Z R h b (cid:16) x j β + f j + h (cid:17) β + b (cid:16) x j β + f j + h (cid:17)(cid:16) β ′ β x j + β ′ f j + β ′ h (cid:17)i φ x ( x, w n,x d x + N X j =1 ε b α ′ α Z R (cid:16) x j β + f j + h (cid:17) φ ( x, w n,x d x + N X j =1 ε b Z R (cid:16) x j β + f j + h (cid:17) φ z ( x, w n,x d x = O ( ε µ ) , and also Z R D ( φ ( x, Z n d x = O ( ε µ )The term D ( φ ) on the boundary integrated against w n,x and Z n in the variable x n are of size of order O ( ε ).5.3. The system involving ( f , e ) . As done in [22] and [23], we can estimate the terms that involve φ in(5.1) and (5.4) integrated against the functions w n,x and Z n in the variable x n in the similar ways. As aconclusion, for n = 1 , · · · , N , there holds the following estimate − ε ς h H f ′′ n + H ′ f ′ n + (cid:0) H ′ − H + α ( z ) (cid:1) f n i + e − β ( f n − f n − ) − e − β ( f n +1 − f n ) + M n = 0 . (5.14)Moreover, M n can be decomposed in the following way M n = M n ( θ, f , f ′ , f ′′ , e , e ′ , e ′′ ) + M n ( θ, f , f ′ , e , e ′ ) , (5.15)where M n and M n are continuous of their arguments. Functions M n and M n satisfy the followingproperties for n = 1 , · · · , N k M n k L (0 , ≤ Cε µ , k M n k L (0 , ≤ Cε µ , where µ is a positive constant. For n = 1 , · · · , N , there also holds the following estimate ε h β e ′′ n + εh λ e n + ε ~ e ′ n + h ̺ (cid:2) e − β ( f n − f n − ) − e − β ( f n +1 − f n ) (cid:3) + ε ̺ ( f ′ n ) + ε ρ + ε ρ + ε ρ + M n = 0 . (5.16) ‡ AND JUN YANG § Moreover, M n can be decomposed in the following way M n = M n ( θ, f , f ′ , f ′′ , e , e ′ , e ′′ ) + M n ( θ, f , f ′ , e , e ′ ) , (5.17)where M n and M n are continuous of their arguments. Functions M n and M n satisfy the followingproperties for n = 1 , · · · , N k M n k L (0 , ≤ Cε τ , k M n k L (0 , ≤ Cε τ , where ˆ τ is a positive constant.Therefore, using θ = εz and defining the operators L n ( f ) ≡ ε ς ( θ ) h H ( θ ) f ′′ n + H ′ ( θ ) f ′ n + (cid:0) H ′ ( θ ) − H ( θ ) + α ( θ/ε ) (cid:1) f n i − e − β ( θ )( f n − f n − ) + e − β ( θ )( f n +1 − f n ) , n = 1 , · · · , N, (5.18)and L ( e ) ≡ − ε h ( θ ) e ′′ − ε ˜ α ( θ ) e ′ − | β ( θ ) | h ( θ ) λ e, (5.19)we derive the following nonlinear system of differential equations for the parameters f and e L n ( f ) = M n , n = 1 , · · · , N, (5.20) L ( e n ) = α + α ,n + ε − β M n , n = 1 , · · · , N, (5.21)with the boundary conditions, n = 1 , · · · , N , b f ′ n (1) − b f n (1) + M ,n ( f , e ) = 0 , b f ′ n (0) − b f n (0) + M ,n ( f , e ) = 0 , (5.22) e ′ n (1) + ˜ b e n (1) + M ,n ( f , e ) = 0 , e ′ n (0) + ˜ b e n (0) + M ,n ( f , e ) = 0 . (5.23)The constants ˜ b and ˜ b are given by ˜ b = α ′ (1) α (1) , ˜ b = α ′ (0) α (0) , where M ij,n ’s are some terms of order O ( ε / ). The functions ς ( θ ) = γ ( θ ) ̺ h ( θ ) >
0, and α ( θ/ε ) are definedin (5.10) by the relation θ = εz . Moreover, we have denoted˜ α ( θ ) = β ( θ ) ~ ( θ ) , (5.24) α ( θ ) = β ( θ ) (cid:2) ε ρ ( θ ) + ερ ( θ ) + ε ρ ( θ ) (cid:3) , (5.25) α ,n ( θ ) = ε ̺ β ( θ ) | f ′ n ( θ ) | + ε − ̺ β ( θ ) h ( θ ) (cid:2) e − β ( θ )( f n − f n − ) − e − β ( θ )( f n +1 − f n ) (cid:3) , (5.26)where ~ ( θ ), ρ ( θ ), ρ ( θ ), ρ ( θ ) are defined in (C.17), (C.7), (C.12), (C.14).6. Suitable choosing of parameters
Solving the system of reduced equations.
Before solving (5.20)-(5.23), some basic facts about theinvertibility of corresponding operators will be derived.Firstly, we consider the following problem L ( e ) = ˜ g ( θ ) , ∀ < θ < ,e ′ (1) + ˜ b e (1) = 0 , e ′ (0) + ˜ b e (0) = 0 . (6.1) Proposition 6.1. If ˜ g ∈ L (0 , , then for all small ε satisfying (1.23) there is a unique solution e ∈ H (0 , to problem (6.1), which satisfies k e k ∗∗ ≤ C ε − k ˜ g k L (0 , . Moreover, if ˜ g ∈ H (0 , , then ε k e ′′ k L (0 , + ε k e ′ k L (0 , + k e k L ∞ (0 , ≤ C k ˜ g k H (0 , . (6.2) Proof.
The proof is similar as that for Lemma 8.1 in [22]. (cid:3)
MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 53
Secondly, we consider the following problem L n ( f ) = ε ˜ h n , (6.3) b f ′ n (1) − b f n (1) = 0 , b f ′ n (0) − b f n (0) = 0 , (6.4)for n = 1 , · · · , N , where f = −∞ , f N +1 = ∞ . Proposition 6.2.
For given ˜ h = (˜ h , · · · , ˜ h N ) T ∈ L (0 , , there exists a sequence { ε l : l ∈ N } from those ε satisfying the gap condition (1.23) and approaching such that problem (6.3)-(6.4) admits a solution f = ( f , · · · , f N ) T with the form: f = 1 β ( ρ ε l (cid:16) − N + 12 , − N + 12 , · · · , N − N + 12 (cid:17) T + ¨f + P T ˆ u + P T ˜w + P T ˇ u ) , (6.5) where the invertible matrix P is defined in (6.33) and the function ρ ε l ( θ ) satisfies e − ρ εl ( θ ) = ε ς ( θ ) β ( θ ) τ ( θ ) ρ ε l ( θ ) , (6.6) with τ given in (6.17) , and in particular ρ ε l ( θ ) = 2 | ln ε l | − ln(2 | ln ε l | ) − ln (cid:16) ς ( θ ) τ ( θ ) β ( θ ) (cid:17) + O (cid:16) ln (cid:0) | ln ε l | (cid:1) | ln ε l | (cid:17) . The vectors ¨f = ( ¨ f , · · · , ¨ f N ) T defined in Lemma 6.4, ˆ u = (ˆ u , · · · , ˆ u N ) T defined by (6.40) and ˜w =( ˜w , · · · , ˜w N ) T in (6.53)-(6.55) do not depend on ˜h . There hold the estimates ¨ f j = O (1) , ˆ u j ( θ ) = O (cid:16)(cid:16) | ε l | − ln(ln | ε l | ) (cid:17) (cid:17) , j = 1 , · · · , N, | ln ε l | k ˜w ′′ j k L (0 , + k ˜w ′ j k L (0 , + k ˜w j k L (0 , ≤ C | ln ε l | , j = 1 , · · · , N − , (6.7) k ˜w N k H (0 , ≤ C. (6.8) For the vector ˇ u = (ˇ u , · · · , ˇ u N ) T , we have | ln ε l | k ˇ u ′′ k L (0 , + 1 p | ln ε l | k ˇ u ′ k L (0 , + k ˇ u k L (0 , ≤ C ε µ k ˜ h k L (0 , + C | ln ε l | . (cid:3) The proof of Proposition 6.2 will be provided in Section 6.2. By accepting this, we here want to finishthe proof of Theorem 1.1.
Proof of Theorem 1.1.
The profile of the solution given in (1.10) can be determined by the approximatesolution given in Section 4, see (4.59). The properties of the parameters f j ’s in (1.11)-(1.12) can be derivedfrom Proposition 6.2.As we have stated in Section 3, we shall complete the last step of suitable choosing the parameters f and e by solving (5.20)-(5.23). If ˆ e solves L (ˆ e ) = α ( θ ) , ∀ < θ < , ˆ e ′ (1) + ˜ b ˆ e (1) = 0 , ˆ e ′ (0) + ˜ b ˆ e (0) = 0 , (6.9)from the definition of α ( θ ) in (5.25), we get k ˆ e k H (0 , ≤ Cε . Replacing e n by ˆ e + ˜ e n , the system (5.20)-(5.23) keeps the same form except that the term α ( θ ) disappear.Moreover, let ˜ e n solves L (˜ e n ) = α ,n ( θ ) , ∀ < θ < , ˜ e ′ n (1) + ˜ b ˜ e n (1) = 0 , ˜ e ′ n (0) + ˜ b ˜ e n (0) = 0 , (6.10)then it derives k ˜ e n k H (0 , ≤ Cε µ . ‡ AND JUN YANG § Define the set D = n f , e ∈ H ( S ) : k f k H (0 , ≤ D | ln ε | , k e k ∗∗ ≤ Cε µ o . For (¯ f , ¯ e ) ∈ D , we can set for n = 1 , · · · , N ˜ h n ( f , e ) ≡ ε − M n ( f , f ′ , f ′′ , e , e ′ , e ′′ ) + ε − M n (¯ f , ¯ f ′ , ¯ e , ¯ e ′ ) , ˜ g n ( f , e ) ≡ ε − β M n ( f , f ′ , f ′′ , e , e ′ , e ′′ ) + ε − β M n (¯ f , ¯ f ′ , ¯ e , ¯ e ′ ) . We now use Contraction Mapping Principle and Schauder Fixed Point Theorem to solve (5.20)-(5.23) withthe right hand replacing by ˜ h n and ˜ g n . Whence, by the fact that M n , M n are contractions on D , makinguse of the argument developed in Propositions 6.1 and 6.2, and the Contraction Mapping Principle, we find f and e for a fixed ¯ f and ¯ e . In this way, we define a mapping Z (¯ f , ¯ e ) = ( f , e ) and the solution of our problemis simply a fixed point of Z . Continuity of M n and M n , n = 1 , · · · , N , with respect to its parametersand a standard regularity argument allows us to conclude that Z is compact as mapping from H (0 ,
1) intoitself. The Schauder Fixed Point Theorem applies to yield the existence of a fixed point of Z as required.This ends the proof of Theorem 1.1. (cid:3) Proof of Proposition 6.2.
Note that (6.3)-(6.4) can be concerned as a small perturbation of a simplerproblem in the form, for n = 1 , · · · , N , ε ς h H f ′′ n + H ′ f ′ n + (cid:0) H ′ − H (cid:1) f n i − e − β ( f n − f n − ) + e − β ( f n +1 − f n ) = ε µ ˜ h n , (6.11) b f ′ n (1) − b f n (1) = 0 , b f ′ n (0) − b f n (0) = 0 , (6.12)where H , H and H are functions defined in (2.111)-(2.113). By similar arguments as done in Section 6 of[62], we can finish the proof of Proposition 6.2 if we can solve (6.11)-(6.12).We now focus on the resolution theory for (6.11)-(6.12), whose proof basically follows the methods in [27]and [65]. However, in this paper, the homogeneous boundary conditions in (6.12) make the procedure muchmore complicated, which will be divided into three steps. In the first step, we will find an approximatesolution by solving an algebraic system and then derive the improved equivalent nonlinear system of (6.11)-(6.12), see (6.26)-(6.27). In step 2, by the decomposition method, the problem can be further transformedinto (6.38)-(6.39). To cancel the boundary error terms ˜ G and ˜ G (see (6.39)), we need to find more boundarycorrection terms ˆ u n , n = 1 , · · · , N (see (6.40)) in the expansions of u n ’s, which directly leads to the system(6.43)-(6.44). Finally, after giving the linear resolution theory in Lemma 6.5, the proof can be finished bythe Contraction Mapping Principle in Step 3. Step 1:
By setting ˇ f n ( θ ) = β ( θ ) f n ( θ ) , (6.13)we get ε ςβ (cid:2) H ˇ f ′′ n + τ ˇ f ′ n + τ ˇ f n (cid:3) − e − ( ˇ f n − ˇ f n − ) + e − ( ˇ f n +1 − ˇ f n ) = ε µ ˜ h n , (6.14)ˇ f ′ n (1) + K ˇ f n (1) = 0 , ˇ f ′ n (0) + K ˇ f n (0) = 0 , (6.15)where ˇ f = −∞ , ˇ f N +1 = ∞ . Here we have denoted τ ( θ ) = H ′ ( θ ) − β ′ ( θ ) β ( θ ) H ( θ ) , (6.16) τ ( θ ) = H ′ ( θ ) − H ( θ ) + 2 | β ′ ( θ ) | β ( θ ) H ( θ ) − β ′′ ( θ ) β ( θ ) H ( θ ) − β ′ ( θ ) β ( θ ) H ′ ( θ ) , (6.17) K = β ′ (0) β (0) + b b , K = β ′ (1) β (1) + b b , (6.18)so that τ ( θ ) > , K = K = 0 , due to the assumptions in (1.8)-(1.9). MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 55
Recall that the assumption (1.8) implies that ς ( θ ) τ ( θ ) /β ( θ ) >
0. Let us define two positive functions ρ ε ( θ ) and δ ( θ ) by e − ρ ε ( θ ) = ε ς ( θ ) β ( θ ) τ ( θ ) ρ ε ( θ ) , δ ( θ ) = τ ( θ ) ρ ε ( θ ) . (6.19)We can easily obtain that ρ ε ( θ ) = 2 | ln ε | − ln (cid:0) | ln ε | (cid:1) − ln (cid:16) ς ( θ ) τ ( θ ) β ( θ ) (cid:17) + O (cid:16) ln (cid:0) | ln ε | (cid:1) | ln ε | (cid:17) , (6.20)1 δ ( θ ) = τ ( θ ) " | ln ε | − ln (cid:0) | ln ε | (cid:1) − ln (cid:16) ς ( θ ) τ ( θ ) β ( θ ) (cid:17) + O (cid:16) ln (cid:0) | ln ε | (cid:1) | ln ε | (cid:17) . (6.21)Then multiplying equation (6.14) by ε − δ ( θ ) and settingˇ f n ( θ ) = (cid:16) n − N − (cid:17) ρ ε ( θ ) + ˆ f n ( θ ) , n = 1 , · · · , N, we get an equivalent system, for n = 1 , · · · , N , δ (cid:2) H ˆ f ′′ n + τ ˆ f ′ n + τ ˆ f n (cid:3) − e − ( ˆ f n − ˆ f n − ) + e − ( ˆ f n +1 − ˆ f n ) = ε µ δ βς ˜ h n − δ (cid:16) n − N − (cid:17) ρ ′′ ε − δ τ (cid:16) n − N − (cid:17) ρ ′ ε − (cid:16) n − N − (cid:17) , (6.22)where ˆ f = −∞ , ˆ f N +1 = ∞ . The boundary conditions becomeˆ f ′ n (1) = − (cid:16) n − N − (cid:17) ρ ′ ε (1) , ˆ f ′ n (0) = − (cid:16) n − N − (cid:17) ρ ′ ε (0) . (6.23) Remark 6.3.
Note that the terms of order O ( | ln ε | ) in the right hand sides of the equations in (6.23)disappear so that they are of O (1) due to the assumptions K = K = 0 which are exactly given in (1.9). (cid:3) First, we want to cancel the terms of O (1) in right hand side of (6.22). To this end, we will introduce thefollowing lemma Lemma 6.4.
There exists a solution ¨ f = ( ¨ f , · · · , ¨ f N ) T to the following nonlinear algebraic system − e − ( ¨ f n − ¨ f n − ) + e − ( ¨ f n +1 − ¨ f n ) = − (cid:16) n − N − (cid:17) (6.24) with n running from to N , where ¨ f = −∞ , ¨ f N +1 = ∞ .Proof. By setting a = a N = 0 , a n = e − ( ¨ f n +1 − ¨ f n ) , n = 1 , · · · , N − , (6.25)the proof can be found in the solving method for equation (7.10) in [65]. (cid:3) We set ˆ f n = ¨ f n + ˜ f n , n = 1 , · · · , N , where ¨ f n = O (1) satisfies the system (6.24). It is obvious that system(6.22)-(6.23) is equivalent to the following nonlinear system of equations, δ h H ˜ f ′′ n + τ ˜ f ′ n + τ ˜ f n i + a n − (cid:0) ˜ f n − ˜ f n − (cid:1) − a n (cid:0) ˜ f n +1 − ˜ f n (cid:1) = δ ε µ βς ˜ h n + δ ˜ g n + N n ( ˜f ) , (6.26)with boundary conditions ˜ f ′ n (0) = G ,n , ˜ f ′ n (1) = G ,n , (6.27)where we have denoted ˜ g n = − (cid:16) n − N − (cid:17) ρ ′′ ε − τ (cid:16) n − N − (cid:17) ρ ′ ε − τ ¨ f n , (6.28) G ,n = − (cid:16) n − N − (cid:17) ρ ′ ε (0) , G ,n = − (cid:16) n − N − (cid:17) ρ ′ ε (1) . (6.29) ‡ AND JUN YANG § Moreover, the nonlinear terms N n , n = 1 , · · · , N , are given by N n ( ˜f ) = a n − (cid:2) e − ( ˜ f n − ˜ f n − ) − (cid:0) ˜ f n − ˜ f n − (cid:1)(cid:3) − a n (cid:2) e − ( ˜ f n +1 − ˜ f n ) − (cid:0) ˜ f n +1 − ˜ f n (cid:1)(cid:3) , (6.30)and ˜ f = −∞ , ˜ f N +1 = ∞ . Step 2:
The first try is to decompose the above system. We will denote: ˜f = ( ˜ f , · · · , ˜ f N ) T , ˜h = (˜ h , · · · , ˜ h N ) T , ˜g = (˜ g , · · · , ˜ g N ) T , N ( ˜f ) = (cid:0) N ( ˜f ) , · · · , N N ( ˜f ) (cid:1) T , G = ( G , , · · · , G ,N ) T , G = ( G , , · · · , G ,N ) T . Then system (6.26) becomes: δ I h H d d θ + τ dd θ + τ i ˜f + A ˜f = δ ε µ βς ˜h + δ ˜g + N ( ˜f ) , (6.31)where I is a N × N unit Matrix and the Matrix A defined as A = a − a · · · − a ( a + a ) − a · · · · · · − a N − ( a N − + a N − ) − a N − · · · − a N − a N − . (6.32)For the symmetric matrix A , using elementary matrix operations it is easy to prove that there exists aninvertible matrix Q such that Q A Q T = diag( a , · · · , a N − , . Since a , · · · , a N − are positive constants defined in (6.25), then all eigenvalues of the matrix A are λ ≥ λ ≥ · · · ≥ λ N − > λ N = 0 . Moreover, since A is a symmetric matrix, there exists another invertible matrix P independent of θ with theform PP T = I , P = p · · · p N − √ N p · · · p N − √ N ... ... ... ... p N · · · p NN − √ N , (6.33)in such a way that P T A P = diag( λ , λ , · · · , λ N − , λ N ) . (6.34)We denote κ = 12 | ln ε | − ln(2 | ln ε | ) , δ − ( θ ) κ = τ ( θ ) + ˜ σ ( θ ) . (6.35)By (6.21) we have ˜ σ ( θ ) = O (cid:16) | ln ε | (cid:17) . (6.36)Multiplying (6.31) by τ ( θ ) + ˜ σ ( θ ), we get the following system κ I h H d d θ + τ dd θ + τ i ˜f + (cid:0) τ + ˜ σ (cid:1) A ˜f = κε µ βς ˜h + κ ˜g + (cid:0) τ + ˜ σ (cid:1) N ( ˜f ) . (6.37)Now, define six new vectors u = ( u , · · · , u N ) T = P T ˜f , ˜ h = (˜ h , · · · , ˜ h N ) T = βς P T ˜h , ˜ g = (˜ g , · · · , ˜ g N ) T = P T ˜g , ˜ N ( u ) = (cid:0) ˜ N ( u ) , · · · , ˜ N N ( u ) (cid:1) T = (cid:0) τ + ˜ σ (cid:1) P T N ( ˜f ) = (cid:0) τ + ˜ σ (cid:1) P T N ( P u ) , MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 57 ˜ G = ( ˜ G , , · · · , ˜ G ,N ) T = P T G , ˜ G = ( ˜ G , , · · · , ˜ G ,N ) T = P T G . Note that the form of P in (6.33) and the expressions of G and G in (6.29) imply that˜ G ,N = ˜ G ,N = 0 . Therefore (6.26)-(6.27) become κ h H u ′′ + τ u ′ + τ u i + diag( λ , · · · , λ N ) ( τ + ˜ σ ) u = κ ε µ ˜ h + κ ˜ g + ˜ N ( u ) , (6.38)with boundary conditions u ′ (0) = ˜ G , u ′ (1) = ˜ G . (6.39)For the convenience of notation, we denote ℓ n = (cid:16) κλ n (cid:17) , Π( s ) = τ ( s ) + ˜ σ ( s ) ,ϑ ( θ ) = Z θ Π( s ) d s, l = Z Π( s ) d s. In order to cancel the error terms on the boundary in (6.39), we introduce the following functionsˆ u n ( θ ) = χ ( θ ) ˜ G ,n ℓ n p Π(0) sin (cid:16) ϑ ( θ ) ℓ n (cid:17) − (cid:0) − χ ( θ ) (cid:1) ˜ G ,n ℓ n p Π(1) sin (cid:16) l − ϑ ( θ ) ℓ n (cid:17) , n = 1 , · · · , N − , and ˆ u N ( θ ) = 0 . In the above, χ is a smooth cut-off function with the properties χ ( θ ) = 1 if | θ | < / χ ( θ ) = 0 if | θ | > / . It is easy to show k ˆ u n k L (0 , ≤ C p | ln ε | . (6.40)For later use, we compute, for n = 1 , · · · , N − u ′ n ( θ ) = χ ( θ ) ˜ G ,n p Π(0) p Π( θ ) cos (cid:16) ϑ ( θ ) ℓ n (cid:17) + (cid:0) − χ ( θ ) (cid:1) ˜ G ,n p Π(1) p Π( θ ) cos (cid:16) l − ϑ ( θ ) ℓ n (cid:17) + ˜ G ,n ℓ n p Π(0) χ ′ ( θ ) sin (cid:16) ϑ ( θ ) ℓ n (cid:17) + ˜ G ,n ℓ n p Π(1) χ ′ ( θ ) sin (cid:16) l − ϑ ( θ ) ℓ n (cid:17) . This implies that ˆ u n satisfies the following boundary conditionsˆ u ′ n (0) = ˜ G ,n , ˆ u ′ n (1) = ˜ G ,n , n = 1 , · · · , N. (6.41)For n = 1 , · · · , N −
1, there holdsˆ u ′′ n ( θ ) = − χ ( θ ) ˜ G ,n p Π(0) Π( θ ) ℓ n sin (cid:16) ϑ ( θ ) ℓ n (cid:17) + (cid:0) − χ ( θ ) (cid:1) ˜ G ,n p Π(1) Π( θ ) ℓ n sin (cid:16) l − ϑ ( θ ) ℓ n (cid:17) + χ ( θ ) ˜ G ,n p Π(0) Π ′ ( θ ) p Π( θ ) cos (cid:16) ϑ ( θ ) ℓ n (cid:17) + (cid:0) − χ ( θ ) (cid:1) ˜ G ,n p Π(1) Π ′ ( θ ) p Π( θ ) cos (cid:16) l − ϑ ( θ ) ℓ n (cid:17) + χ ′ ( θ ) ˜ G ,n p Π(0) Π( θ ) cos (cid:16) ϑ ( θ ) ℓ n (cid:17) − χ ′ ( θ ) ˜ G ,n p Π(1) Π( θ ) cos (cid:16) l − ϑ ( θ ) ℓ n (cid:17) + χ ′′ ( θ ) ˜ G ,n ℓ n p Π(0) sin (cid:16) ϑ ( θ ) ℓ n (cid:17) + χ ′′ ( θ ) ˜ G ,n ℓ n p Π(1) sin (cid:16) l − ϑ ( θ ) ℓ n (cid:17) . Whence, we obtain, for n = 1 , · · · , N , (cid:13)(cid:13)(cid:13) κ h H ˆ u ′′ n + τ ˆ u ′ n + τ ˆ u n i + λ n ( τ + ˜ σ )ˆ u n (cid:13)(cid:13)(cid:13) L (0 , ≤ C | ln ε | . (6.42) ‡ AND JUN YANG § Letting u = ˜ u + ˆ u with ˆ u = (ˆ u , · · · , ˆ u N ) T , the system (6.38)-(6.39) is equivalent to the following system,for n = 1 , · · · , N , κ h H ˜ u ′′ n + τ ˜ u ′ n + τ ˜ u n i + λ n ( τ + ˜ σ )˜ u n = κε µ h n + κ ˜ g n + ˆ g n + ˜ N n (˜ u + ˆ u ) , (6.43)with boundary conditions ˜ u ′ n (0) = 0 , ˜ u ′ n (1) = 0 , (6.44)where ˆ g n = − κ h H ˆ u ′′ n + τ ˆ u ′ n + τ ˆ u n i − λ n ( τ + ˜ σ )ˆ u n , n = 1 , · · · , N − , and also ˜ g N = 0 . For later use, we will estimate the terms in the right hand of (6.43). For n = 1 , · · · , N −
1, there hold˜ N n (˜ u + ˆ u ) = (cid:0) τ + ˜ σ (cid:1)(cid:0) P T N ( P (˜ u + ˆ u )) (cid:1) n = (cid:0) τ + ˜ σ (cid:1) N X i =1 p in N i ( P (˜ u + ˆ u )) , and also ˜ h N = 1 √ N N X i =1 ˜ h i , ˜ g N = 1 √ N N X i =1 ˜ g i , ˜ N N (˜ u + ˆ u ) = (cid:0) τ + ˜ σ (cid:1)(cid:0) P T N ( P (˜ u + ˆ u )) (cid:1) N = (cid:0) τ + ˜ σ (cid:1) √ N N X i =1 N i ( P (˜ u + ˆ u )) = 0 . According to the definitions of ˜ h n ’s and ˜ g n ’s, we can easily get k ˜ h n k L (0 , ≤ C, k ˜ g n k L (0 , ≤ C, n = 1 , · · · , N. (6.45) Step 3:
For the purpose of using a fixed point argument to solve (6.43)-(6.44), we concern the followingresolution theory for the linear differential equations.
Lemma 6.5. (1).
Assume that the non-degeneracy condition (2.120) holds. For any small ε , there existsa unique solution v to the equation h H d d θ + τ dd θ + τ i v = h, v ′ (0) = 0 , v ′ (1) = 0 , (6.46) with the estimate k v k H (0 , ≤ C k h k L (0 , . (6.47) (2). Consider the following system, for n = 1 , · · · , N − , κ h H d d θ + τ dd θ + τ i v n + λ n (cid:16) τ + ˜ σ (cid:17) v n = p n ,v ′ n (0) = 0 , v ′ n (1) = 0 . (6.48) There exists a sequence { ε l , l ∈ N } approaching and satisfying the gap condition (1.23) such that problem(6.48) has a unique solution v = v ( p ) and | ln ε l | k v ′′ k L (0 , + 1 p | ln ε l | k v ′ k L (0 , + k v k L (0 , ≤ C p | ln ε l | k p k L (0 , , (6.49) where v = ( v , · · · , v N − ) T and p = ( p , · · · , p N − ) T . Moreover, if p ∈ H (0 , then | ln ε l | k v ′′ k L (0 , + k v ′ k L (0 , + k v k L (0 , ≤ C k p k H (0 , . (6.50) MBROSETTI-MALCHIODI-NI CONJECTURE: CLUSTERING CONCENTRATION LAYERS 59
Proof.
We can use the inverse of the transformation in (6.13), i.e., v = β ( θ ) ˜ v , and then obtaindd θ v = β ′ ˜ v + β dd θ ˜ v, d d θ v = β ′′ ˜ v + 2 β ′ dd θ ˜ v + β d d θ ˜ v. (6.51)Therefore, we get an equivalent problem of (6.46) H ˜ v θθ + H ′ ˜ v θ + (cid:0) H ′ − H (cid:1) ˜ v = 1 β h, b ˜ v ′ (0) − b ˜ v (0) = 0 , b ˜ v ′ (1) − b ˜ v (1) = 0 . (6.52)Recalling the definition of H , H and H in (2.111)-(2.113) and applying the non-degeneracy condition(2.120), we can solve (6.52) directly. The proof of the second part is similar as that for Claim 1 and Claim2 in [65]. The details are omitted here. (cid:3) In order to solve (6.43)-(6.44), we first concern the system, for n = 1 , · · · , N − κ h H ˜w ′′ n + τ ˜w ′ n + τ ˜w n i + λ n (cid:0) τ + ˜ σ (cid:1) ˜w n = κ ˜ g n + ˆ g n , (6.53) H ˜w ′′ N + τ ˜w ′ N + τ ˜w N = ˜ g N , (6.54)with boundary conditions ˜w ′ n (0) = 0 , ˜w ′ n (1) = 0 , n = 1 , · · · , N. (6.55)Using Lemma 6.5, we can solve the above system and get the estimates as in (6.7)-(6.8). The substituting˜ u n = ˜w n + ˇ u n , n = 1 , · · · , N, will then imply that the nonlinear problem (6.43)-(6.44) can be transformed into the following system for n = 1 , · · · , N , κ h H ˇ u ′′ n + τ ˇ u ′ n + τ ˇ u n i + λ n (cid:0) τ + ˜ σ (cid:1) ˇ u n = κε µ h n + ˜ N n (ˇ u + ˜w + ˆ u ) , (6.56)with boundary conditions ˇ u ′ n (0) = 0 , ˇ u ′ n (1) = 0 n = 1 , · · · , N. (6.57)Finally, we claim that problem (6.56)-(6.57) can be solved by using Lemma 6.5 and a Contraction MappingPrinciple in the set X = ( ˇ u ∈ H (0 ,
1) : 1 | ln ε | k ˇ u ′′ k L (0 , + 1 p | ln ε | k ˇ u ′ k L (0 , + k ˇ u k L (0 , ≤ C p | ln ε | ) . In fact, this can be done in the following way. According to the definition of ˜ N , we obtain, for any ˇ u ∈ X ˜ N (ˇ u + ˜w + ˆ u ) = (cid:0) τ + ˜ σ (cid:1) P T N (cid:0) P (ˇ u + ˜w + ˆ u ) (cid:1) ... N N (cid:0) P (ˇ u + ˜w + ˆ u ) (cid:1) . This implies that (cid:13)(cid:13) ˜ N (ˇ u + ˜w + ˆ u ) (cid:13)(cid:13) L (0 , ≤ C N X n =1 (cid:13)(cid:13) N n (cid:0) P (ˇ u + ˜w + ˆ u ) (cid:1)(cid:13)(cid:13) L (0 , , where the expression of N n is N n (cid:0) P (ˇ u + ˜w + ˆ u ) (cid:1) = a n − ( e − h ( P (ˇ u + ˜w +ˆ u )) n − (cid:0) P (ˇ u + ˜w +ˆ u ) (cid:1) n − i − h ( P (ˇ u + ˜w + ˆ u )) n − ( P (ˇ u + ˜w + ˆ u )) n − i) − a n ( e − [ ( P (ˇ u + ˜w +ˆ u )) n +1 − ( P (ˇ u + ˜w +ˆ u )) n ] − ‡ AND JUN YANG § + h ( P (ˇ u + ˜w + ˆ u )) n +1 − ( P (ˇ u + ˜w + ˆ u )) n i) = O (cid:16)(cid:12)(cid:12)(cid:12)(cid:0) P (ˇ u + ˜w + ˆ u )) n − ( P (ˇ u + ˜w + ˆ u ) (cid:1) n − (cid:12)(cid:12)(cid:12) (cid:17) + O (cid:16)(cid:12)(cid:12)(cid:12)(cid:0) P (ˇ u + w + ˆ u )) n +1 − ( P (ˇ u + ˜w + ˆ u ) (cid:1) n (cid:12)(cid:12)(cid:12) (cid:17) . The definitions of ˆ u in (6.40) and P in (6.33) will imply that (cid:13)(cid:13) | ( P ˆ u ) n | (cid:13)(cid:13) L (0 , ≤ C | ln ε | , (cid:13)(cid:13) | ( P ˇ u ) n | (cid:13)(cid:13) L (0 , ≤ C | ln ε | , n = 1 , · · · , N. Gathering the above estimates, we get the following estimate (cid:13)(cid:13) ˜ N (ˇ u + ˜w + ˆ u ) (cid:13)(cid:13) L (0 , ≤ C | ln ε | . Therefore, for any n = 1 , · · · , N −
1, using (6.45) and Lemma 6.5, we can get a solution ˇ u n to κ h H ˇ u ′′ n + τ ˇ u ′ n + τ ˇ u n i + λ n (cid:0) τ + ˜ σ (cid:1) ˇ u n = κε µ ˜ h n + ˜ N n (ˇ u + ˜w + ˆ u ) , ˇ u ′ n (0) = 0 , ˇ u ′ n (1) = 0 , with the following estimate 1 | ln ε l | k ˇ u ′′ n k L (0 , + 1 p | ln ε l | k ˇ u ′ n k L (0 , + k ˇ u n k L (0 , ≤ C p | ln ε l | (cid:13)(cid:13)(cid:13)(cid:13) κε µ ˜ h n + ˜ N n (ˇ u + ˜w + ˆ u ) (cid:13)(cid:13)(cid:13)(cid:13) L (0 , ≤ ε µ k ˜ h n k L (0 , + C p | ln ε l | ≤ C p | ln ε l | . Concerning the N -th equation in (6.43)-(6.44), i.e., H ˇ u ′′ N + τ ˇ u ′ N + τ ˇ u N = ε µ ˜ h N , ˇ u ′ N (0) = 0 , ˇ u ′ N (1) = 0 , using (6.45) and Lemma 6.5, we can also find a solution satisfying k ˇ u N k H (0 , ≤ ε µ k ˜ h N k L (0 , ≤ C p | ln ε l | . Now, the result follows by a straightforward application of Contraction Mapping Principle and Lemma 6.5.The proof of Proposition 6.2 is complete. (cid:3)
Acknowledgements:
S. Wei was supported by NSFC (No. 12001203) and Guangdong Basic and AppliedBasic Research Foundation (No. 2020A1515110622); J. Yang was supported by NSFC (No. 11771167 andNo. 11831009). (cid:3)
Appendices A. The derivation of the equation for h The computations of (4.56) can be showed as follows. Since S ,j , S ,j , M ,j ( x, z ), M ,j ( x, z ), M ,j ( x, z )are even functions of x j , then integration against w j,x therefore just vanish. This gives thatLHS of (4.56) = Z R (cid:2) ε S ,j + ε S ,j + M ,j ( x, z ) + M ,j ( x, z ) + M ,j ( x, z ) (cid:3) w j,x d x ≡ J + J + J + J + J . These terms can be computed in the sequel. • Recalling the expression of S ,j in (4.5), direct computation leads to J = − ε (cid:16) − h β h + h h ′′ β + h β ′ β h ′ + h α ′ αβ h ′ (cid:17) Z R w j,x d x − ε h β ′ β h ′ Z R x j w j,x w j,xx d x + ε h β h Z R x j w j,x w j,xx d x − ε V tt (0 , εz ) β h Z R x j w j w j,x d x = ε ̺ β h − h h ′′ − h ( β ′ β + 2 α ′ α ) h ′ + ( h − h + σV tt (0 , εz ) β − ) h i , (A.1)where we have used the relations (4.12) and ̺ ≡ Z R w x d x = − Z R xw x w xx d x. (A.2) • According to the definition of S ,j in (4.5) and (A.2), it follows that J = ε αβ Z R (cid:2) h (cid:16) − αβh ′ + αβ ′ h (cid:17) x j w j,xx + h (cid:16) α ′ βh + αβ ′ h (cid:17) w j,x − h αβh ′ w j,x (cid:3) w j,x d x = ε ̺ β h h (cid:16) β ′ β + α ′ α (cid:17) h + 12 h h ′ − h h ′ i . (A.3) • By the definition of M ,j ( x, z ) in (4.45), the facts ω ,j , ω ,j , w j,x are odd functions of x j and ω ,j , ω ,j are even functions of x j , we obtain J = ε h Z R n h β (cid:0) a ω ,j,x + a ω ,j,x (cid:1) + h β (cid:0) a x j ω ,j,xx + a x j ω ,j,xx (cid:1) − V t (0 , εz ) β (cid:0) a x j ω ,j + a x j ω ,j (cid:1) + h (cid:0) a ω ,j,xx + a ω ,j,xx (cid:1) − V t (0 , εz ) β (cid:0) a ω ,j + a ω ,j (cid:1)o w j,x d x = ε h Z R n h a (cid:0) a ω ,j,x + a ω ,j,x (cid:1) + h a (cid:0) a x j ω ,j,xx + a x j ω ,j,xx (cid:1) + h ( a − a ) (cid:0) a σ − x j ω ,j + a σ − x j ω ,j (cid:1)o w j,x d x + ε h Z R n h a (cid:0) a ω ,j,xx + a ω ,j,xx (cid:1) + h a (cid:0) a ω ,j + a ω ,j (cid:1)o w j,x d x. (A.4)Here, we have used the (4.13) and the definitions of a , a , a , a as in (4.19)-(4.20). • By the definition of M ,j ( x, z ) in (4.47), we get J = ε ξ ( εz ) β h Z R (cid:0) h β ′ β + h (cid:1) h A ′ (cid:0) ˜ d ( εz ) (cid:1) ε Q Z j,x + φ ,j,xz i w j,x d x + ε ξ ( εz ) β Z R n h h A (cid:0) ˜ d ( εz ) (cid:1) Z j,x + φ ,j,x i + h β (cid:16) x j β (cid:17) h A (cid:0) ˜ d ( εz ) (cid:1) Z j,xx + φ ,j,xx i − V t (0 , εz ) β (cid:16) x j β (cid:17) h A (cid:0) ˜ d ( εz ) (cid:1) Z j + φ ,j io w j,x d x = ε ̺ β α ( z ) h + ε ̺ β G ( z ) , (A.5)where α ( z ) = 2 ξ ( εz ) ̺ Z R h ε A ′ (cid:0) ˜ d ( εz ) (cid:1) β ( εz ) Z j,x + φ ,j,xz ( x, z ) i w j,x d x, (A.6) G ( z ) = ε ξ ( εz ) ̺ Z R n h h A (cid:0) ˜ d ( εz ) (cid:1) Z j,x + φ ,j,x i + h β (cid:16) x j β (cid:17) h A (cid:0) ˜ d ( εz ) (cid:1) Z j,xx + φ ,j,xx i − V t (0 , εz ) β (cid:16) x j β (cid:17) h A (cid:0) ˜ d ( εz ) (cid:1) Z j + φ ,j io w j,x d x. (A.7) • The definition of M ,j is given in (4.51). Since ω ,j , ω ,j and w j,x are odd functions of x j , while ω ,j , ω ,j , φ ,j and φ ,j are even functions of x j , we then obtain that J = ε h Z R p ( p − w j p − h h (cid:0) a a ω ,j ω ,j + a a ω ,j ω ,j + a a ω ,j ω ,j + a a ω ,j ω ,j (cid:1) + 2 (cid:0) a ω ,j + a ω ,j (cid:1) ξ ( εz ) (cid:0) A (cid:0) ˜ d ( εz ) (cid:1) Z j + φ ,j (cid:1)i w j,x d x ≡ ε h h Z R p ( p − w jp − h a a ω ,j ω ,j + a a ω ,j ω ,j + a a ω ,j ω ,j + a a ω ,j ω ,j i w j,x d x + ε ̺ β G ( z ) , (A.8)where G ( z ) = β̺ ξ ( εz ) h p ( p − Z R w j p − (cid:0) a ω ,j + a ω ,j (cid:1)h A (cid:0) ˜ d ( εz ) (cid:1) Z j + φ ,j ( x, z ) i w j,x d x. (A.9)By differentiating the equation (4.21) and using equations (4.23), (4.24), we obtain Z R p ( p − w jp − w j,x ω ,j ω ,j d x = − Z R w j,x ω ,j d x + Z R h w j,x + 1 σ x j w j i ω ,j,x d x, (A.10) Z R p ( p − w jp − w j,x ω ,j ω ,j d x = − Z R w j,xxx ω ,j d x + Z R h w j,x + 1 σ x j w j i ω ,j,x d x. (A.11)Similarly, by differentiating the equation (4.22) and using equations (4.23), (4.24), we obtain Z R p ( p − w jp − w j,x ω ,j ω ,j d x = − Z R w j,x ω ,j d x + Z R h − σ x j w j + x j w j,xx i ω ,j,x d x, (A.12) Z R p ( p − w jp − w j,x ω ,j ω ,j d x = − Z R w j,xxx ω ,j d x + Z R h − σ x j w j + x j w j,xx i ω ,j,x d x. (A.13)Adding (A.4), (A.8) and using (A.10)-(A.13), we have J + J = ε h h ( a a Z R w j,x ω ,j,x d x − a a σ Z R w j ω ,j d x + a a σ Z R w j ω ,j d x + a a Z R w j,x ω ,j,x d x − a a σ Z R w j ω ,j d x + a a σ Z R w j ω ,j d x − a a Z R w j,x ω ,j,x d x − a a Z R w j,x ω ,j,x d x ) = ε h h n − a a (cid:16) p − (cid:17) − a a (cid:16) − p − (cid:17) + a a (cid:16) − p − (cid:17) − a a + 12 a a + a a (cid:16) p − (cid:17)o Z R w x d x + ε ̺ β G ( z )= ε ρ h h n − a a + 12 a a − a a + 12 a a o + ε ̺ β G ( z )= ε ρ h h − σ V t (0 , θ ) β (cid:16) V t (0 , θ ) V (0 , θ ) − h ( θ ) h ( θ ) (cid:17)i + ε ̺ β G ( z ) , (A.14)where we have used (2.99) and the following integral identities2 Z R ω ,j,x w j,x d x = − σ Z R ( w j,x ) d x = − σ Z R w x d x, (A.15) σ − Z R ω ,j w j d x = (cid:16) − p − (cid:17) Z R ( w j,x ) d x = (cid:16) − p − (cid:17) Z R w x d x, (A.16)2 Z R ω ,j,x w j,x d x = − Z R w j,x d x = − Z R w x d x, (A.17) Z R ω ,j w j d x = (cid:16) p − (cid:17) Z R w j,x d x = σ Z R w x d x. (A.18)Finally, denote ζ ( θ ) = 1 α β √ h , (A.19) ~ ( θ ) = h h β ′ β + 2 α ′ α i + h − h = ζ ( θ ) H ′ ( θ ) , (A.20)and ~ ( θ ) = − h h − h + σ V tt (0 , θ ) β i − h (cid:16) β ′ β + α ′ α (cid:17) + σ V t (0 , θ ) β (cid:16) V t (0 , θ ) V (0 , θ ) − h h (cid:17) = ζ ( θ ) (cid:2) H ′ ( θ ) − H ( θ ) (cid:3) , (A.21)where the last equalities in (A.20) and (A.21) will be verified in Appendix D. We infer that equation (4.56)becomesLHS of (4.56) = − ε ̺ β h h h ′′ + ~ ( εz ) h ′ + ( ~ ( εz ) + α ( z )) h − G ( z ) − G ( z ) i = − ε ̺ β ζ ( εz ) ( H ( εz ) h ′′ + H ′ ( εz ) h ′ + h(cid:0) H ′ ( εz ) − H ( εz ) (cid:1) + α ( z ) ζ ( εz ) i h − G ( z ) + G ( z ) ζ ( εz ) ) = 0 . B. The first projection of error
We do estimates for the term R S E w n,x d x given in Section 5, where E is defined in (4.60) and w n,x isan odd function of x n . Integration against all even terms of x n , say E ,n and S ,n , M ,n , M ,n in E ,therefore just vanish. We have Z S E w n,x d x = n Z S n + Z S \ S n o E w n,x d x. (B.1)We begin with Z S n E w n,x d x = N X j =1 Z S n ε S ,j w n,x d x + N X j =1 Z S n ε S ,j w n,x d x + Z S n B ( v ) w n,x d x + Z S n ε h β φ ,zz w n,x d x + Z S n ε h β φ ,zz w n,x d x + Z S n M ( x, z ) w n,x d x + Z S n M ( x, z ) w n,x d x + Z S n M ( x, z ) w n,x d x + Z S n M ( x, z ) w n,x d x + Z S n M ( x, z ) w n,x d x + Z S n (cid:2) B ( ε φ ) + B ( ε φ ) (cid:3) w n,x d x + O ( ε ) ≡ I + I + I + I + I + I + I + I + I + I + I + O ( ε ) . (B.2)These terms will be estimated as follows. • By repeating the same computation used in (A.1) and (A.3), we getI = ε ̺ β (cid:26) − h f ′′ n − h h β ′ β + 2 α ′ α i f ′ n + h h − h + σ V tt (0 , εz ) β i f n (cid:27) + O ( ε ) N X j =1 ( f j + f ′ j + f ′′ j ) , (B.3)where ̺ is a positive constant defined in (A.2). • There also holds I = ε ̺ β h h (cid:16) β ′ β + α ′ α (cid:17) f n + 12 h f ′ n − h f ′ n i + O ( ε ) N X j =1 ( f j + f ′ j ) . (B.4) • Recall the expression of B ( v ) in (4.6), thenI = Z S n n αβ (cid:2) ˆ B ( v ) + a ( εs, εz ) ε s v (cid:3) + h p ( w n ) p − ( v − w n ) − h X j = n w jp + h p ( p − w n ) p − ( v − w n ) + max j = n O ( e − | βf j − x | ) o w n,x d x = ̺ h h e − β ( f n − f n − ) − e − β ( f n +1 − f n ) i + ε µ max j = n O ( e − β | f j − f n | ) + ε N X j =1 (cid:0) b εj f ′′ j + b εj (cid:1) , (B.5)where µ is a small positive constant, and ̺ is a positive constant given by ̺ = p C p Z ∞ w p − w x ( e − x − e x ) d x. (B.6) • Recall the expression of ε φ in (4.14), thenI = Z S n ε h β φ ,zz w n,x d x = ε N X j =1 (cid:0) b εj f ′′ j + b εj (cid:1) . (B.7) • It can be derived that I = Z S n ε h β φ ,zz w n,x d x = O ( ε ) . (B.8) • From the definition of M ( x, z ) in (4.48), we can estimate the term I as the followingI = ε ξ ( εz ) 1 β f n Z S n (cid:0) h β ′ β + h (cid:1)h A ′ (cid:0) ˜ d ( εz ) (cid:1) ε Q Z n,x + φ ,n,xz i w n,x d x + ε N X j =1 (cid:0) b εj f ′′ j + b εj (cid:1) ≡ − ε ̺ β α ( z ) f n + ε N X j =1 (cid:0) b εj f ′′ j + b εj (cid:1) + O ( ε ) N X j =1 (cid:0) f j + f ′ j (cid:1) , (B.9)where α ( z ) and ̺ are defined in (A.6) and (A.2). • From the definition of M ( x, z ) in (4.50), we need only consider the odd terms and the higher order termsinvolving e ′ j and e ′′ j , so we getI = N X j =1 ε β e j Z S n h h Z j,x + h x j Z j,xx − V t (0 , εz ) β x j Z j i w n,x d x + N X j =1 ε αβ h e ′′ j ( εz ) Z S n x j w n,x Z j d x + ε N X j =1 (cid:0) b εj e ′ j + b εj f ′′ j + b εj (cid:1) ≡ ε ̺ β h ~ ( εz ) e n + ε ~ ( εz ) e ′′ n i + ε N X j =1 (cid:0) b εj e ′ j + b εj f ′′ j + b εj (cid:1) , (B.10)where ~ ( εz ) and ~ ( εz ) are defined like the following ~ ( εz ) = − ̺ Z R h h Z x + h xZ xx − V t β − x Z i w x d x, (B.11) ~ ( εz ) = − ̺ αβ h Z R x w x Z d x. (B.12) • Recalling the definitions of M ( x, z ) and M ( x, z ) as in (4.46), (4.52), we can getI + I = ε ρ h − σ V t (0 , εz ) β (cid:16) V t (0 , εz ) V (0 , εz ) − h h (cid:17)i f n + ε N X j =1 (cid:0) b εj f ′′ j + b εj (cid:1) . • According to the fact that the terms in B ( ε φ ) and B ( ε φ ) are of order O ( ε ), it follows thatI + I = ε N X j =1 (cid:0) b εj f ′′ j + b εj (cid:1) . (B.13)The above computations lead to the estimate Z S n E w n,x d x = − ε ̺ β ζ ( εz ) n H ( εz ) f ′′ n + H ′ ( εz ) f ′ n + h H ′ ( εz ) − H ( εz ) + α ( z ) ζ ( εz ) i f n o + ε ̺ β (cid:2) ~ ( εz ) e n + ε ~ ( εz ) e ′′ n (cid:3) + h ̺ (cid:2) e − β ( f n − f n − ) − e − β ( f n +1 − f n ) (cid:3) + ε µ max j = n O ( e − β | f j − f n | ) + ε N X j =1 (cid:0) b εj e ′ j + b εj f ′′ j + b εj (cid:1) + O ( ε ) N X j =1 (cid:0) f j + f ′ j + f ′′ j + f j (cid:1) . (B.14)On the other hand, to compute R S \ S n E w n,x d x for fixed n = 1 , · · · , N , we notice that for ( x, z ) ∈ S δ/ε \ A n with S δ/ε = (cid:8) − δ/ε < x < δ/ε, < z < /ε (cid:9) , there holds w n,x = max j = n O ( e β | f j − f n | ) . Thus we can estimate Z S \ S n E w n,x d x = ε max j = n O ( e − β | f j − f n | ) + O ( ε ) X i =1 I i . (B.15)C. The second projection of error
We estimate the term R S E Z n d x in Section 5, where E and its decomposition are defined in (4.60) and(4.62), and Z n is an odd function of x n . We have Z S E Z n d x = Z S E Z n d x + Z S E Z n d x, where Z S E Z n d x = ε h β e ′′ n + εh λ e n + O ( ε ) , and Z S E Z n d x = n Z S n + Z S \ S n o E Z n d x. (C.16)Since S ,j , S ,j , M ,j ( x, z ), M ,j ( x, z ), M ,j ( x, z ) are even functions of x j , then integration against w j,x just vanish. This gives that Z S n E Z n d x = N X j =1 Z S n ε S ,j Z n d x + Z S n B ( v ) Z n d x + Z S n ε h β φ ,zz Z n d x + Z S n ε h β φ ,zz Z n d x + Z S n M ( x, z ) Z n d x + Z S n M ( x, z ) Z n d x + Z S n M ( x, z ) Z n d x + Z S n M ( x, z ) Z n d x + Z S n M ( x, z ) Z n d x + Z S n (cid:0) M ( x, z ) + M ( x, z ) (cid:1) Z n d x + Z S n B ( ε φ ) Z n d x + Z S n B ( ε φ ) Z n d x ≡ II + II + II + II + II + II + II + II + II + II + II + II . (C.1)Here are the details of computations. • According to the expression of S ,j in (4.5) and the assumption of f j , it follows thatII = N X j =1 ε Z S n h(cid:16) h f ′ j + 2 h f ′ j h ′ − h f j f ′ j − h f ′ j h + h f j (cid:17) w n,xx − β − V tt (0 , εz ) f n w n i Z n d x = ε ̺ (cid:0) h f ′ n + 2 h f ′ n h ′ − h f n f ′ n − h f ′ n h + h f n (cid:1) + O ( ε ) N X j =1 (cid:0) f j + f ′ j + f ′ j f j (cid:1) , (C.2)where ̺ = 2 Z R w xx Z d x. (C.3) • Recall the expression of B ( v ) in (4.6), thenII = Z S n n αβ (cid:2) ˆ B ( v ) + a ( εs, εz ) ε s v (cid:3) + h p ( w n ) p − ( v − w n ) − h X j = n w j p + h p ( p − w n ) p − ( v − w n ) + max j = n O ( e − | βf j − x | ) o Z n d x = ̺ h h e − β ( f n − f n − ) − e − β ( f n +1 − f n ) i + ε ˆ τ max j = n O ( e − β | f j − f n | ) + ε N X j =1 (cid:0) b εj f ′′ j + b εj (cid:1) , (C.4)where ˆ τ is a small positive constant with < ˆ τ < ̺ is positive constant given by ̺ = p C p Z ∞ w p − ( x ) Z ( x )[ e − x − e x ] d x. (C.5) • According to the definition of ε φ ( x, z ) as in (4.14), we obtainII = N X j =1 h ε β − a ( εz ) f ′′ j Z S n ω ,j Z n d x + ε β − a ( εz ) f ′′ j Z S n ω ,j Z n d x + ε b εj i = ε β − f ′′ n Z S n (cid:2) a ( εz ) ω ,n + a ( εz ) ω ,n (cid:3) Z n d x + ε N X j =1 (cid:0) b εj f ′′ j + b εj (cid:1) = ε ρ ( εz ) + ε N X j =1 (cid:0) b εj f ′′ j + b εj (cid:1) , (C.6)where ρ ( εz ) = β − f ′′ n Z R h a ( εz ) ω ,n + a ( εz ) ω ,n i Z n d x. (C.7) • It is easy to prove that II = ε β − h Z S n φ ,zz Z n d x = O ( ε ) . (C.8) • According to the expression of M ( x, z ) and M ( x, z ) as in (4.40) and (4.43), hence, it is easy to obtainthat II = N X j =1 ε β h ξ ′′ ( εz ) Z S n (cid:2) A ′ (cid:0) ˜ d ( εz ) (cid:1) Z j + φ ,j (cid:3) Z n d x, = ε h β ξ ′′ ( εz ) A ′ (cid:0) ˜ d ( εz ) (cid:1) + O ( ε ) . (C.9) • Similarly, there holds II = Z S n M ( x, z ) Z n d x = O ( ε ) . (C.10) • The estimate of II can be proved by the same way, i.e.,II = ε f n Z S n h h (cid:0) a ω ,n,xx + a ω ,n,xx (cid:1) − V t (0 , εz ) β (cid:0) a ω ,n + a ω ,n (cid:1) i Z n d x + ε N X j =1 (cid:0) b εj f ′′ j + b εj (cid:1) = ε ρ ( εz ) + ε N X j =1 (cid:0) b εj f ′′ j + b εj (cid:1) , (C.11)where ρ ( εz ) = f n Z R h h (cid:0) a ω ,xx + a ω ,xx (cid:1) − V t (0 , εz ) β (cid:0) a ω + a ω (cid:1)i Z d x. (C.12) • On the other hand, from the definition of M ( x, z ) in (4.48), we can compute thatII = N X j =1 ε β f j ξ ( εz ) Z S n n h β (cid:2) A (cid:0) ˜ d ( εz ) (cid:1) Z j,xx + φ ,j,xx (cid:3) − V t (0 , εz ) β (cid:2) A (cid:0) ˜ d ( εz ) (cid:1) Z j + φ ,j (cid:3)o Z n d x + O ( ε )= ε ρ ( εz ) + O ( ε ) , (C.13)where ρ ( εz ) = 1 β f n ξ ( εz ) Z R n h β (cid:2) A (cid:0) ˜ d ( εz ) (cid:1) Z xx + φ ,xx (cid:3) − V t (0 , εz ) β (cid:2) A (cid:0) ˜ d ( εz ) (cid:1) Z + φ (cid:3)o Z d x. (C.14) • We need only to compute those parts of M ( x, z ) in (4.50), which are even in x n . It is easy to check thatII = N X j =1 ε β e j Z S n h h Z j,x + h x j Z j,xx − V t (0 , εz ) β x j Z j i Z n d x + O ( ε ) = O ( ε ) . (C.15)Additionally, we also need to consider some higher order terms in II . The ones involving first derivative of e j are ε N X j =1 e ′ j (cid:16) h β ′ β + h β (cid:17) Z S n x j Z j,x Z n d x + ε N X j =1 e ′ j (cid:16) α ′ αβ + h β (cid:17) Z S n Z j Z n d x = ε h (cid:16) α ′ αβ + h β (cid:17) − (cid:16) h β ′ β + h β (cid:17) i e ′ n + ε N X j =1 b εj e ′ j ( εz ) ≡ ε ~ ( εz ) e ′ n + ε N X j =1 b εj e ′ j ( εz ) , (C.16)where ~ ( εz ) = (cid:16) α ′ αβ + h β (cid:17) − (cid:16) h β ′ β + h β (cid:17) . (C.17)Moreover, the ones involving second derivative of e j in II are ε f n β − ~ ( εz ) e ′′ n ( εz ) + O ( ε ) N X j =1 e ′′ j ( εz ) , (C.18)with O ( ε ) uniform in ε and ~ ( εz ) is a smooth functions of their argument. • In the terms of II and II , we need only to consider those parts which are even in x . It is good that theeven (in x ) terms in II and II are of order O ( ε ). Moreover, the terms in B ( ε φ ) are of order O ( ε ).Consequently, we deduce that II + II + II = O ( ε ) . (C.19)To compute R S \ S n E Z n d x , we notice that for ( x, z ) ∈ S δ/ε \ A n , Z ( x n ) = max j = n O ( e − p β | f j − f n | ) , and thus we can estimate Z S \ S n E Z n d x = ε max j = n O ( e − β | f j − f n | ) + O ( ε ) X i =1 II i . (C.20)D. The computations of (4.13) , (A.20) and (A.21)We first show the validity of (4.13) under the assumption of stationary condition for Γ in (A3) of Section1. In fact, the stationary assumption means that (c.f. (2.100)), i.e.,12 f = − σ V t (0 , θ ) V (0 , θ ) f . (D.21)This gives that 1 h f β − h h f β = 12 1 h f β − h h f β − σ V t (0 , θ ) β . According to the expressions of h , h as in (2.59) and (2.62), we can obtain that h β = 12 h β − σ V t (0 , θ ) β , which is exactly the formula (4.13).Recalling the expression as in (2.57)-(2.58), we then have H ( θ ) = V p − (0 , θ ) q V (cid:0) , θ (cid:1) √ f w = α β p h h . (D.22)This gives that H ′ ( θ ) = α β p h h h β ′ β + 1 √ h (cid:16) √ h w (cid:17) ′ + 2 α ′ α h i . Recalling the definitions of h , h , h as in (2.57)-(2.58), (2.59) and (2.60), the coefficient of f ′ is h h β ′ β + 2 α ′ α i + h − h = h h β ′ β + 2 α ′ α i + 1 √ h (cid:16) √ h w (cid:17) ′ = 1 α β √ h H ′ , which is exactly the formula (A.20).According to the definition of β in (3.24) and σ = p +1 p − − , then there holds H ( θ ) = V σ (cid:0) , θ (cid:1) √ f l = α β √ h l . This implies that H ′ ( θ ) = α h β ′ √ h l + β √ h ∂ θ l −
12 1( √ h ) ∂ θ h β l i + 2 αα ′ β √ h l . Using the fact (D.21) and (2.57)-(2.58), then H ′ ( θ ) − H ( θ ) = α h β ′ √ h l + β √ h ∂ θ l −
12 1( √ h ) ∂ θ h β l i + 2 αα ′ β √ h l − α V (cid:0) , θ (cid:1) √ f f − α σ V tt (0 , θ ) V (cid:0) , θ (cid:1) p f − α σ ( σ − (cid:12)(cid:12) V t (0 , θ ) (cid:12)(cid:12) V (0 , θ ) p f − α σ V t (0 , θ ) V (cid:0) , θ (cid:1) √ f f + α V (cid:0) , θ (cid:1)(cid:0) √ f (cid:1) f . = α β p h ( β ′ β h l + 1 h ∂ θ l −
12 1 h ∂ θ h l + 2 α ′ α h l − h f − σ V tt (0 , θ ) β − σ | V t (0 , θ ) | V β − σ V t (0 , θ ) β f f ) . On the other hand, according to the definitions of h , h , h , h , h as in (2.57)-(2.58), (2.60), (2.61), and(2.62), the coefficient of f is − h h − h + σ V tt (0 , θ ) β i − h (cid:16) β ′ β + α ′ α (cid:17) + σ V t (0 , θ ) β (cid:16) V t (0 , θ ) V (0 , θ ) − h ( θ ) h ( θ ) (cid:17) = − h f + 32 h h f − h h f + h h f + 1 √ h h √ h l i ′ + h h f + h h f − (cid:16) h h + h h (cid:17) f i − σ V tt (0 , θ ) β β + 2 1 h l (cid:16) β ′ β + α ′ α (cid:17) − σ V t (0 , θ ) β h V t (0 , θ ) V (0 , θ ) − f f + h h i = − h f −
12 1 h ∂ θ h l + 1 h ∂ θ l + β ′ β h l + 2 α ′ α h l − σ V tt (0 , θ ) β + σ | V t (0 , θ ) | V β − σ V t (0 , θ ) β f f . (D.23)Therefore, we obtain that h h − h + σ V tt (0 , θ ) β i + h (cid:16) β ′ β + α ′ α (cid:17) − σ V t (0 , θ ) β h V t (0 , θ ) V (0 , θ ) − h h i = H ′ − H α β √ h , which is exactly the formula (A.21).E. The computations of (2.116) and (2.118)Due to the assumptions in (1.5), we obtain a (0 ,
0) = a (0 , , a (0 ,
1) = a (0 , , ˜ a (0) = ˜ a (0) = ˜ a (1) = ˜ a (1) = 1 √ , and − a (0 , | ˜ a ′ (0) | + a (0 , | ˜ a ′ (0) | = 0 , − a (0 , | ˜ a ′ (1) | + a (0 , | ˜ a ′ (1) | = 0 . Using the above facts and expressions of w , l as in (2.56), (2.55), we can derive that w (0) = a (0 , a (0) | n (0) | + a (0 , a (0) | n (0) | = a (0 , , (E.24)and l (0) = (cid:2) a (0 , | n (0) | + a (0 , | n (0) | (cid:3) Θ tt (0 ,
0) + (cid:2) a (0 , a (0)˜ a ′ (0) | n (0) | + a (0 , a (0)˜ a ′ (0) | n (0) | (cid:3) − k (cid:2) − a (0 , | ˜ a ′ (0) | + a (0 , | ˜ a ′ (0) | (cid:3) n (0) n (0) + (cid:2) − ∂ t a (0 , a (0) + ∂ t a (0 , a (0) (cid:3) n (0) n (0)= a (cid:0) , (cid:1) k a (0 , √ (cid:20) ˜ a ′ (0) | n (0) | + ˜ a ′ (0) | n (0) | (cid:21) + 1 √ (cid:20) − ∂ t a (0 ,
0) + ∂ t a (0 , (cid:21) n (0) n (0) , (E.25)where we have used (2.16)-(2.17) and (2.11). The formulas (2.36), (2.40) and (2.41) imply that h (0) = (cid:2) ˜ a (0) | n (0) | + ˜ a (0) | n (0) | (cid:3) = ˜ a (0) = 12 , g (0) = 1 h (0) (cid:2) ˜ a (0) − ˜ a (0) (cid:3) n (0) n (0) = 0 , g (0) = 1 h (0) (cid:2) ˜ a (0) | n (0) | + ˜ a (0) | n (0) | (cid:3) = 1 , g (0) = − h (0) (cid:2) ˜ a (0)˜ a ′ (0) | n (0) | + ˜ a (0)˜ a ′ (0) | n (0) | (cid:3) + k h (0) (cid:2) − ˜ a (0) + ˜ a (0) (cid:3) n (0) n (0) − h (0) (cid:2) − q (0) n (0) + q (0) n (0) (cid:3) + g (cid:2) ˜ a (0) − ˜ a (0) (cid:3) n (0) n (0)= − √ (cid:16) ˜ a ′ (0) | n (0) | + ˜ a ′ (0) | n (0) | (cid:17) − k , where we have used (2.22), (2.16)-(2.17) and (2.11). Recalling the definition of y , p as in (2.67), (2.75), itis easy to obtain y (0) = p g (0) = 1 , p (0) = a (0 , a (0) | n (0) | + a (0 , a (0) | n (0) | = a (0 , a (0) = a (0 , √ . Then by using (E.24), we get b = 1 p h (0) y (0) p (0) = a (0 ,
0) = 2 w (0) . (E.26)On the other hand, recalling the definition of p , p , as in (2.72), (2.73), it is easy to obtain p (0) = a (0 , a (0) | n | + a (0 , a (0) | n | = a (0 , a (0) = a (0 , √ , p (0) = − (cid:2) ∂ t a (0 , − ∂ t a (0 , (cid:3) n (0) n (0) . The term y given in (2.66) will be evaluated as the following y (0) = g g − g g g = −√ (cid:16) ˜ a ′ (0) | n (0) | + ˜ a ′ (0) | n (0) | (cid:17) − k . Then by using (E.25), we have b = 1 p h (0) y (0) p (0) + 1 p h (0) y (0) p (0)= − √ (cid:2) ∂ t a (0 , − ∂ t a (0 , (cid:3) n (0) n (0) − √ (cid:16) ˜ a ′ (0) | n (0) | + ˜ a ′ (0) | n (0) | (cid:17) a (0 , − k a (0 , − l (0) . (E.27)The formulas (E.26) and (E.27) will lead to V σ (cid:0) , θ (cid:1) √ f h w (0) f ′ (0) + l (0) f (0) i = V σ (cid:0) , θ (cid:1) √ f (cid:2) b f ′ (0) − b f (0) (cid:3) . (E.28)This is the formula (2.118). Formula (2.116) can be verified in the identically same way. References [1] A. Ambrosetti, M. Badiale and S. Cingolani,
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