A diffused interface with the advection term in a Sobolev space
AA DIFFUSED INTERFACE WITH THE ADVECTIONTERM IN A SOBOLEV SPACE
YOSHIHIRO TONEGAWA AND YUKI TSUKAMOTO
Abstract.
We study the asymptotic limit of diffused surface energyin the van der Waals–Cahn–Hillard theory when an advection term isadded and the energy is uniformly bounded. We prove that the limit in-terface is an integral varifold and the generalized mean curvature vectoris determined by the advection term. As the application, a prescribedmean curvature problem is solved using the min-max method. Introduction
The object of study in this paper is the energy functional appearing inthe van der Waals–Cahn–Hillard theory [2, 5], E ε ( u ) = (cid:90) Ω ε |∇ u | W ( u ) ε , (1.1)where u : Ω ⊂ R n → R ( n ≥
2) is the normalized density distribution oftwo phases of a material, |∇ u | = (cid:80) nk =1 ( ∂u/∂x k ) and W : R → [0 , ∞ ) is adouble-well potential with two global minima at ±
1. In the thermodynamiccontext, W corresponds to the Helmholtz free energy density and the typicalexample is W ( u ) = (1 − u ) . When the positive parameter ε is smallrelative to the size of the domain Ω and E ε ( u ) is bounded, it is expectedthat u is close to +1 or − ± O ( ε ) thickness which we maycall the diffused interface of u . In this case, the quantity E ε ( u ) is expectedto be proportional to the surface area of the diffused interface. Due to theimportance of the surface area in calculus of variations, it is interesting toinvestigate the validity of such expectation and other salient properties of E ε .In this direction, there have been a number of works studying the as-ymptotic behavior of E ε as ε →
0+ under various assumptions. For theenergy minimizers with appropriate side conditions, it is well-known that itΓ-converges to the area functional of the limit interface [8, 10, 11, 12, 18].On the other hand, due in part to the non-convex nature of the functional,there may exist multiple and even infinite number of critical points of E ε different from the energy minimizers. For general critical points, Hutchinsonand the first author [6] proved that the limit is an integral stationary varifold[1]. For general stable critical points, the first author and Wickramasekera[24] proved that the limit is an embedded real-analytic minimal hypersur-face except for a closed singular set of codimension seven. More recently, The first author is partially supported by JSPS KAKENHI Grant Numbers (A)25247008 and (S) 26220702. a r X i v : . [ m a t h . A P ] A p r DIFFUSED INTERFACE WITH THE ADVECTION TERM 2
Guaraco [4] showed that a uniform Morse index bound is sufficient to con-clude the same regularity for n ≥ E ε have direct links tothe minimal surface theory as above, more generally, it turned out thatsuitable controls of the first variation of E ε guarantee the analogous goodasymptotic behaviors. For example, under the assumption thatlim inf ε → (cid:0) E ε ( u ε ) + (cid:107) f ε (cid:107) W ,p (Ω) (cid:1) < ∞ with f ε := − ε ∆ u ε + W (cid:48) ( u ε ) /ε and p > n/
2, the first author [20, 23] provedthat the limit interface is an integral varifold whose generalized mean cur-vature belongs to L q ( q = p ( n − / ( n − p ) > n −
1) with respect to thesurface measure. Here W ,p (Ω) := { u ∈ L p (Ω) : ∇ u ∈ L p (Ω) } . The meancurvature of the limit interface is characterized by the weak W ,p limit of f ε [16]. Another example concerns one of De Giorgi’s conjectures. Under theassumption that (with f ε as above)lim inf ε → (cid:0) E ε ( u ε ) + ε − (cid:107) f ε (cid:107) L (Ω) (cid:1) < ∞ and n = 2 ,
3, R¨oger-Sch¨atzle [15] (independently [13] for the case of n =2) proved the similar result. In this case, the limit interface has an L generalized mean curvature.In this paper, along the line of research described above, we investigatethe asymptotic behavior of u ε satisfying − ε ∆ u ε + W (cid:48) ( u ε ) ε = εv ε · ∇ u ε , (1.2)where v ε is considered here as a given vector field and we assume thatlim inf ε → (cid:0) E ε ( u ε ) + (cid:107) v ε (cid:107) W ,p (Ω) (cid:1) < ∞ and p > n/
2. The problem is related to (parabolic) Allen-Cahn-type equa-tions studied in [9, 19], for example. It is also natural to investigate theeffect of advection term as ε → L q (the sameas above) generalized mean curvature which is characterized by the weak W ,p limit of v ε . Using this result, we give some existence theorem for a vectorial prescribed mean curvature problem , as described in Theorem 2.2.Despite the simplicity of the problem, this is the first existence result in thesetting of the min-max method, with minimal regularity assumptions on theprescribed vector field.As for the proof, just as in the case of [6, 20, 23], the key point is toprove a certain monotonicity-type formula which is the essential tool in thesetting of Geometric Measure Theory. We wish to treat εv ε · ∇ u ε as aperturbative term, and to do so, we need to control a certain “trace” normof v ε on diffused interface. If an ε -independent upper density ratio estimateof diffused surface measure is available, then we can control εv ε · ∇ u ε by the W ,p -norm of v ε . For this purpose, we establish the key estimate, Theorem DIFFUSED INTERFACE WITH THE ADVECTION TERM 3
Assumptions and main results
We use the notation that U r ( a ) := { x ∈ R n : | x − a | < r } , B r ( a ) := { x ∈ R n : | x − a | ≤ r } , U r := U r (0) and B r := B r (0).2.1. Assumptions.
Throughout the paper, we assume that:(a) The function W : R → [0 , ∞ ) is C and has two strict minima W ( ±
1) = W (cid:48) ( ±
1) = 0.(b) For some γ ∈ ( − , W (cid:48) > − , γ ) and W (cid:48) < γ, α ∈ (0 ,
1) and κ > W (cid:48)(cid:48) ( x ) ≥ κ for all | x | ≥ α .Let Ω ⊂ R n be a bounded domain. We assume that we are given W , (Ω)functions { u i } ∞ i =1 , W ,p (Ω; R n ) vector fields { v i } ∞ i =1 and positive constants { ε i } ∞ i =1 satisfying − ε i ∆ u i + W (cid:48) ( u i ) ε i = ε i v i · ∇ u i (2.1)weakly on Ω for each i ∈ N . In addition, assume thatlim i →∞ ε i = 0 , n < p < n (2.2)and that there exist constants c , E and λ such that, for all i ∈ N , wehave: (cid:107) u i (cid:107) L ∞ (Ω) ≤ c , (2.3) (cid:90) Ω (cid:18) ε i |∇ u i | W ( u i ) ε i (cid:19) ≤ E , (2.4) (cid:107) v i (cid:107) L npn − p (Ω) + (cid:107)∇ v i (cid:107) L p (Ω) ≤ λ . (2.5)The condition (2.3) is not essential and can be often derived from thePDE or the proof of existence. Here we assume (2.3) for simplicity. Next,define Φ( s ) := (cid:90) s − (cid:112) W ( t ) / dt, w i ( x ) := Φ( u i ( x )) . By the Cauchy–Schwarz inequality and (2.4), we obtain (cid:90) Ω |∇ w i | ≤ (cid:90) Ω (cid:18) ε i |∇ u i | W ( u i ) ε i (cid:19) ≤ E . Hence, by the compactness theorem for BV functions [26, Corollary 5.3.4],there exist a converging subsequence (which we denote by the same notation) { w i } in the L norm and the limit BV function w . Define u ( x ) := Φ − ( w ( x )) . DIFFUSED INTERFACE WITH THE ADVECTION TERM 4 where Φ − is the inverse function of Φ. It follows that u i converges to u a.e.on Ω. By Fatou’s Lemma and (2.4), we have (cid:90) Ω W ( u ) = (cid:90) Ω lim i →∞ W ( u i ) ≤ lim inf i →∞ (cid:90) Ω W ( u i ) = 0 . This shows that u = ± u is a BV function. For simplicity wewrite ∂ ∗ { u = 1 } as the reduced boundary [26] of { u = 1 } and (cid:107) ∂ ∗ { u = 1 }(cid:107) as the boundary measure.2.2. The associated varifolds.
We associate to each solution of (1.2) avarifold in a natural way in the following. We refer to [1, 17] for a compre-hensive treatment of varifolds.Let G ( n, n −
1) be the Grassmannian, i.e. the space of unoriented ( n − R n . We also regard S ∈ G ( n, n −
1) as the n × n matrix representing the orthogonal projection of R n onto S . For two givensquare-matrices S and S , we write S · S := trace( S t ◦ S ), where theupper-script t indicates the transpose of the matrix and ◦ is the matrixmultiplication. We say that V is an ( n − ⊂ R n if V is a Radon measure on G n − (Ω) := Ω × G ( n, n − V n − (Ω) bethe set of all ( n − V ∈ V n − (Ω),we let (cid:107) V (cid:107) be the weight measure of V . For V ∈ V n − (Ω), we define thefirst variation of V by δV ( g ) := (cid:90) G n − (Ω) ∇ g ( x ) · S dV ( x, S ) (2.6)for any vector field g ∈ C c (Ω; R n ). We let (cid:107) δV (cid:107) be the total variation of δV . If (cid:107) δV (cid:107) is absolutely continuous with respect to (cid:107) V (cid:107) , then the Radon-Nikodym derivative δV / (cid:107) V (cid:107) exists as a vector-valued (cid:107) V (cid:107) measurable func-tion. In this case, we define the generalized mean curvature vector of V by − δV / (cid:107) V (cid:107) and we use the notation H V .We associate to each function u i a varifold V i as follows. First, we definea Radon measure µ i on Ω by dµ i := 1 σ (cid:16) ε i |∇ u i | W ( u i ) ε i (cid:17) d L n , (2.7)where L n is the n -dimensional Lebesgue measure and σ := (cid:82) − (cid:112) W ( s ) ds .Define V i ∈ V n − (Ω) by V i ( φ ) := (cid:90) {|∇ u i |(cid:54) =0 } φ (cid:16) x, I − ∇ u i ( x ) |∇ u i ( x ) | ⊗ ∇ u i ( x ) |∇ u i ( x ) | (cid:17) dµ i ( x ) (2.8)for φ ∈ C c ( G n − (Ω)), where I is the n × n identity matrix and ⊗ is thetensor product of the two vectors. Note that I − ∇ u i ( x ) |∇ u i ( x ) | ⊗ ∇ u i ( x ) |∇ u i ( x ) | representsthe orthogonal projection to the ( n − { a ∈ R n : a · ∇ u i ( x ) = 0 } . By definition, we have (cid:107) V i (cid:107) = µ i {|∇ u i |(cid:54) =0 } DIFFUSED INTERFACE WITH THE ADVECTION TERM 5 and by (2.6), we have δV i ( g ) = (cid:90) {|∇ u i |(cid:54) =0 } ∇ g · (cid:16) I − ∇ u i |∇ u i | ⊗ ∇ u i |∇ u i | (cid:17) dµ i (2.9)for each g ∈ C c (Ω , R n ).2.3. Main Theorems.
With the above assumptions and notation, we show:
Theorem 2.1.
Suppose that u i , v i , ε i satisfy (2.1) - (2.5) and let V i be thevarifold associated with u i as in (2.8) . On passing to a subsequence we canassume that v i → v weakly in W ,p , u i → u a.e., V i → V in the varifold sense.Then we have the following properties. (1) For each φ ∈ C c (Ω) , (cid:107) V (cid:107) ( φ ) = lim i →∞ σ (cid:90) Ω ε i |∇ u i | φ = lim i →∞ σ (cid:90) Ω W ( u i ) ε i φ = lim i →∞ σ (cid:90) Ω |∇ w i | φ. (2) spt (cid:107) ∂ ∗ { u = 1 }(cid:107) ⊂ spt (cid:107) V (cid:107) and { u i } converges locally uniformly to ± on Ω \ spt (cid:107) V (cid:107) . (3) For each < b < , {| u i | ≤ − b } locally converges to spt (cid:107) V (cid:107) in theHausdorff distance sense in Ω . (4) V is an integral varifold and the density θ ( x ) of V satisfies θ ( x ) = (cid:40) o dd H n − a.e. x ∈ ∂ ∗ { u = 1 } , e ven H n − a.e. x ∈ spt (cid:107) V (cid:107)\ ∂ ∗ { u = 1 } , (5) the generalized mean curvature vector H V of V is given by H V ( x ) = S ⊥ ( v ( x )) , for ( x, S ) ∈ G n − (Ω) for V a.e., where S ⊥ ∈ G ( n, is the projec-tion to the orthogonal complement of S , i.e., S ⊥ = I − S . (6) For ˜Ω ⊂⊂ Ω , there exists a constant λ depending only on c , λ , n , p , W , E and dist( ˜Ω , ∂ Ω) such that (cid:90) ˜Ω | H V ( x ) | p ( n − n − p d (cid:107) V (cid:107) ( x ) ≤ (cid:90) ˜Ω | v ( x ) | p ( n − n − p d (cid:107) V (cid:107) ( x ) ≤ λ . Note that p ( n − n − p > n − due to (2.2) . Since V is integral and the generalized mean curvature vector is in thestated class, V satisfies various good properties described in [17, Section 17].In particular, there exists a closed countably ( n − ⊂ Ω(which we can take as the support of (cid:107) V (cid:107) , see [17, Section 17.9(1)]) suchthat, for any φ ∈ C c ( G n − (Ω)), (cid:90) G n − (Ω) φ ( x, S ) dV ( x, S ) = (cid:90) Γ φ ( x, T x Γ) θ ( x ) d H n − ( x ) . Here, T x Γ ∈ G ( n, n −
1) is the approximate tangent space of Γ at x whichexists H n − a.e. x ∈ Γ. With this notation, (5) implies that H V ( x ) = DIFFUSED INTERFACE WITH THE ADVECTION TERM 6 ( T x Γ) ⊥ ( v ( x )) for H n − a.e. x ∈ Γ, i.e., the generalized mean curvaturevector of V coincides with the projection of v to the orthogonal subspace( T x Γ) ⊥ for H n − a.e. x ∈ Γ. If we additionally assume that θ = 1 for H n − a.e. x ∈ Γ, then because of the integrability of H V and the Allardregularity theorem [1], except for a closed H n − -null set, Γ is locally a C , − np hypersurface. Without the assumption θ = 1, we can still conclude thatspt (cid:107) V (cid:107) is C , − np hypersurface on a dense open set of spt (cid:107) V (cid:107) , even thoughwe do not know if the complement is H n − -null or not.2.4. A vectorial prescribed mean curvature problem.
As an applica-tion of Theorem 2.1, we prove the following: Theorem 2.2.
Let Ω ⊂ R n be a bounded domain with Lipschitz bound-ary and let ρ ∈ W ,p (Ω) be a given function, where p > n . Then, thereexists a non-zero integral varifold V such that H V ( x ) = S ⊥ ( ∇ ρ ( x )) for V a.e. ( x, S ) ∈ G n − (Ω) .Proof. We may assume p < n . Consider the following functional for ε > u ∈ W , (Ω): F ε ( u ) := (cid:90) Ω (cid:16) ε |∇ u | W ( u ) ε (cid:17) exp( ρ ) . By the Sobolev embedding, ρ ∈ C , − np (Ω) and thus 0 < exp(min ρ ) ≤ exp( ρ ) ≤ exp(max ρ ) < ∞ . By considering the path space in W , (Ω)connecting u ≡ u ≡ −
1, the standard min-max method gives a non-trivial critical point u ε for each ε >
0, with uniform strictly positive lowerand upper bounds of F ε ( u ε ) (see for example [4] for the detail). The criticalpoint satisfies (2.1) with v = ∇ ρ and | u ε | ≤
1. Take a sequence ε i → u i . Then the sequence u i , ∇ ρ, ε i satisfy all the assumptions of Theorem 2.1. The limit varifold V thus hasthe desired property. (cid:3) For more remarks on the main results, see Section 5.3.
The estimate for the upper density ratio
In this section, we prove Theorem 3.8-3.10, which give ε -independentestimates of the upper and lower density ratios of the energy. Throughoutthis section, we drop the index i and set Ω = U = {| x | < } since theresult is local. Assume u ∈ W , ( U ) and v ∈ W ,p ( U ; R n ) satisfy (2.1)with a positive ε and (2.3)-(2.5) are satisfied for a given set of c , E , λ .The exponent p satisfies (2.2). We first derive two preliminary propertiesfor u , Lemma 3.1 and 3.2. Lemma 3.1.
There exists c > depending only on c , λ , n, p and W suchthat sup x ∈ U − ε ε |∇ u ( x ) | ≤ c (3.1) and sup x,x (cid:48) ∈ U − ε ε − np |∇ u ( x ) − ∇ u ( x (cid:48) ) || x − x (cid:48) | − np ≤ c (3.2) The authors thank Nick Edelen for a discussion which inspired this application.
DIFFUSED INTERFACE WITH THE ADVECTION TERM 7 for < ε < / . If ε ≥ / , then we have for any < s < x ∈ U s |∇ u ( x ) | ≤ c (3.3) where c depends additionally on s . In both cases, we have u ∈ W , np n − p ) loc ( U ) ∩ W , npn − p loc ( U ) ∩ C , − np loc ( U ) . Proof.
Consider the case 0 < ε < /
2. Define ˜ u ( x ) := u ( εx ) and ˜ v ( x ) := εv ( εx ) for x ∈ U ε − . After this change of variables, we obtain from (2.1)that − ∆˜ u + W (cid:48) (˜ u ) = ˜ v · ∇ ˜ u weakly on U ε − . (3.4)Under the change of variables, we obtain from (2.5) (cid:107) ˜ v (cid:107) L npn − p ( U ε − ) + (cid:107)∇ ˜ v (cid:107) L p ( U ε − ) ≤ λ ε − np . (3.5)For any U ( x ) ⊂ U ε − , let φ ∈ C c ( U ( x )) be a function such that 0 ≤ φ ≤ φ = 1 on B ( x ) and |∇ φ | ≤ U ( x ). Use (3.4) with the test function˜ uφ . Using also (2.3), we obtain (cid:90) |∇ ˜ u | φ ≤ c (cid:90) (2 φ |∇ φ (cid:107)∇ ˜ u | + | W (cid:48) | φ + | ˜ v (cid:107)∇ ˜ u | φ ) ≤ (cid:90) |∇ ˜ u | φ + (cid:90) (4 c |∇ φ | + c | W (cid:48) | φ + c | ˜ v | φ ) . (3.6)Since npn − p >
2, (3.5) and (3.6) givesup B ( x ) ⊂ U ε − (cid:90) B ( x ) |∇ ˜ u | ≤ c ( c , λ , n, p, W ) . (3.7)We next note that the function ˜ uφ weakly satisfies the following equation: − ∆(˜ uφ ) = − ˜ u ∆ φ − ∇ φ · ∇ ˜ u + (˜ v · ∇ ˜ u − W (cid:48) (˜ u )) φ. (3.8)Using the standard L p theory [3, Theorem 9.11] to (3.8), we may start abootstrapping argument as follows. Staring with q = 2, we have ∇ ˜ u ∈ L qloc = ⇒ ˜ v · ∇ ˜ u ∈ L npqnp + q ( n − p ) loc = ⇒ ˜ u ∈ W , npqnp + q ( n − p ) loc = ⇒ ∇ ˜ u ∈ L npqnp − q (2 p − n ) loc with the corresponding estimates relating these norms. Note that the ex-ponent of integrability of ∇ ˜ u is raised from q to q · npnp − q (2 p − n ) , with thefactor strictly larger than one. Thus, in a finite number of bootstrapping,we obtain the W ,sloc (with s > n ) estimate for ˜ u , and by the Sobolev in-equality, the L ∞ loc estimate for ∇ ˜ u . Again by the L p theory, we obtain the W , npn − p loc estimate of ˜ u . In particular, by the Sobolev inequality, we obtain(3.1) and (3.2). Since the right-hand side of (3.8) is in W , np n − p ) loc (note that˜ v · ∇ ˜ u ∈ L np n − p ) loc and np n − p ) > u ∈ W , np n − p ) loc and theweak third-derivatives of ˜ u exist. The case of ε ≥ / (cid:3) DIFFUSED INTERFACE WITH THE ADVECTION TERM 8
Lemma 3.2.
Given < s < , there exist constants < ε , η < dependingonly on c , λ , W, n, p and s such that sup x ∈ B s | u ( x ) | ≤ ε η (3.9) for ε ≤ ε .Proof. Let q = npn − p − φ ∈ C ∞ c ( B s +12 ) with φ ≥
0. Multiplying (2.1)by [( u − + ] q φ , we have − ε (cid:90) q [( u − + ] q − |∇ u | φ + 2[( u − + ] q φ ∇ φ · ∇ u = (cid:90) W (cid:48) ε [( u − + ] q φ − (cid:90) ε ∇ u · v [( u − + ] q φ . (3.10)By W (cid:48) ( u ) ≥ κ ( u −
1) for u ≥ κε (cid:90) [( u − + ] q +1 φ + (cid:90) εq [( u − + ] q − |∇ u | φ ≤ ε (cid:90) [( u − + ] q φ |∇ φ ||∇ u | + c (cid:90) | v | [( u − + ] q φ ≤ qε (cid:90) [( u − + ] q − |∇ u | φ + 8 εq (cid:90) [( u − + ] q +1 |∇ φ | + κ ε (cid:90) [( u − + ] q +1 φ + ε q c ( q, c ) κ q (cid:90) | v | q +1 φ , (3.11)which shows κ ε (cid:90) [( u − + ] q +1 φ ≤ εq (cid:90) [( u − + ] q +1 |∇ φ | + ε q c ( q, c ) κ q (cid:90) | v | q +1 φ . (3.12)By (2.3) and iterating the computation above with suitable φ , we obtain (cid:90) B s [( u − + ] q +1 ≤ c ( s, q, λ , n, p, W, c , c ) ε q +1 . (3.13)To derive a contradiction, assume that u ( x ) − ≥ ε η for some x ∈ B s . By(3.1), for y ∈ B ε η c ( x ), u ( y ) − ≥ u ( x ) − − sup |∇ u | ε η c ≥ ε η . (3.14)Then we have c ε q +1 ≥ (cid:90) B ε η c ( x ) [( u − + ] q +1 ≥ (cid:18) ε η (cid:19) q +1 ω n (cid:18) ε η c (cid:19) n , (3.15)which show by q = npn − p − ε η npn − p − npn − p + n + nη ≤ c ( s, q, λ , n, p, W, c , c ) . (3.16)This is a contradiction if η and ε are sufficiently small. u ≥ − − ε η is provedsimilarly. (cid:3) The next Lemma 3.3 is the starting point of the ultimate establishmentof the monotonicity formula.
DIFFUSED INTERFACE WITH THE ADVECTION TERM 9
Lemma 3.3.
For B r ( x ) ⊂ U , we have ddr (cid:40) r n − (cid:90) B r ( x ) (cid:18) ε |∇ u | W ( u ) ε (cid:19)(cid:41) = 1 r n (cid:90) B r ( x ) (cid:18) W ( u ) ε − ε |∇ u | (cid:19) + εr n +1 (cid:90) ∂B r ( x ) (( y − x ) · ∇ u ) + εr n (cid:90) B r ( x ) ( v · ∇ u )(( y − x ) · ∇ u ) . (3.17) Proof.
Multiply both sides of (2.1) by ∇ u · g , where g = ( g , · · · , g n ) ∈ C c ( U ; R n ). By integration by parts, we obtain (cid:90) (cid:16)(cid:16) ε |∇ u | Wε (cid:17) div g − ε (cid:88) i,j u y i u y j g iy j + ε ( v · ∇ u )( ∇ u · g ) (cid:17) = 0 . (3.18)We assume that x = 0 after a suitable translation and let g j ( y ) = y j ρ ( | y | ).Writing r = | y | , (3.18) becomes (cid:90) (cid:16)(cid:16) ε |∇ u | Wε (cid:17) (cid:0) rρ (cid:48) + nρ (cid:1) − ε ρ (cid:48) r ( y · ∇ u ) − ε |∇ u | ρ + ε ( ∇ u · v )( ∇ u · y ) ρ (cid:17) = 0 . We choose ρ which is a smooth approximation of χ B r , the characteristicfunction of B r , and then we take a limit ρ → χ B r . Then we have − ( n − (cid:90) B r (cid:18) ε |∇ u | Wε (cid:19) + r (cid:90) ∂B r (cid:18) ε |∇ u | Wε (cid:19) = (cid:90) B r (cid:18) Wε − ε |∇ u | (cid:19) + εr (cid:90) ∂B r ( y · ∇ u ) + ε (cid:90) B r ( ∇ u · v )( ∇ u · y ) . By dividing the above equation by r n , the lemma follows. (cid:3) We need the following lemma to control the negative contribution of theright-hand side of (3.17).
Lemma 3.4.
Given < s < , there exist constants < β < and < ε < which depend only on c , λ , W , n , p and s such that, if ε ≤ ε , sup B s (cid:18) ε |∇ u | − W ( u ) ε (cid:19) ≤ ε − β . (3.19)The proof of Lemma 3.4 is deferred to the end of this section. Next, for B r ( x ) ⊂ U and 0 < r < dist( x, ∂U ), define E ( r, x ) := 1 r n − (cid:90) B r ( x ) (cid:18) ε |∇ u | W ( u ) ε (cid:19) . Using Lemma 3.3 and Lemma 3.4, we prove:
Lemma 3.5.
Given < s < , there exist constants < ε , c , c < whichdepend only on c , λ , W, n, p and s such that, if B ε β ( x ) ⊂ U s , | u ( x ) | ≤ α and ε ≤ ε , then E ( r, x ) ≥ c for all ε ≤ r ≤ c ε β . (3.20) DIFFUSED INTERFACE WITH THE ADVECTION TERM 10
Proof.
By integrating (3.17) over [ ε, r ], we have E ( r, x ) − E ( ε, x ) ≥ − (cid:90) rε dττ n (cid:90) B τ ( x ) (cid:18) ε |∇ u | − W ( u ) ε (cid:19) + + (cid:90) rε dττ n (cid:90) B τ ( x ) ε ( ∇ u · v )( ∇ u · ( y − x )) . (3.21)By (3.19) and B r ( x ) ⊂ U s , we have (cid:90) rε dττ n (cid:90) B τ ( x ) (cid:18) ε |∇ u | − W ( u ) ε (cid:19) + ≤ ω n rε − β . (3.22)By (3.1) and (2.5), we have (cid:90) rε dττ n (cid:90) B τ ( x ) ε ( ∇ u · v )( ∇ u · ( y − x )) ≤ (cid:90) r dττ n − (cid:90) B τ ( x ) c ε − | v |≤ c ( λ , n, p, c ) r − np ε − . (3.23)Since | u ( x ) | ≤ α , using (3.1), we have | u ( y ) | ≤ α +12 for all y ∈ B (1 − α ) ε c ( x ).By choosing a larger c if necessary, we may assume (1 − α )2 c ≤
1. Define c := ω n − α ) n (2 c ) n min | t |≤ α W ( t ) > . With this choice, we obtain E ( ε, x ) ≥ ε n − (cid:90) B (1 − α ) ε c ( x ) W ( u ) ε ≥ ω n (1 − α ) n (2 c ) n min | t |≤ α W ( t ) = 2 c . (3.24)Since a larger β satisfies (3.19) as well, we may assume (3 − np ) β − > β < c for sufficiently small ε and c if r ≤ c ε β . Then, (3.20) follows from (3.21)-(3.24). (cid:3) Theorem 3.6.
Given < s < , there exist constants < ε , β < and < c which depend only on c , λ , W , n , p and s such that, if B r ( x ) ⊂ U s , c ε β < r and ε ≤ ε , then r n (cid:90) B r ( x ) (cid:18) ε |∇ u | − W ( u ) ε (cid:19) + ≤ c r − β ( E ( r, x ) + 1) . (3.25) Proof.
The proof is similar to [23, Proposition 3.4] with a minor modifica-tion. Define β := − β β and β := β . β and β are chosen so that β β = β − β , (3.26)0 < β < , < β < β < . (3.27) DIFFUSED INTERFACE WITH THE ADVECTION TERM 11
We estimate the integral of (3.25) by separating B r ( x ) into three disjointsets. Define A := { x ∈ B r ( x ) \ B r − ε β ( x ) } , B := { y ∈ B r − ε β ( x ) : dist( {| u | ≤ α } , y ) < ε β } , C := { y ∈ B r − ε β ( x ) : dist( {| u | ≤ α } , y ) ≥ ε β } . Note that r > c ε β > ε β for small ε .The estimates of the integral over A and B are exactly the same as in[23]. Namely, for A , we use L n ( A ) ≤ nω n r n − ε β and (3.19) as well as r − ≤ ( c ε β ) − . For B , we use (3.20) and prove that L n ( B ) ≤ c ( n ) ε nβ N ,where N is an integer satisfying B ⊂ ∪ Ni =1 B ε β ( x i ). Here we only considerthe estimate on C and refer the reader to the proof of [23, Proposition 3.4].Define φ ( x ) := min { , ε − β dist( {| y | ≥ r } ∪ {| u | ≤ α } , x ) } .φ is a Lipschitz function and is 0 on {| y | > r } ∪ {| u | < α } , 1 on C and |∇ φ | ≤ ε − β . Differentiate (2.1) with respect to x j , multiply it by u x j φ andsum over j . Then we have (cid:90) (cid:88) j εu x j ∆ u x j φ = (cid:90) W (cid:48)(cid:48) ε |∇ u | φ − ε (cid:88) j (cid:90) ( v · ∇ u ) x j φ u x j . (3.28)By integration by parts, the Cauchy–Schwarz inequality and (3.1), we obtain (cid:90) ε |∇ u | φ + W (cid:48)(cid:48) ε |∇ u | φ = (cid:90) − (cid:88) i,j εu x j u x i x j φφ x i − ε ∇ u · v (∆ uφ + 2 φ ∇ u · ∇ φ ) ≤ (cid:90) ε |∇ u | φ + c (cid:90) ( ε |∇ u | |∇ φ | + | v | φ ε − ) , (3.29)where c depends on c . Since | u | ≥ α on the support of φ , we have W (cid:48)(cid:48) ≥ κ .Thus (cid:90) ε |∇ u | φ + κε |∇ u | φ ≤ c (cid:90) ( ε |∇ u | |∇ φ | + | v | φ ε − ) . (3.30)By |∇ φ | ≤ ε − β and (2.5), we have (cid:90) κε |∇ u | φ ≤ c (cid:18) ε − β (cid:90) B r ε |∇ u | + ε − (cid:107) v (cid:107) L npn − p ( ω n r n ) np − n − p ) np (cid:19) ≤ c (cid:18) ε − β (cid:90) B r ε |∇ u | + ε − r n − n − p ) p (cid:19) , (3.31)where c = c + c ( n, p ) λ . Since φ = 1 on C , multiplying (3.31) by ε κr n ,1 r n (cid:90) C ε |∇ u | ≤ c κ (cid:18) ε − β r E ( r, x ) + εr − np (cid:19) . (3.32)By the definitions of β , β , β and r ≥ c ε β , we have ε − β r ≤ ε β β r − β c β , (3.33) DIFFUSED INTERFACE WITH THE ADVECTION TERM 12 and using ε ≤ r , εr − np ≤ r np − , (3.34)where np − <
1. Hence, we obtain1 r n (cid:90) C ε |∇ u | ≤ c κ (cid:32) ε β β r − β c β E ( r, x ) + 1 r np − (cid:33) . (3.35)By re-defining β = min { β , − np } and the estimates of integrals over A , B and C , we proved (3.25). (cid:3) To proceed, we need the following theorem (see [26, Theorem 5.12.4]).
Theorem 3.7.
Let µ be a positive Radon measure on R n satisfying K ( µ ) := sup B r ( x ) ⊂ R n r n − µ ( B r ( x )) < ∞ . Then there exists a constant c ( n ) such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R n φ dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( n ) K ( µ ) (cid:90) R n |∇ φ | d L n for all φ ∈ C c ( R n ) . Theorem 3.8.
There exists a constant < c which depends only on c , λ , W , E , n and p such that, if < ε < / and U r ( x ) ⊂ U − ε , then dist( x, ∂U − ε ) n − E ( r, x ) ≤ c . (3.36) Proof.
Define E := sup U r ( x ) ⊂ U − ε dist( x, ∂U − ε ) n − E ( r, x ) . By Lemma 3.1, we have sup x ∈ U − ε ε |∇ u ( x ) | ≤ c . Thus for any U r ( x ) ⊂ U − ε , we have E ( r, x ) ≤ ω n r ( c ε + sup | t |≤ c W ( t ) ε ) ≤ c ( n, c , W ) ε and we have E < ∞ for each ε . In the following, we give an estimate of E depending only on c , λ , n, p, W and E . Let U r ( x ) ⊂ U − ε be fixedsuch that dist( x , ∂U − ε ) n − E ( r , x ) > E . (3.37)For simplicity, define l := dist ( x , ∂U − ε ) = 1 − ε − | x | and change variables by ˜ x = ( x − x ) /l , ˜ r = r/l , ˜ ε = ε/l , ˜ u (˜ x ) = u ( x ) and˜ v (˜ x ) = lv ( x ). Note that U l + ε ( x ) ⊂ U . In particular, we write˜ r := r /l ≤ / . DIFFUSED INTERFACE WITH THE ADVECTION TERM 13
By (2.1), (2.4) and (2.5), we have − ˜ ε ∆˜ u + W (cid:48) (˜ u )˜ ε = ˜ ε ˜ v · ∇ ˜ u for ˜ x ∈ U ε , (3.38) (cid:90) U ε (cid:18) ˜ ε |∇ ˜ u | W (˜ u )˜ ε (cid:19) ≤ l − n E , (3.39) (cid:107) ˜ v (cid:107) L npn − p ( U ε ) + (cid:107)∇ ˜ v (cid:107) L p ( U ε ) ≤ l − np λ . (3.40)Define for B ˜ r (˜ x ) ⊂ U ε ˜ E (˜ r, ˜ x ) := 1˜ r n − (cid:90) B ˜ r (˜ x ) (cid:18) ˜ ε |∇ ˜ u | W (˜ u )˜ ε (cid:19) . (3.41)Under the above change of variables, note that we have E ( r, x ) = ˜ E (˜ r, ˜ x ).Next, for any x ∈ B l/ ( x ), we have dist ( x, ∂U − ε ) ≥ l/
4. Hence for any x ∈ B l/ ( x ) and r < l/ ≤ dist ( x, ∂U − ε ) /
2, by the definition of E , wehave dist( x, ∂U − ε ) n − E ( r, x ) ≤ E . This shows (again using dist( x, ∂U − ε ) ≥ l/ ˜ x ∈ B , < ˜ r< ˜ E (˜ r, ˜ x ) ≤ n − l − n E . (3.42)We next let c , c , c , c , ε , ε , ε , ε , β , β be constants obtained in Lemma3.1–3.5 and Theorem 3.6 corresponding to the same c , λ , n, p, W and s =3 /
4. Then note that the estimates up to Theorem 3.6 hold for ˜ u and ˜ v for U / and with respect to the new variables (˜ x , ˜ r , ˜ ε etc.) if˜ ε ≤ ˆ ε := min { ε , ε , ε , ε , / } (3.43)due to (3.38) and (3.40). It is important to note that ˆ ε depends only on c , λ , n, p and W . A priori, we do not know if (3.43) holds or not and weprove the desired estimate for E by exhausting all the possibilities.First consider the case ˜ ε ≥ ˆ ε . We use (3.3) and (3.1), respectively, for˜ ε > / / ≥ ˜ ε ≥ ˆ ε . Suppose that ˜ ε > /
2. By (3.37) and (3.3), wehave 34 l n − E < ˜ E (˜ r , ≤ ω n ˜ r (˜ εc + 2 sup | x |≤ c W ( x ))and since l ˜ ε = ε ≤ l < r ≤ /
2, we obtain E < ω n ( c + 2 sup | x |≤ c W ( x )) . (3.44)If 1 / ≥ ˜ ε ≥ ˆ ε , again by (3.37), (3.1) and ˜ r ≤ /
2, we have34 l n − E < ˜ E (˜ r , ≤ ω n ( c + sup | x |≤ c W ( x )) ˜ r ˜ ε ≤ ω n ( c + sup | x |≤ c W ( x )) 12ˆ ε and we obtain E < ω n ( c + sup | x |≤ c W ( x )) 23ˆ ε . (3.45) DIFFUSED INTERFACE WITH THE ADVECTION TERM 14
Thus by (3.44) and (3.45), if ˜ ε ≥ ˆ ε , E is bounded by a constant whichdepends only on c , λ , n, p and W .For the rest of the proof, consider the case ˜ ε < ˆ ε and consider the followingfour cases (a)–(d) depending on how large ˜ r = r /l is relative to ˜ ε and ˜ r ,where ˜ r will be determined shortly depending only on c , λ , n, p, W and E :( a ) ˜ r < ˜ r ≤ , ( b ) c ˜ ε β < ˜ r ≤ ˜ r , ( c ) ˜ ε < ˜ r ≤ c ˜ ε β , ( d ) 0 < ˜ r ≤ ˜ ε. To control the term involving v in (3.17), define a Radon measure µ ( A ) := (cid:90) A ∩ B (cid:18) ˜ ε |∇ ˜ u | W (˜ u )˜ ε (cid:19) . By Theorem 3.7 and (3.42) (note that (3.42) has the restriction ˜ r < / n − by a larger constant dependingonly on n ), we have (cid:12)(cid:12)(cid:12) (cid:90) B φ dµ (cid:12)(cid:12)(cid:12) ≤ c ( n ) l − n E (cid:90) R n |∇ φ | d L n (3.46)for all φ ∈ C c ( R n ). By (3.17) and (3.25), if c ˜ ε β < ˜ r ≤ , we have dd ˜ r ˜ E (˜ r, ≥ − c ˜ r − β ( ˜ E (˜ r,
0) + 1) − r n − (cid:90) B ˜ r ˜ ε | ˜ v ||∇ ˜ u | + 1˜ r n (cid:90) B ˜ r (cid:18) W (˜ u )˜ ε − ˜ ε |∇ ˜ u | (cid:19) + . (3.47)Let φ ∈ C c ( B r ) be such that 0 ≤ φ ≤ φ ( y ) = 1 in B ˜ r and |∇ φ | ≤ r . Weuse (3.46) with (3.40) by smoothly approximating | ˜ v | as (cid:90) B ˜ r ˜ ε | ˜ v ||∇ ˜ u | ≤ (cid:90) B r ˜ εφ | ˜ v ||∇ ˜ u | ≤ c ( n ) l − n E (cid:90) B r |∇ ( φ | ˜ v | ) |≤ c ( n ) l − n E (cid:90) B r r | ˜ v | + |∇ ˜ v | ≤ c ( n ) l − n − np λ ˜ r n − np E . (3.48)Hence, for c ˜ ε β < ˜ r ≤ , (3.47) with (3.48) and (3.42) give dd ˜ r ˜ E (˜ r, ≥ − c ˜ r β − ( c ( n ) l − n E + 1) − c ( n ) l − n − np λ ˜ r − np E + 1˜ r n (cid:90) B ˜ r (cid:18) W (˜ u )˜ ε − ˜ ε |∇ ˜ u | (cid:19) + . (3.49)By integrating (3.49) over ˜ r ∈ (˜ s , ˜ s ) with c ˜ ε β < ˜ s < ˜ s ≤ , we obtain˜ E (˜ s , − ˜ E (˜ s , ≥ − c (˜ s β + l − np ˜ s − np ) l − n E − c ˜ s β + (cid:90) ˜ s ˜ s d ˜ r ˜ r n (cid:90) B ˜ r (cid:18) W (˜ u )˜ ε − ˜ ε |∇ ˜ u | (cid:19) + , (3.50) DIFFUSED INTERFACE WITH THE ADVECTION TERM 15 where c depends only on c , λ , n, p and W . At this point, we choose˜ r < / c , λ , n, p and W so that c (˜ r β + ˜ r − np ) < . This in particular implies from (3.50) that if c ˜ ε β < ˜ s < ˜ s ≤ ˜ r , then˜ E (˜ s , − ˜ E (˜ s , ≥ − c − l − n E . (3.51)With this ˜ r being fixed, we proceed to check that E is bounded in termsof c , λ , n, p, W, E in each case (a)–(d).Case (a): By (3.37), (3.39) and ˜ r < ˜ r , we have34 l − n E ≤ ˜ E (˜ r , ≤ ˜ r − n l − n E ≤ ˜ r − n l − n E . Hence E ≤
34 ˜ r − n E and E is bounded by a constant depending only on c , λ , n, p, W and E .Case (b): Since c ˜ ε β < ˜ r ≤ ˜ r , we may use (3.51) with ˜ s = ˜ r and ˜ s = ˜ r .Then we obtain ˜ E (˜ r , − ˜ E (˜ r , ≥ − c − l − n E . Then, by (3.37) and (3.39), we obtain E ≤
4( ˜ E (˜ r ,
0) + c ) l n − ≤ r − n E + c ) , which depends only on c , λ , n, p, W and E .Case (c): By the same estimate used in the proof of Lemma 3.5, we have˜ E ( c ˜ ε β , − ˜ E (˜ r , ≥ − c . (3.52)We use (3.51) with ˜ s = c ˜ ε β and ˜ s = ˜ r to obtain˜ E (˜ r , − ˜ E ( c ˜ ε β , ≥ − c − l − n E , (3.53)and (3.52) and (3.53) combined with (3.37) give E ≤ l n − ( ˜ E (˜ r ,
0) + c + c ) ≤ r − n E + 4( c + c ) , which depends only on c , λ , n, p, W and E .Case (d): Since ˜ r ≤ ˜ ε , we use (3.1) to obtain˜ E (˜ r , ≤ ω n ( c + sup | x |≤ c W ( x )) ˜ r ˜ ε ≤ ω n ( c + sup | x |≤ c W ( x )) . (3.54)Then (3.54) and (3.37) gives E < ω n ( c + sup | x |≤ c W ( x )) l n − ≤ ω n ( c + sup | x |≤ c W ( x )) . This completes the estimate for E . (cid:3) Once we obtain the upper density estimate, we may obtain the followingmonotonicity formula.
DIFFUSED INTERFACE WITH THE ADVECTION TERM 16
Theorem 3.9.
Given < s < , there exist constants < c and < ε < depending only on c , λ , n, p, W, E and s , such that, if c ε β ≤ s < s , B s ( x ) ⊂ U s and ε < ε , then E ( s , x ) − E ( s , x ) ≥ − c ( s − np + s β )+ (cid:90) s s dττ n (cid:90) B τ ( x ) (cid:16) W ( u ) ε − ε |∇ u | (cid:17) + . (3.55) Proof.
Let ε = min { ε , ε , ε , ε , (1 − s ) / } corresponding to the given s andsuppose that ε < ε . For any x ∈ U s and 0 < r < (1 − s ) /
2, by Theorem3.8, E ( r, x ) ≤ c (1 − s − ε ) − n , where the right-hand side is bounded by aconstant depending only on c , λ , n, p, W, E and s . For B s ( x ) ⊂ U s , wehave (3.25) and (3.17). Arguing as (3.46)-(3.50) without change of variables(so l = 1) and with µ restricted to B s in place of B / , we obtain (3.55). (cid:3) Theorem 3.10.
Given < s < , there exist constants < c dependingonly on c , λ , n, p, W, E and s such that, if ε < ε , | u ( x ) | ≤ α and B r ( x ) ⊂ U s , then E ( r, x ) ≥ c . (3.56) Proof.
By Lemma 3.5, we may assume c ˜ ε β ≤ r and E ( c ε β , x ) ≥ c .In (3.55), let s = c ε β and s = r . Fix r > c , λ , n, p, W, E and s so that c ( r − np + r β ) ≤ c /
2. Then for c ε β ≤ r ≤ r , (3.55) shows that E ( r, x ) ≥ c /
2. For 1 > r > r , E ( r, x ) ≥ r n − E ( r , x ) ≥ r n − c /
2. Thus, setting c = r n − c /
2, we have (3.56). (cid:3)
For the rest of the present section, we finish the proof of Lemma 3.4. Weuse the following result proved in [23, Lemma 3.9].
Lemma 3.11.
Given < η, β < , η ≤ β , < c , there exist ε > , c > depending only on η , β , c , n and W with the following properties:Suppose f ∈ C ( U ε − β ) , g ∈ C ( U ε − β ) and < ε ≤ ε satisfy − ∆ f + W (cid:48) ( f ) = g on U ε − β and sup U ε − β | f | ≤ ε η , sup U ε − β (cid:18) |∇ f | − W ( f ) (cid:19) ≤ c . Then sup B ε − β (cid:18) |∇ f | − W ( f ) (cid:19) ≤ c ( ε − β (cid:107) g (cid:107) W ,n ( B ε − β ) + ε η ) . (3.57)We remark that the assumptions on W are essentially used for the proofof Lemma 3.11. Proof of Lemma 3.4.
As in the proof of Lemma 3.1, define ˜ u ( x ) := u ( εx ),˜ v ( x ) := εv ( εx ), and subsequently drop ˜ · for simplicity. We have − ∆ u + W (cid:48) ( u ) = ∇ u · v DIFFUSED INTERFACE WITH THE ADVECTION TERM 17 on U ε − . With respect to the new variables, we need to provesup U sε − (cid:18) |∇ u | − W ( u ) (cid:19) ≤ ε − β (3.58)for some 0 < β < ε . Let φ λ be the standardmollifier, namely, define φ ( x ) := (cid:40) C exp (cid:16) | x | − (cid:17) for | x | <
10 for | x | ≥ , where the constant C > (cid:82) R n φ = 1, and define φ λ ( x ) := λ n φ ( xλ ). For 0 < β < n and p later,define for x ∈ U ε − − f ( x ) := ( u ∗ φ ε β )( x ) = (cid:90) u ( x − y ) φ ε β ( y ) dy. (3.59)By (3.1) and (3.2), we have sup U ε − − | f − u | ≤ c ε β , (3.60)sup U ε − − |∇ f − ∇ u | ≤ c ε β (2 − np ) . (3.61)We next define g to be g := ( ∇ u · v ) ∗ φ ε β + ( W (cid:48) ( f ) − W (cid:48) ( u ) ∗ φ ε β ) , (3.62)so that we have − ∆ f + W (cid:48) ( f ) = g. (3.63)To use Lemma 3.11, we next estimate the W ,n norm of g on U ε − β ( x ) with x ∈ U sε − , where 0 < β < n and p .In the following, let us write U ε − β ( x ) as U ε − β and U ε − β ( x ) as U ε − β forsimplicity. The first term of (3.62) can be estimated as (cid:107) ( ∇ u · v ) ∗ φ ε β (cid:107) W ,n ( U ε − β ) ≤ c (1 + ε − β ) (cid:107) v (cid:107) L n ( U ε − β ) (3.64)where c depends only on φ , n and c . By (3.5), we obtain (cid:107) v (cid:107) L n ( U ε − β ) ≤ (cid:107) v (cid:107) L npn − p ( U ε − β ) { ω n (2 ε − β ) n } p − nnp ≤ λ ε (2 − np )(1 − β ) (2 n ω n ) p − nnp . (3.65)Thus (3.64) and (3.65) show (cid:107) ( ∇ u · v ) ∗ φ ε β (cid:107) W ,n ( U ε − β ) ≤ c ε (2 − np )(1 − β ) − β (3.66)where c depends only on φ, n, p, λ and c . We next consider the secondterm of (3.62). By (3.60), (3.61) and W (cid:48) ( f ) − W (cid:48) ( u ) ∗ φ ε β = ( W (cid:48) ( f ) − W (cid:48) ( u )) + ( W (cid:48) ( u ) − W (cid:48) ( u ) ∗ φ ε β ) , we computesup | W (cid:48) ( f ) − W (cid:48) ( u ) | ≤ sup | W (cid:48)(cid:48) | sup | u − f | ≤ c ε β , (3.67) DIFFUSED INTERFACE WITH THE ADVECTION TERM 18 sup |∇ ( W (cid:48) ( f ) − W (cid:48) ( u )) | ≤ sup | W (cid:48)(cid:48) | sup |∇ f − ∇ u | + sup |∇ u | sup | W (cid:48)(cid:48)(cid:48) | sup | u − f |≤ c ε β (2 − np ) , (3.68)sup | W (cid:48) ( u ) − W (cid:48) ( u ) ∗ φ ε β | ≤ c ε β , (3.69)sup |∇ ( W (cid:48) ( u ) − W (cid:48) ( u ) ∗ φ ε β ) | ≤ c ε β (2 − np ) . (3.70)Here c depends only on φ, n, λ , c and W . Hence, (3.67)-(3.70) show (cid:107) W (cid:48) ( f ) − W (cid:48) ( u ) ∗ φ ε β (cid:107) W ,n ( U ε − β ) ≤ c ε β (2 − np ) − β . (3.71)By (3.62), (3.66) and (3.71), we have (cid:107) g (cid:107) W ,n ( U ε − β ) ≤ c ε (2 − np )(1 − β ) − β + 4 c ε β (2 − np ) − β . (3.72)We use Lemma 3.11 to f and g . Due to Lemma 3.2, we have | f | ≤ | u | ≤ ε η on U ε − β and we may choose smaller η if necessary. Because of (3.1), wehave c = sup U ε − β ( |∇ f | − W ( f )) for a constant depending only on c and W . Then we have all the assumptions for Lemma 3.11 and obtainsup U ε − β (cid:16) |∇ f | − W ( f ) (cid:17) ≤ c ( ε (2 − np )(1 − β ) − β − β + ε β (2 − np ) − β + ε η ) . (3.73)At this point, we fix sufficiently small 0 < β , β < n and p such that(2 − np )(1 − β ) − β − β > , β (2 − np ) − β > . This shows that we may choose a 0 < β < U ε − β (cid:18) |∇ f | − W ( f ) (cid:19) ≤ ε − β (3.74)for all sufficiently small ε >
0. We may take the center of U ε − β (= U ε − β ( x ))to be any x ∈ U sε − so that we have the estimate on U sε − . By (3.61), (3.74)and sup | W ( f ) − W ( u ) | ≤ sup | W (cid:48) | sup | u − f | ≤ c ε β , we may also replace f by u in (3.74) by choosing a larger 0 < β < (cid:3) Rectifiability and integrality of the limit varifold
In this section, we recover the index i and assume that { u i } ∞ i =1 , { v i } ∞ i =1 and { ε i } ni =1 satisfy (2.1)-(2.5). Define µ i and V i as in (2.7) and (2.8). Bythe standard weak compactness theorem of Radon measures, there exists asubsequence (denoted by the same index) and a Radon measure µ and avarifold V such that µ i → µ, V i → V. Lemma 4.1.
For x ∈ spt µ , there exists a subsequence x i ∈ U such that u i ( x i ) ∈ [ − α, α ] and lim i →∞ x i = x . DIFFUSED INTERFACE WITH THE ADVECTION TERM 19
Proof.
We prove this by contradiction and assume that there exists some r > | u i | ≥ α on U r ( x ) for all large i . Without loss of general-ity, assume u i ≥ α on U r ( x ). Then we repeat the same argument leadingto (3.30) with φ there replaced by C c ( U r ( x )). The argument shows thatlim i →∞ (cid:82) ε i |∇ u i | φ = 0. Next, multiplying u i − W (cid:48) ( u i )( u i − ≥ κ ( u i − , we obtain (cid:90) Wε i φ ≤ c ( W ) (cid:90) ( u i − ε i φ ≤ c ( W ) κ (cid:90) W (cid:48) ( u i )( u i − ε i φ = 2 c ( W ) κ (cid:90) ( ε i ∆ u i ( u i −
1) + ε i v i · ∇ u i ) φ . (4.1)By integration by parts and (2.5), the right-hand side of (4.1) converges to0. This shows that µ ( U r ( x )) = 0 and contradicts x ∈ spt µ . (cid:3) Theorem 4.2.
There exist constants < D ≤ D < ∞ which depend onlyon c , λ , n, p, W, E and s such that, for x ∈ spt µ ∩ U s and B r ( x ) ⊂ U s , wehave D r n − ≤ µ ( B r ( x )) ≤ D r n − . (4.2) Proof.
This follows immediately from Theorem 3.8, 3.10 and Lemma 4.1. (cid:3)
For the subsequent use, define ξ i := ε i |∇ u i | − W ( u i ) ε i . Once we have the monotonicity formula (3.55), we may prove the following“equi-partition of energy” by the same proof as in [20, Proposition 4.3]:
Theorem 4.3. ξ i , ε i |∇ u i | − |∇ w i | and W ( u i ) ε i − |∇ w i | all converge to zeroin L loc ( U ) .Proof of Theorem 2.1. Recall that (cid:107) V i (cid:107) = µ i {|∇ u i |(cid:54) =0 } . Since σµ i {|∇ u i | =0 } ≤ | ξ i | d L n → µ i {|∇ u i |(cid:54) =0 } converges to µ . We also know that (cid:107) V i (cid:107) con-verges to (cid:107) V (cid:107) by definition, thus we have (cid:107) V (cid:107) = µ . This proves (1). Theclaims (2) and (3) follows from Theorem 3.8, 3.10 and Lemma 4.1. Next,by (2.9), (3.18) and Theorem 4.3, δV i ( g ) = (cid:90) {|∇ u i |(cid:54) =0 } div g dµ i − σ (cid:90) {|∇ u i |(cid:54) =0 } ∇ g · (cid:16) ∇ u i |∇ u i | ⊗ ∇ u i |∇ u i | (cid:17) ( ε i |∇ u i | − ξ i )= − σ (cid:90) ε i ( v i · ∇ u i )( ∇ u i · g ) + o (1) (4.3)for g ∈ C c ( U ; R n ), where lim i →∞ o (1) = 0. By Theorem 3.8 and spt g ⊂ U s for some 0 < s <
1, we have a uniform bound on E ( r, x ) (corresponding to DIFFUSED INTERFACE WITH THE ADVECTION TERM 20 u i ) for B r ( x ) ⊂ U s . Hence, by Theorem 3.7, we have (cid:90) ε i (( v i − v ) · ∇ u i )( ∇ u i · g ) ≤ c (cid:18)(cid:90) | v i − v | p | g | p ε i |∇ u i | (cid:19) p ≤ c (cid:18)(cid:90) |∇ ( | v i − v | p | g | p ) | (cid:19) p ≤ c (cid:16) (cid:107)∇ v i − ∇ v (cid:107) L p (cid:107) v i − v (cid:107) p − L p + (cid:107) v i − v (cid:107) pL p (cid:17) p (4.4)where the integrations are over spt g . The above converges to 0 since wemay choose a further subsequence of v i which converges to v strongly in L ploc . Thus in the right-hand side of (4.3), we may replace v i by v . Let (cid:15) > v be a smooth vector field such that (cid:107) v − ˜ v (cid:107) W ,p ( U s ) < (cid:15) .By the varifold convergence V i → V , we have1 σ (cid:90) ε i (˜ v · ∇ u i )( ∇ u i · g ) = (cid:90) S ⊥ (˜ v ) · g dV i ( x, S ) + o (1) → (cid:90) S ⊥ (˜ v ) · g dV ( x, S ) . (4.5)We may arbitrarily approximate the quantities in (4.5) by v by the sameargument in (4.4), hence by (4.3)-(4.5), we obtain δV ( g ) = − (cid:90) S ⊥ ( v ) · g dV ( x, S ) . (4.6)Hence, (cid:107) δV (cid:107) is a Radon measure on U . By (4.2) and Allard’s rectifiabilitytheorem [1, 5.5.(1)], V is rectifiable. Next from (4.6), (cid:107) δV (cid:107) is absolutelycontinuous with respect to (cid:107) V (cid:107) and H V ( x ) = S ⊥ ( v ( x )) holds for V a.e. for( x, S ). This proves (5). The proof of (4) is the same as [23] for the followingreason. We may set f = ε ∇ u · v in [23] and we have (cid:107) ε ∇ u · v (cid:107) L npn − p ( U s ) ≤ c λ due to Lemma 3.1. In the proof, as long as we have the monotonicityformula (3.55) and the estimate Lemma 3.4, all the argument goes through.The point is that we do not need to take a derivative of f for the proof ofintegrality and we only need the control of L npn − p norm as well as the estimate(3.2). Finally, by arguing as in (4.4) and the H¨older inequality, we have (cid:90) U s φ p ( n − n − p d (cid:107) V (cid:107) ≤ c (cid:107)∇ φ (cid:107) L p (cid:107) φ (cid:107) n ( p − / ( n − p ) L np/ ( n − p ) for any function φ ∈ C c ( U s ; R + ) and we have the same inequality for v ∈ W ,p ( U ) by the density argument. Thus we have (6). (cid:3) Concluding remarks
In [9, 19], we studied the singular perturbation problem for ∂ t u ε + v ε · ∇ u ε = ∆ u ε − W (cid:48) ( u ε ) ε (5.1)and proved that the time-parametrized family of limit varifolds satisfies themotion law of “normal velocity = mean curvavture vector + v ⊥ ” in a weakformulation (see [7, 21] for the case of v ε = 0). In these works, we assumedthat the prescribed initial data satisfies a boundedness of the upper density DIFFUSED INTERFACE WITH THE ADVECTION TERM 21 ratio. Part of the difficulty was to show that the upper density ratio boundcan be controlled locally in time and uniformly with respect to ε . For theequilibrium problem, it is certainly not natural to assume such an upperdensity ratio estimate. It is interesting to see if one can drop the upperdensity ratio assumption for the initial data in the proof of [9, 19].The vectorial prescribed mean curvature problem as in Theorem 2.2 seems,as far as we know, little studied so far. Traditionally, the prescription is thescalar version, i.e. given a scalar function (or constant) f , one looks for ahypersurface satisfying H · ν = f , where ν is the normal unit vector. Thevectorial version is physically natural from the view point of force balance,in that the problem seeks the equality between the surface tension force andan external force acting on the surface. It must be said that the prescribedvector field in Theorem 2.2 is the gradient of a potential ρ , and not a generalvector field. This is rather restrictive for applications and it is interestingto know if there can be a remedy for generalizations. If there may not exista variational framework such as the min-max method to find solutions of(1.2), it should be still useful to have this diffused interface approach sincethe functional is well-behaved functional-analytically. As a further question,it is also interesting to investigate the asymptotic behavior of stable criticalpoints of F ε in the proof of Theorem 2.2, since we have a very successfulanalogy in [22, 24].Though we did not attempt to do so in this paper, we expect that all theanalysis and main results can be transplanted to the setting of general Rie-mannian manifold with smooth metric, since the analysis is local in nature.One would need to treat extra error terms coming from the metric whichcan be controlled. References [1] W. Allard,
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Department of Mathematics, Tokyo Institute of Technology, 152-8551,Tokyo, Japan
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