The spectral gap to torsion problem for some non-convex domains
aa r X i v : . [ m a t h . A P ] F e b THE SPECTRAL GAP TO TORSION PROBLEM FOR SOMENON-CONVEX DOMAINS
HUA CHEN AND PENG LUO
Abstract.
In this paper we study the following torsion problem ( − ∆ u = 1 in Ω ,u = 0 on ∂ Ω . Let Ω ⊂ R be a bounded, convex domain and u ( x ) be the solution of above problemwith its maximum y ∈ Ω. Steinerberger [14] proved that there are universal constants c , c > λ max (cid:0) D u ( y ) (cid:1) ≤ − c exp (cid:18) − c diam(Ω)inrad(Ω) (cid:19) . And in [14] he proposed following open problem: “Does above result hold true on domains that are not convex but merely simply connectedor perhaps only bounded? The proof uses convexity of the domain Ω in a very essentialway and it is not clear to us whether the statement remains valid in other settings.” Here by some new idea involving the computations on Green’s function, we compute thespectral gap λ max D u ( y ) for some non-convex smooth bounded domains, which gives anegative answer to above open problem. Keywords:
Spectral gap, torsion problem, Green’s function · · Introduction and main results
In this paper, we consider the following torsion problem ( − ∆ u = 1 in Ω ,u = 0 on ∂ Ω . (1.1)Problem (1.1) is a classical topic in PDEs, with references dating back to St. Venant(1856).From then, many results are devoted to analysis the qualitative properties of the positivesolutions. A very interest problem is the location and the number of the critical points ofabove positive solutions. This is related with the level sets of the positive solutions. For amore general case, the following nonlinear problem ( − ∆ u = f ( u ) in Ω ,u = 0 on ∂ Ω . Date : February 10, 2021. has also been considered widely. For example, one can refer to [3, 7, 8, 10, 11, 13] and therelated references.A well-known and seminal result is the fundamental theorem in Gidas, Ni and Nirenberg[5] by moving plane. Gidas-Ni-Nirenberg’s Theorem shows that the uniqueness of thecritical points is related to the shape of the superlevel sets. Although there are someconjectures on the uniqueness of the critical point in more general convex domains, thisseems to be a very difficult problem. And another important result is [4], which holds for awide class of nonlinearities f without the symmetry assumption on Ω and for semi-stablesolutions. For further results, we can refer to [2, 9, 12] and references therein.When f ( u ) ≡
1, the torsion function seems to be the classical object in the study of levelsets of elliptic equations. First from [13], we know that the level sets are convex and thereis a unique global maximum of the torsion function on planar convex domains. And thenthe eccentricity of the level sets close to the (unique) maximum point y is determined bythe eigenvalues of the Hessian D u ( y ). Let λ and λ are two eigenvalues of D u ( y ), thendirectly, we have λ , λ ≤ λ + λ = tr D u ( y ) = ∆ u ( y ) = − . This gives us that the level sets will be highly eccentric if one of the two eigenvalues isclose to 0. In this aspect, Steinerberger [14] gave a beautiful description, which shows thatthe level sets aren’t highly eccentric for any convex domain Ω and can be stated as follows.
Theorem A.
Let Ω ⊂ R be a bounded, convex domain and u ( x ) be the solution of problem (1.1) with its maximum y ∈ Ω . There are universal constants c , c > such that λ max (cid:0) D u ( y ) (cid:1) ≤ − c exp (cid:18) − c diam(Ω)inrad(Ω) (cid:19) . (1.2)Also Steinerberger [14] gave some details to show that the above result has the sharpscaling. Above Theorem A was proved by Fourier analysis in [14] and highly depends onthe convexity of the domain Ω. Next at page 1616 of [14], Steinerberger proposed thefollowing open problem : Problem A . Convexity of the Domain.
Does Theorem A also hold true on domains thatare not convex but merely simply connected or perhaps only bounded? The proof usesconvexity of the domain Ω in a very essential way and it is not clear to us whether thestatement remains valid in other settings. In this paper, we devote to give some answer to above Problem A. To study ProblemA, we will compute the Hessian of the torsion function at the maximum point on a simplenon-convex domain. For example, we suppose that Ω ε = Ω \ B ( x , ε ) with x ∈ Ω and B ( x , ε ) denote the ball centered at x and radius ε , u ε is the solution of ( − ∆ u = 1 in Ω ε ,u = 0 on ∂ Ω ε . (1.3)And then we have following result. RITICAL POINTS OF POSITIVE SOLUTIONS 3
Theorem 1.1.
Let Ω ⊂ R be a bounded and convex domain, y is the maximum pointof u ( x ) as in Theorem A. Suppose that u ε ( x ) is the solution of problem (1.3) with itsmaximum x ε ∈ Ω ε . Let λ and λ be two eigenvalues of D u ( x ) at y , then lim ε → λ max (cid:0) D u ε ( x ε ) (cid:1) = ( max (cid:8) λ , λ (cid:9) if x = y , max (cid:8) λ , λ , −| λ − λ | (cid:9) if x = y . Remark 1.2.
Taking Ω ⊂ R a bounded and convex domain, Ω ε = Ω \ B ( x , ε ) with x = y and ε small, if we suppose that (1.2) is true for Ω ε , then there exist two positive constants c and c , which is independent with ε , such that λ max (cid:0) D u ε ( x ε ) (cid:1) ≤ − c exp (cid:18) − c diam (Ω ε ) inrad (Ω ε ) (cid:19) ≤ − c exp (cid:18) − c diam (Ω) inrad (Ω) (cid:19) . (1.4) On the other hand, moreover if we suppose λ = λ (for example Ω = B (0 , ), thenTheorem 1.1 gives us lim ε → λ max (cid:0) D u ε ( x ε ) (cid:1) = 0 , which is a contradiction with (1.4) . Hence we deduce that (1.2) doesn’t hold for above non-convex domain Ω ε , which gives a negative answer to above Problem A in [14]. And thenin this case, we find that the level sets of the torsion function are highly eccentric. Remark 1.3.
Our crucial ideas are as follows. To compute the eigenvalues of the Hessianof u ε ( x ) at the maximum point on Ω ε , a first step is to find the location of the maximumpoint x ε . And then we need to analyze the asymptotic behavior of D u ε ( x ε ) . It is wellknown that u ( x ) and u ε ( x ) are represented by corresponding Green’s function. Hence wewrite u ε ( x ) by the basic Green’s function and then analyze the properties of Green’s functionon Ω ε . To be specific, we will establish the basic estimate near ∂B ( x , ε ) : u ε ( x ) = u ( x ) + log | x − x || log ε | (cid:16) u (cid:0) x (cid:1) + o (1) (cid:17) + o (cid:0) (cid:1) . Furthermore, another crucial result is to derive that u ε ( x ) and u ( x ) + log | x − x || log ε | u ( x ) areclose in the C -topology in B ( x , d ) \ B ( x , ε ) for some small fixed d > , which can befound in Proposition 3.6 below. Remark 1.4.
Now we would like to point out that Ω ε in Theorem 1.1 can be replaced by Ω ′ ε = Ω \ A ε with A ε = ε (cid:0) A − x (cid:1) + x and x ∈ A ∩ Ω , where A is a convex domain in R and A − x = { x, x + x ∈ A } . Since this is not essential,we omit the details. Remark 1.5.
We point out one possible application in the study of Brownian motion,which is also stated in [14]: we recall that the torsion function u ( x ) also describes theexpected lifetime of Brownian motion ω x ( t ) started in x until it first touches the boundary.If one moves away from the point in which lifetime is maximized, then the expected lifetimein a neighborhood is determined by the eccentricity of the level set. H. CHEN AND P. LUO
The paper is organized as follows. In Section 2, we recall some properties of the Green’sfunction and split our solution u ε in different parts which will be estimated in the nextsection. In Section 3, we compute the terms u ε , ∇ u ε and ∇ u ε . Section 4 is devoted tothe proof of Theorem 1.1.2. Properties of the Green’s function and splitting of the solution u ε First we recall that, for ( x, y ) ∈ Ω × Ω, x = y , the Green’s function G ( x, y ) verifies ( − ∆ x G ( x, y ) = δ ( y ) in Ω ,G ( x, y ) = 0 on ∂ Ω , in the sense of distribution. Next we recall the classical representation formula, G ( x, y ) = − π log (cid:12)(cid:12) x − y (cid:12)(cid:12) − H ( x, y ) , (2.1)where H ( x, y ) is the regular part of the Green’s function . Since in the paper we need toconsider the Green’s function in different domains, we would like to denote by G U ( x, y ) asthe Green’s function on U . And we have following facts on the harmonic function whichcan be found in [6]. Lemma 2.1.
Let u ( x ) be a harmonic function in U ⊂⊂ R , then (cid:12)(cid:12) ∇ u ( x ) (cid:12)(cid:12) ≤ r sup ∂B ( x,r ) | u | , for B ( x, r ) ⊂⊂ U. (2.2) Lemma 2.2 (Green’s representation formula) . If u ∈ C ( U ) , then it holds u ( x ) = − Z ∂U u ( y ) ∂G U ( x, y ) ∂ν y dσ ( y ) − Z U ∆ u ( y ) G U ( x, y ) dy, for x ∈ U, (2.3) where ν y is the outer normal vector on ∂U . Let us denote by G ( w, s ) the Green’s function of R \ B (0 ,
1) given by (see [1]) G ( w, s ) = − π (cid:18) log (cid:12)(cid:12) w − s (cid:12)(cid:12) − log (cid:12)(cid:12) | w | s − w | w | (cid:12)(cid:12)(cid:19) . By a straightforward computation, we have ∂G ( w, s ) ∂ν s = 1 − | w | π | w − s | , for | w | > , | s | = 1 and ν s = − s. (2.4) Remark 2.3.
Let us point out that the
Green’s function G ( w, s ) of R \ B (0 , and the Poisson kernel of B (0 , has the same formula (see [1]). This will be used to compute someintegral in R \ B (0 , . Next lemma will be basic and useful in the following computations in next section.
RITICAL POINTS OF POSITIVE SOLUTIONS 5
Lemma 2.4.
Let v ε ( x ) be the function which verifies ∆ v ε ( x ) = 0 in Ω \ B ( x , ε ) ,v ε ( x ) = 0 on ∂ Ω ,v ε ( x ) = 1 on ∂B ( x , ε ) . Then we have that v ε ( x ) = − π log ε (cid:16) − πH ( x , x )log ε (cid:17) G ( x, x ) + O (cid:18) | log ε | (cid:19) . Proof.
First we define w ε ( x ) = 1 H ( x , x ) (cid:20) log ε π v ε ( x ) + G ( x, x ) (cid:21) . Then it holds ∆ w ε ( x ) = 0 in Ω \ B ( x , ε ) ,w ε ( x ) = 0 on ∂ Ω ,w ε ( x ) = H ( x ,x ) (cid:16) log ε π + G ( x, x ) (cid:17) = − O ( ε ) on ∂B ( x , ε ) . Hence repeating above procedure, we can find ∆ (cid:2) log ε π w ε ( x ) − G ( x, x ) (cid:3) = 0 in Ω \ B ( x , ε ) , log ε π w ε ( x ) − G ( x, x ) = 0 on ∂ Ω , log ε π w ε ( x ) − G ( x, x ) = H ( x, x ) + O ( ε | log ε | ) on ∂B ( x , ε ) . (2.5)Then by the maximum principle and (2.5), we get thatlog ε π w ε ( x ) − G ( x, x ) = O (1) in Ω \ B ( x , ε ) , which gives w ε ( x ) = 2 π log ε G ( x, x ) + O (cid:18) | log ε | (cid:19) . Hence coming back to v ε ( x ), we find v ε ( x ) = 2 π log ε (cid:16) H ( x , x ) w ε ( x ) − G ( x, x ) (cid:17) = − π log ε (cid:16) − πH ( x , x )log ε (cid:17) G ( x, x ) + O (cid:18) | log ε | (cid:19) , which gives the claim. (cid:3) H. CHEN AND P. LUO
Let u and u ε be solutions of (1.1) and (1.3) respectively, then we can write down theequation satisfied by u ε − u as follows − ∆ (cid:0) u ε − u (cid:1) = 0 in Ω ε ,u ε − u = 0 on ∂ Ω ,u ε − u = − u on ∂B ( x , ε ) . (2.6)Now by Green’s representation formula (2.3), we get u ε ( x ) = u ( x ) + Z ∂B ( x ,ε ) ∂G ε ( x, z ) ∂ν z u ( z ) dσ ( z ) , (2.7)where ν z = − z − x | z − x | is the outer normal vector of ∂ (cid:0) R \ B ( x , ε ) (cid:1) and G ε ( x, z ) is the Green’sfunction of − ∆ in Ω ε with zero Dirichlet boundary condition. Now we set x = x + εw, z = x + εs and F ε ( w, s ) = G ε ( x + εw, x + εs ) , then (2.7) becomes u ε ( x ) = u ( x ) + K ε ( w ) + L ε ( w ) , (2.8)where K ε ( w ) := Z ∂B (0 , ∂G ( w, s ) ∂ν s u ( x + εs ) dσ ( s ) , and L ε ( w ) := Z ∂B (0 , (cid:18) ∂F ε ( w, s ) ∂ν s − ∂G ( w, s ) ∂ν s (cid:19) u ( x + εs ) dσ ( s ) , with ν s = − s | s | the outer normal vector of ∂ (cid:0) R \ B (0 , (cid:1) .3. Asymptotic analysis on u ε To compute λ max (cid:0) D u ε ( x ) (cid:1) at the maximum point of u ε , the first thing is to find thelocation of the maximum point of u ε . Here we divide Ω ε into the following two cases: (1): x is far away from x , namely | x − x | ≥ C > (2): x is near x , namely | x − x | = o (1).And we will find that the behavior of u ε ( x ) near ∂B ( x , ε ) is crucial and a key point is tounderstand the limit of G ε ( x, y ) according to the location of x . Lemma 3.1.
Let u and u ε be solutions of (1.1) and (1.3) respectively. Then for any fixed r > , it holds u ε ( x ) → u ( x ) uniformly in C (cid:0) Ω \ B ( x , r ) (cid:1) . RITICAL POINTS OF POSITIVE SOLUTIONS 7
Proof.
First for any x ∈ Ω \ B ( x , r ), we know u ε ( x ) = u ( x ) + Z ∂B ( x ,ε ) ∂G ε ( x, z ) ∂ν z u ( z ) dσ ( z )= u ( x ) + Z ∂B ( x ,ε ) ∂G ε ( x, z ) ∂ν z (cid:16) u ( x ) + O (cid:0) ε (cid:1)(cid:17) dσ ( z )= u ( x ) + u ( x ) Z ∂B ( x ,ε ) ∂G ε ( x, z ) ∂ν z dσ ( z ) + O (cid:0) ε (cid:1) Z ∂B ( x ,ε ) (cid:12)(cid:12)(cid:12) ∂G ε ( x, z ) ∂ν z (cid:12)(cid:12)(cid:12) dσ ( z ) . (3.1)Now let x = x + εw and z = x + εs , then using (2.4), we have ∂G ε ( x, z ) ∂ν z = 1 ε (cid:18) ∂G ( w, s ) ∂ν s + (cid:16) ∂F ε ( w, s ) ∂ν s − ∂G ( w, s ) ∂ν s (cid:17)(cid:19) = O (cid:18) ε (cid:16) (cid:12)(cid:12) ∂F ε ( w, s ) ∂ν s − ∂G ( w, s ) ∂ν s (cid:12)(cid:12)(cid:17)(cid:19) . (3.2)On the other hand, we can verify that ∆ w (cid:16) ∂F ε ( w,s ) ∂ν s − ∂G ( w,s ) ∂ν s (cid:17) = 0 in Ω − x ε \ B (0 , , (cid:16) ∂F ε ( w,s ) ∂ν s − ∂G ( w,s ) ∂ν s (cid:17) = 0 on ∂B (0 , , (cid:16) ∂F ε ( w,s ) ∂ν s − ∂G ( w,s ) ∂ν s (cid:17) = | w | − π | w − s | = O (cid:0) (cid:1) on ∂ Ω − x ε . (3.3)By the maximum principle and (3.3), we get that ∂F ε ( w, s ) ∂ν s − ∂G ( w, s ) ∂ν s = O (1) in Ω − x ε \ B (0 , . (3.4)Hence from (3.2) and (3.4), we find ∂G ε ( x, z ) ∂ν z = O (cid:18) ε (cid:19) . (3.5)Also defining v ( x ) := Z ∂B ( x ,ε ) ∂G ε ( x, z ) ∂ν z dσ ( z ), then it holds ∆ x v ( x ) = 0 in Ω \ B ( x , ε ) ,v ( x ) = 0 on ∂ Ω ,v ( x ) = − ∂B ( x , ε ) . Then using Lemma 2.4, we have v ( x ) = 2 π log ε (cid:16) − πH ( x , x )log ε (cid:17) G ( x, x ) + O (cid:18) | log ε | (cid:19) . (3.6) H. CHEN AND P. LUO
Hence from (3.1), (3.5) and (3.6), we find u ε ( x ) = u ( x ) + u ( x ) (cid:18) π log ε (cid:16) − πH ( x , x )log ε (cid:17) G ( x, x ) + O (cid:16) | log ε | (cid:17)(cid:19) + O (cid:0) ε (cid:1) = u ( x ) + O (cid:18) | log ε | (cid:19) uniformly in C (cid:0) Ω \ B ( x , r ) (cid:1) . On the other hand, for any fixed x ∈ Ω \ B ( x , r ), by (2.1), we can verify that G ( x, z ) , (cid:12)(cid:12) ∂G ( x, z ) ∂ν z (cid:12)(cid:12) , |∇ x G ( x, z ) | and (cid:12)(cid:12) ∇ x ∂G ( x, z ) ∂ν z (cid:12)(cid:12) are bounded for z ∈ ∂B ( x , ε ) . (3.7)And by (2.6), it follows − ∆ x (cid:16) ∂G ε ( x, z ) ∂ν z − ∂G ( x, z ) ∂ν z (cid:17) = 0 in Ω ε . Since B (cid:0) x, r (cid:1) ⊂⊂ Ω ε , using Lemma 2.1, (3.5) and (3.7), we get, for x ∈ Ω \ B ( x , r ) and z ∈ ∂B ( x , ε ), (cid:12)(cid:12)(cid:12)(cid:12) ∇ x (cid:16) ∂G ε ( x, z ) ∂ν z − ∂G ( x, z ) ∂ν z (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18)(cid:12)(cid:12)(cid:12) ∂G ε ( x, z ) ∂ν z − ∂G ( x, z ) ∂ν z (cid:12)(cid:12)(cid:12)(cid:19) = O (cid:18) ε (cid:19) , (3.8)and (cid:12)(cid:12)(cid:12)(cid:12) ∇ x (cid:16) ∂G ε ( x, z ) ∂ν z − ∂G ( x, z ) ∂ν z (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18)(cid:12)(cid:12)(cid:12) ∂G ε ( x, z ) ∂ν z − ∂G ( x, z ) ∂ν z (cid:12)(cid:12)(cid:12)(cid:19) = O (cid:18) ε (cid:19) . (3.9)Then (3.7), (3.8) and (3.9) give us that (cid:12)(cid:12)(cid:12)(cid:12) ∇ x ∂G ε ( x, z ) ∂ν z (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) ε (cid:19) and (cid:12)(cid:12)(cid:12)(cid:12) ∇ x ∂G ε ( x, z ) ∂ν z (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) ε (cid:19) . Hence from above estimates, it follows u ε ( x ) = u ( x ) + O (cid:18) | log ε | (cid:19) uniformly in C (cid:0) Ω \ B ( x , r ) (cid:1) . (cid:3) In the rest of this section, we devote to analyze the asymptotic behavior on u ε , ∇ u ε and ∇ u ε near ∂B ( x , ε ). And using (2.8), we need to compute the terms K ε , ∇ w K ε , ∇ w K ε , L ε , ∇ w L ε and ∇ w L ε . Lemma 3.2.
Let w = x − x ε and if | x − x | → , then it holds K ε ( w ) = − u (cid:16) x + εw | w | (cid:17) + ε (cid:16) − | w | (cid:17) , (3.10) ∂K ε ( w ) ∂w i = O (cid:16) ε | w | (cid:17) and ∂ K ε ( w ) ∂w i ∂w j = O (cid:16) ε | w | (cid:17) . (3.11) RITICAL POINTS OF POSITIVE SOLUTIONS 9
Proof.
First taking τ = w | w | = ε ( x − x ) | x − x | and using (2.4), we get Z ∂B (0 , ∂G ( w, s ) ∂ν s u ( x + εs ) dσ ( s ) = − π Z ∂B (0 , − | τ | | τ − s | u ( x + εs ) dσ ( s ) . Lemma 2.2 and (2.4) give us that for any φ ∈ C (cid:0) B (0 , (cid:1) , it holds φ ( s ) = 12 π Z ∂B (0 , − | s | | s − y | φ ( y ) dσ ( y ) − Z B (0 , ∆ φ ( y ) G ( s, y ) dy. (3.12)Hence for | τ | < φ ( τ ) = u ( x + ετ ) in (3.12) we find u ( x + ετ ) = 12 π Z ∂B (0 , − | τ | | τ − s | u ( x + εs ) dσ ( s ) + ε (cid:0) − | τ | (cid:1) . From the above computations we get K ε ( w ) = Z ∂B (0 , ∂G ( w, s ) ∂ν s u ( x + εs ) dσ ( s )= − u (cid:16) x + εw | w | (cid:17) + ε (cid:16) − | w | (cid:17) . (3.13)And then by differentiating (3.13) with respect to w i , we have ∂K ε ( w ) ∂w i = − ε | w | ∂u ( x + εw | w | ) ∂x i − w i | w | X j =1 ∂u ( x + εw | w | ) ∂x j w j ! + ε w i | w | . (3.14)Next differentiating (3.14) with respect to w i , we find ∂ K ε ( w ) ∂w i ∂w j = O (cid:18) ε | w | (cid:19) . (3.15)Hence (3.10) and (3.11) follow by (3.13), (3.14) and (3.15). (cid:3) Lemma 3.3.
Let w = x − x ε and if | x − x | → , then it holds L ε ( w ) = log | w || log ε | (cid:16) u ( x ) + o (1) (cid:17) . (3.16) Proof.
First we define M ε, ( w, s ) = X i =1 (cid:18) ∂G ( w, s ) ∂s i − ∂F ε ( w, s ) ∂s i (cid:19) s i − π M ε, ( w, s ) , with ∆ w M ε, ( w, s ) = 0 in Ω − x ε \ B (0 , ,M ε, ( w, s ) = 0 on ∂B (0 , ,M ε, ( w, s ) = 1 on ∂ Ω − x ε . Then L ε ( w ) can be written as L ε ( w ) = 12 π Z ∂B (0 , M ε, ( w, s ) u ( x + εs ) dσ ( s ) | {z } := L ε, ( w ) + Z ∂B (0 , M ε, ( w, s ) u ( x + εs ) dσ ( s ) | {z } := L ε, ( w ) . (3.17)Also for any w ∈ ∂ Ω − x ε and s ∈ ∂B (0 , X i =1 (cid:18) ∂G ( w, s ) ∂s i − ∂F ε ( w, s ) ∂s i (cid:19) s i = X i =1 ∂G ( w, s ) ∂s i s i = 12 π | w | − | w − s | . Hence we can verify ∆ w M ε, ( w, s ) = 0 in Ω − x ε \ B (0 , ,M ε, ( w, s ) = 0 on ∂B (0 , ,M ε, ( w, s ) = π (cid:16) | w | − | w − s | − (cid:17) on ∂ Ω − x ε . (3.18)Since for any w ∈ ∂ Ω − x ε and s ∈ ∂B (0 , | w | − | w − s | − O (cid:16) | w | (cid:17) = O (cid:0) ε (cid:1) . (3.19)Then by the maximum principle, (3.18) and (3.19), we find (cid:12)(cid:12) M ε, ( w, s ) (cid:12)(cid:12) = O ( ε ) for w ∈ Ω − x ε \ B (0 ,
1) and s ∈ ∂B (0 , . (3.20)Hence it follows L ε, ( w ) = O (cid:0) ε (cid:1) for w ∈ Ω − x ε \ B (0 , . (3.21)Next we estimate M ε, ( w, s ). To do this let us introduce the function ψ ε ( x, s ) as follows ψ ε ( x, s ) := 1 − M ε, (cid:18) x − x ε , s (cid:19) for x ∈ Ω \ B ( x , ε ) and s ∈ ∂B (0 , . Then it follows ∆ x ψ ε ( x, s ) = 0 in Ω \ B ( x , ε ) ,ψ ε ( x, s ) = 0 on ∂ Ω ,ψ ε ( x, s ) = 1 on ∂B ( x , ε ) . Hence using Lemma 2.4 we have that ψ ε ( x, s ) = − π log ε (cid:16) − πH ( x , x )log ε (cid:17) G ( x, x ) + O (cid:18) | log ε | (cid:19) . RITICAL POINTS OF POSITIVE SOLUTIONS 11
Coming back to M ε, ( w, s ), we get M ε, ( w, s ) =1 + 2 π log ε (cid:16) − πH ( x , x )log ε (cid:17) G ( εw + x , x ) + O (cid:18) | log ε | (cid:19) = log | w || log ε | (cid:16) o (1) (cid:17) + o (cid:18) | log ε | (cid:19) . In last equality we used that ε | w | = | x − x | →
0. And then L ε, ( w ) = log | w || log ε | (cid:16) u ( x ) + o (1) (cid:17) + o (cid:18) | log ε | (cid:19) . (3.22)Then (3.16) follows by (3.17), (3.21) and (3.22). (cid:3) Then we have following estimate on u ε near ∂B ( x , ε ). Proposition 3.4.
Let w = x − x ε and | x − x | → , then it holds u ε ( x ) = u ( x ) + log | x − x || log ε | (cid:16) u (cid:0) x (cid:1) + o (1) (cid:17) + o (1) . (3.23) Proof.
From (2.8), (3.10) and (3.16), we have u ε ( x ) = u ( x ) − u (cid:16) x + εw | w | (cid:17) + ε (cid:0) − | w | (cid:1) + log | w || log ε | (cid:16) u (cid:0) x (cid:1) + o (1) (cid:17) = u ( x ) + log | x − x || log ε | (cid:16) u (cid:0) x (cid:1) + o (1) (cid:17) + o (1) . (cid:3) Now we continue to compute ∇ w L ε and ∇ w L ε . Lemma 3.5.
Let w = x − x ε and | x − x | → , we have following results: (1) For any fixed C > , if | w | ≥ C , then it holds ∂L ε ( w ) ∂w i = u ( x ) w i | log ε | · | w | + o (cid:18) | w | · | log ε | (cid:19) , (3.24) and ∂ L ε ( w ) ∂w i ∂w j = u ( x ) | log ε | · | w | (cid:16) δ ij − w i w j | w | (cid:17) + o (cid:18) | w | · | log ε | (cid:19) . (3.25)(2) If lim ε → | w | = 1 , then it holds D ∇ w L ε ( w ) , w E ≥ u ( x )2 | log ε | . (3.26) Proof of (3.24) and (3.25) . First for w ∈ Ω − x ε \ B (0 ,
1) and s ∈ ∂B (0 , B (cid:0) w, | w | − (cid:1) ⊂⊂ Ω − x ε \ B (0 , . Then using (2.2), (3.18) and (3.20), we have (cid:12)(cid:12) ∇ w M ε, ( w, s ) (cid:12)(cid:12) = O (cid:16) ε | w | − (cid:17) and (cid:12)(cid:12) ∇ w M ε, ( w, s ) (cid:12)(cid:12) = O (cid:16) ε ( | w | − (cid:17) . (3.27)Hence if ε | w | → | w | ≥ C for any fixed C >
1, then (3.27) implies (cid:12)(cid:12) ∇ w M ε, ( w, s ) (cid:12)(cid:12) = O (cid:16) ε | w | (cid:17) and (cid:12)(cid:12) ∇ w M ε, ( w, s ) (cid:12)(cid:12) = O (cid:16) ε | w | (cid:17) . And then it follows ∂L ε, ( w ) ∂w i = O (cid:18) ε | w | (cid:19) and ∂ L ε, ( w ) ∂w i ∂w j = O (cid:18) ε | w | (cid:19) . (3.28)Also M ε, ( w, s ) − log | w || log ε | is a harmonic function with respect to w in Ω − x ε \ B (0 , ε | w | → | w | ≥ C for any fixed C >
1, using (2.2) we have (cid:12)(cid:12)(cid:12)(cid:12) ∇ w (cid:18) M ε, ( w, s ) − log | w || log ε | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = o (cid:18) | w | · | log ε | (cid:19) , (3.29)and (cid:12)(cid:12)(cid:12)(cid:12) ∇ w (cid:18) M ε, ( w, s ) − log | w || log ε | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = o (cid:18) | w | · | log ε | (cid:19) . (3.30)Hence from (3.29) and (3.30), we have ∂L ε, ( w ) ∂w i = w i | log ε | · | w | (cid:0) u ( x ) + o (1) (cid:1) , (3.31)and ∂ L ε, ( w ) ∂w i ∂w j = 1 | log ε | · | w | (cid:0) u ( x ) + o (1) (cid:1)(cid:16) δ ij − w i w j | w | (cid:17) . (3.32)And then (3.24) and (3.25) follows by (3.17), (3.28), (3.31) and (3.32). (cid:3) Proof of (3.26) . To consider the case lim ε → | w | = 1, we define a new function η ε, ( w, s ) = X i =1 (cid:18) ∂G ( w, s ) ∂s i − ∂F ε ( w, s ) ∂s i (cid:19) s i + log | w | π log( rε ) . Then we can write ∂L ε ( w ) ∂w i = Z ∂B (0 , (cid:18) ∂η ε, ( w, s ) ∂w i − w i π | w | log( rε ) (cid:19) u ( x + εs ) dσ ( s ) . Now we can verify ∆ w η ε, ( w, s ) = 0 in Ω − x ε \ B (0 , ,η ε, ( w, s ) = 0 on ∂B (0 , ,η ε, ( w, s ) = π (cid:16) | w | − | w − z | + log | w | log( rε ) (cid:17) on ∂ Ω − x ε . (3.33) RITICAL POINTS OF POSITIVE SOLUTIONS 13
Setting z = x + εw , for any w ∈ ∂ Ω − x ε , we get that | w | − | w − s | + log | w | log( rε ) = | z − x | − ε | z − x − εs | + log( r | z − x | ) − log( rε )log( rε )= log( r | z − x | )log( rε ) + O ( ε ) . Then taking some appropriate r > r > r | z − x | < z ∈ ∂ Ω), we have | w | − | w − s | + log | w | log( rε ) ≥ . (3.34)Hence by the maximum principle, (3.33) and (3.34), it holds η ε, ( w, s ) ≥ , for any w ∈ Ω − x ε \ B (0 , . Then Hopf’s lemma gives us that ∂η ε, ( τ, s ) ∂ν τ < , for any τ ∈ ∂B (0 ,
1) with ν τ = − τ | τ | . If lim ε → | w | = 1, by the sign-preserving property, for small ε , we find X i =1 ∂η ε, ( w, s ) ∂w i w i = −| w | · ∂η ε, ( w, s ) ∂ν w ≥ , with ν w = − w | w | . (3.35)Hence using (3.35), we can compute that D ∇ w L ε ( w ) , w E = Z ∂B (0 , X i =1 ∂η ε, ( w, s ) ∂w i w i − π log( rε ) ! u ( x + εs ) dσ ( s ) ≥ | log ε | (cid:0) u ( x ) + o (1) (cid:1) ≥ u ( x )2 | log ε | , which completes the proofs of (3.26). (cid:3) From above computations, the precise asymptotic behavior of ∇ u ε and ∇ u ε near ∂B ( x , ε ) can be stated as follows. Proposition 3.6.
Let w = x − x ε and | x − x | → , we have following results: (1). For any fixed C > , if | w | ≥ C , then it holds ∂u ε ( x ) ∂x i = ∂u ( x ) ∂x i + u ( x )( x i − x ,i ) | log ε | · | x − x | + o (cid:16) | log ε | · | x − x | (cid:17) , (3.36) and ∂ u ε ( x ) ∂x i ∂x j = ∂ u ( x ) ∂x i ∂x j + u ( x ) | log ε | · | x − x | (cid:16) δ ij − ( x i − x ,i )( x j − x ,j ) | x − x | (cid:17) + o (cid:16) | log ε | · | x − x | (cid:17) . (3.37)(2). If lim ε → | w | = 1 , then it holds (cid:12)(cid:12) ∇ u ε ( x ) (cid:12)(cid:12) ≥ u ( x )8 ε | log ε | . (3.38) Proof. (1). First by (2.8), we have ∂u ε ( x ) ∂x i = ∂u ( x ) ∂x i + 1 ε ∂K ε ( w ) ∂w i + 1 ε ∂L ε ( w ) ∂w i . (3.39)Then from (3.11), (3.24) and (3.39), we find ∂u ε ( x ) ∂x i = ∂u ( x ) ∂x i + O (cid:16) | w | (cid:17) + u ( x ) w i ε · | log ε | · | w | + o (cid:18) ε · | w | · | log ε | (cid:19) = ∂u ( x ) ∂x i + u ( x )( x i − x ,i ) | log ε | · | x − x | + o (cid:16) | log ε | · | x − x | (cid:17) + O (cid:16) ε | x − x | (cid:17) = ∂u ( x ) ∂x i + u ( x )( x i − x ,i ) | log ε | · | x − x | + o (cid:16) | log ε | · | x − x | (cid:17) . Similarly, by (2.8), we know ∂ u ε ( x ) ∂x i ∂x j = ∂ u ( x ) ∂x i ∂x j + 1 ε ∂ K ε ( w ) ∂w i ∂w j + 1 ε ∂ L ε ( w ) ∂w i ∂w j , which, together with (3.11) and (3.25), implies (3.37).(2). If lim ε → | w | = 1, from (2.8), (3.11) and (3.26), we have (cid:12)(cid:12) ∇ u ε ( x ) (cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:10) ∇ x u ε ( x ) , w (cid:11)(cid:12)(cid:12)(cid:12) ≥ ε (cid:12)(cid:12)(cid:12)(cid:10) ∇ w L ε ( w ) , w (cid:11)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:10) ∇ x u ( x ) , w (cid:11)(cid:12)(cid:12)(cid:12) − ε (cid:12)(cid:12)(cid:12)(cid:10) ∇ w K ε ( w ) , w (cid:11)(cid:12)(cid:12)(cid:12) ≥ u ( x )8 ε | log ε | . (cid:3) Proofs of Theorem 1.1
Firstly, we give the precise location of the maximum point x ε of u ε ( x ) on Ω ε . Proposition 4.1. If x = y , then the maximum point x ε of u ε ( x ) on Ω ε satisfies x ε → y as ε → , where y is the maximum point of u ( x ) . RITICAL POINTS OF POSITIVE SOLUTIONS 15
Proof.
First, for x ∈ Ω ε with | x − x | = o (1), it holds log | x − x | <
0, and then from (3.23),we find u ε ( x ) ≤ u ( x ) + o (1) , in (cid:8) x ∈ Ω ε , | x − x | = o (1) (cid:9) . (4.1)If x = y , then u ( x ) < u ( y ), here we use the uniqueness of the critical point of u ( x )(see[13]). Hence by (4.1), we have u ε ( x ) ≤ u ( x ) + u ( y )2 < u ( y ) in (cid:8) x ∈ Ω ε , | x − x | = o (1) (cid:9) , which gives us that x ε / ∈ (cid:8) x ∈ Ω ε , | x − x | = o (1) (cid:9) .Hence combining Lemma 3.1, we know that there exists a fixed small r > x ε ∈ B ( y , r ) the maximum point x ε of u ε ( x ) will belong to B ( y , r ) and x ε → y as ε → (cid:3) Proposition 4.2. If x = y , then the maximum point x ε of u ε ( x ) on Ω ε can be writtenas x ε = x + r − u ( x ) + o (1) λ p | log ε | v, where λ = max { λ , λ } , λ and λ are the eigenvalues of the matrix D u ( x ) and v anassociated eigenfunction with | v | = 1 .Proof. Since Ω is convex and x = y , then from [13], we know that u ( x ) admits exactone critical point x . This means that for any z = x , there exists r > u ( x )has no critical points in B ( z , r ). And from Lemma 3.1, we know that all critical points of u ε ( x ) belong to D ε := n x ∈ Ω ε , | x − x | = o (1) o . Next, for x ∈ D ε , if | x − x | ε < C , from (3.36) and (3.38), we have (cid:12)(cid:12) ∇ u ε ( x ) (cid:12)(cid:12) ≥ c ε | log ε | for some c > , which implies that ∇ u ε ( x ) = 0 admits no solutions if x ∈ D ε and | x − x | ε < C .Finally, we analyze the critical points of u ε ( x ) on x ∈ D ε and | x − x | ε → ∞ . From (3.36),we can deduce that0 = ∂u ε ( x ε ) ∂x i = ∂u ( x ) ∂x i + u ( x )( x i − x ,i ) | log ε | · | x − x | + o (cid:16) | log ε | · | x − x | (cid:17) = X j =1 (cid:18) ∂ u ( x ) ∂x i ∂x j + o (1) (cid:19) ( x ε,j − x ,j ) + x ε,i − x ,i | log ε | · | x ε − x | (cid:16) u ( x ) + o (1) (cid:17) . (4.2)By (4.2) we immediately get that − | x ε − x | | log ε | → λ as ε →
0. Dividing (4.2) by | x ε − x | and passing to the limit, we find that all critical points x ε of u ε can be written as x ε = x + r − u ( x ) + o (1) λ p | log ε | v, where λ = λ or λ , λ and λ are the eigenvalues of the matrix D u ( x ) and v anassociated eigenfunction with | v | = 1.Now we devote to prove that the maximum point x ε of u ε ( x ) on Ω ε can be written as x ε = x + r − u ( x ) + o (1) λ p | log ε | v, with λ = max { λ , λ } . And if λ = λ , then above result holds automatically. Now let x ε = x + s − u ( x ) + o (1) λ p | log ε | v with D u ( x ) v = λ v . (4.3)Then from (3.37), we know ∂ u ε ( x ε ) ∂x i ∂x j = ∂ u ( x ) ∂x i ∂x j − λ (cid:16) δ ij − v i v j (cid:17) + o (cid:0) (cid:1) . (4.4)Next we take v satisfying D u ( x ) v = λ v with | v | = 1 and v ⊥ v . And then denoting P = (cid:18) v v v v (cid:19) , we have P − D u ε ( x ε ) P = P − D u ( x ) P − (cid:16) λ + o (1) (cid:17) E + λ P − v v T P . (4.5)where E is the unit matrix. Also we compute that P − v v T P = (cid:18) v v v v (cid:19) (cid:18) v v (cid:19) (cid:0) v v (cid:1) (cid:18) v v v v (cid:19) = (cid:18) (cid:19) (cid:0) (cid:1) = (cid:18) (cid:19) . (4.6)Hence from (4.3), (4.4), (4.5) and (4.6), it follows P − D u ε ( x ε ) P = (cid:18) λ + o (1) 00 λ − λ + o (1) (cid:19) . (4.7)This gives us that det D u ε ( x ε ) = (cid:16) λ + o (1) (cid:17)(cid:16) λ − λ + o (1) (cid:17) . If λ < λ , then det D u ε ( x ε ) < x ε is a saddle point of u ε ( x ). Hence in this case, x ε is not a maximum point of u ε ( x ). If λ > λ , then two eigenvalues of D u ε ( x ε ) arenegative and x ε is a maximum point of u ε ( x ). These complete the proof of Proposition4.2. (cid:3) Now we are ready to prove Theorem 1.1.
RITICAL POINTS OF POSITIVE SOLUTIONS 17
Proof of Theorem 1.1 . We divide into following two cases.
Case 1: x = y . First from Proposition 4.1, we know that the maximum point x ε of u ε ( x ) on Ω ε satisfies x ε → y as ε → . Also we recall that all eigenvalues of D u ( y ) are negative. Hence by continuity, (1.2) andLemma 3.1, we find thatlim ε → λ max (cid:0) D u ε ( x ε ) (cid:1) = λ max (cid:0) D u ( y ) (cid:1) = max n λ , λ o < . (4.8) Case 2: x = y . Without loss of generality, we suppose that λ ≤ λ < u ε ( x ) satisfies x ε = x + s − u ( x ) + o (1) λ p | log ε | v with D u ( x ) v = λ v . And then (4.7) gives us thatlim ε → λ max (cid:0) D u ε ( x ε ) (cid:1) = max n λ , λ − λ o ( < , for λ < λ , = 0 , for λ = λ . Hence for general λ , λ , it holdslim ε → λ max (cid:0) D u ε ( x ε ) (cid:1) = max n λ , λ , −| λ − λ | o ( < , for λ = λ , = 0 , for λ = λ , which, together with (4.8), completes the proof of Theorem 1.1. (cid:3) Acknowledgments
Part of this work was done while Peng Luo was visiting the Math-ematics Department of the University of Rome “La Sapienza” whose members he wouldlike to thank for their warm hospitality. Hua Chen was supported by NSFC grants (No.11631011,11626251). Peng Luo was supported by NSFC grants (No.11701204,11831009).
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School of Mathematics and Statistics, Wuhan University, Wuhan 430072,China
Email address : [email protected] (Peng Luo) School of Mathematics and Statistics, Central China Normal University,Wuhan 430079, China
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