A diffused interface with the advection term in a Sobolev space

We study the asymptotic limit of diffused surface energy in the van der Waals--Cahn--Hillard theory when an advection term is added and the energy is uniformly bounded. We prove that the limit interface is an integral varifold and the generalized mean curvature vector is determined by the advection term. As the application, a prescribed mean curvature problem is solved using the min-max method.


Introduction
The object of study in this paper is the energy functional appearing in the van der Waals-Cahn-Hillard theory [2,5], where u : Ω ⊂ R n → R (n ≥ 2) is the normalized density distribution of two phases of a material, |∇u| 2 = n k=1 (∂u/∂x k ) 2 and W : R → [0, ∞) is a double-well potential with two global minima at ±1. In the thermodynamic context, W corresponds to the Helmholtz free energy density and the typical example is W (u) = (1 − u 2 ) 2 . When the positive parameter ε is small relative to the size of the domain Ω and E ε (u) is bounded, it is expected that u is close to +1 or −1 on most of Ω while a spatial change between ±1 occurs within a hypersurface-like region of O(ε) thickness which we may call the diffused interface of u. In this case, the quantity E ε (u) is expected to be proportional to the surface area of the diffused interface. Due to the importance of the surface area in calculus of variations, it is interesting to investigate the validity of such expectation and other salient properties of E ε .
In this direction, there have been a number of works studying the asymptotic behavior of E ε as ε → 0+ under various assumptions. For the energy minimizers with appropriate side conditions, it is well-known that it Γ-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, Hutchinson and 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 hypersurface except for a closed singular set of codimension seven. More recently, Guaraco [4] showed that a uniform Morse index bound is sufficient to conclude the same regularity for n ≥ 3 and gave a new proof of Almgren-Pitts theorem [14] as the application. The new proof significantly simplifies the existence part of the proof even though one needs to use Wickramasekera's hard regularity theorem [25].
While the investigations on the critical points of E ε have direct links to the minimal surface theory as above, more generally, it turned out that suitable controls of the first variation of E ε guarantee the analogous good asymptotic behaviors. For example, under the assumption that lim inf ε→0+ E ε (u ε ) + f ε W 1,p (Ω) < ∞ with f ε := −ε∆u ε + W (u ε )/ε and p > n/2, the first author [20,23] proved that the limit interface is an integral varifold whose generalized mean curvature belongs to L q (q = p(n − 1)/(n − p) > n − 1) with respect to the surface measure. Here W 1,p (Ω) := {u ∈ L p (Ω) : ∇u ∈ L p (Ω)}. The mean curvature of the limit interface is characterized by the weak W 1,p limit of f ε [16]. Another example concerns one of De Giorgi's conjectures. Under the assumption that (with f ε as above) lim inf ε→0+ E ε (u ε ) + ε −1 f ε 2 L 2 (Ω) < ∞ and n = 2, 3, Röger-Schätzle [15] (independently [13] for the case of n = 2) proved the similar result. In this case, the limit interface has an L 2 generalized mean curvature.
In this paper, along the line of research described above, we investigate the asymptotic behavior of u ε satisfying where v ε is considered here as a given vector field and we assume that and p > n/2. The problem is related to (parabolic) Allen-Cahn-type equations studied in [9,19], for example. It is also natural to investigate the effect of advection term as ε → 0+. We prove the analogous result Theorem 2.1 to [20,23], namely, the limit is an integral varifold with L q (the same as above) generalized mean curvature which is characterized by the weak W 1,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 the setting of the min-max method, with minimal regularity assumptions on the prescribed vector field. As for the proof, just as in the case of [6,20,23], the key point is to prove a certain monotonicity-type formula which is the essential tool in the setting of Geometric Measure Theory. We wish to treat εv ε · ∇u ε as a perturbative term, and to do so, we need to control a certain "trace" norm of v ε on diffused interface. If an ε-independent upper density ratio estimate of diffused surface measure is available, then we can control εv ε · ∇u ε by the W 1,p -norm of v ε . For this purpose, we establish the key estimate, Theorem 3.8, which gives a local uniform upper density ratio estimate. Once this part is done, the rest proceeds just like [23] with minor modifications.
The paper is organized as follows. In Section 2 we state our assumptions and explain the main results. Section 3 contains the main estimates which ultimately give a monotonicity-type formula, Theorem 3.9. In Section 4, we prove the main theorem by modifying the proof in [20,23], and in Section 5, we give some concluding remarks.
Let Ω ⊂ R n be a bounded domain. We assume that we are given weakly on Ω for each i ∈ N. In addition, assume that and that there exist constants c 0 , E 0 and λ 0 such that, for all i ∈ N, we have: The condition (2.3) is not essential and can be often derived from the PDE or the proof of existence. Here we assume (2.3) for simplicity. Next, define By the Cauchy-Schwarz inequality and (2.4), we obtain 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 1 norm and the limit BV function w. Define where Φ −1 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 This shows that u = ±1 a.e. on Ω and u is a BV function. For simplicity we write ∂ * {u = 1} as the reduced boundary [26] of {u = 1} and ∂ * {u = 1} as the boundary measure.
2.2. The associated varifolds. We associate to each solution of (1.2) a varifold in a natural way in the following. We refer to [1,17] for a comprehensive treatment of varifolds. Let G(n, n − 1) be the Grassmannian, i.e. the space of unoriented (n − 1)dimensional subspaces in 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 given square-matrices S 1 and S 2 , we write S 1 · S 2 := trace(S t 1 • S 2 ), where the upper-script t indicates the transpose of the matrix and • is the matrix multiplication. We say that V is an (n − 1)-dimensional varifold in Ω ⊂ R n if V is a Radon measure on G n−1 (Ω) := Ω × G(n, n − 1). Let V n−1 (Ω) be the set of all (n − 1)-dimensional varifolds in Ω. Convergence in the varifold sense means convergence in the usual sense of measures. For V ∈ V n−1 (Ω), we let V be the weight measure of V . For V ∈ V n−1 (Ω), we define the first variation of V by for any vector field g ∈ C 1 c (Ω; R n ). We let δV be the total variation of δV . If δV is absolutely continuous with respect to V , then the Radon-Nikodym derivative δV / V exists as a vector-valued V measurable function. In this case, we define the generalized mean curvature vector of V by −δV / V and we use the notation H V .
We associate to each function u i a varifold V i as follows. First, we define a Radon measure µ i on Ω by where L n is the n-dimensional Lebesgue measure and σ := for φ ∈ C c (G n−1 (Ω)), where I is the n × n identity matrix and ⊗ is the tensor product of the two vectors. Note that |∇u i (x)| represents the orthogonal projection to the (n − 1)-dimensional subspace {a ∈ R n : a · ∇u i (x) = 0}. By definition, we have and by (2.6), we have for each g ∈ C 1 c (Ω, R n ). 2.3. Main Theorems. With the above assumptions and notation, we show: ) and let V i be the varifold associated with u i as in (2.8). On passing to a subsequence we can assume that v i → v weakly in W 1,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), for (x, S) ∈ G n−1 (Ω) for V a.e., where S ⊥ ∈ G(n, 1) is the projection to the orthogonal complement of S, i.e., S ⊥ = I − S. (6) ForΩ ⊂⊂ Ω, there exists a constant λ 1 depending only on c 0 ,λ 0 ,n,p,W , Since V is integral and the generalized mean curvature vector is in the stated class, V satisfies various good properties described in [17,Section 17]. In particular, there exists a closed countably (n − 1)-rectifiable set Γ ⊂ Ω (which we can take as the support of V , see [17, Section 17.9(1)]) such that, for any φ ∈ C c (G n−1 (Ω)), Here, T x Γ ∈ G(n, n − 1) is the approximate tangent space of Γ at x which exists H n−1 a.e. x ∈ Γ. With this notation, (5) x ∈ Γ, i.e., the generalized mean curvature vector of V coincides with the projection of v to the orthogonal subspace (T x Γ) ⊥ for H n−1 a.e. x ∈ Γ. If we additionally assume that θ = 1 for H n−1 a.e. x ∈ Γ, then because of the integrability of H V and the Allard regularity theorem [1], except for a closed H n−1 -null set, Γ is locally a C 1,2− n p hypersurface. Without the assumption θ = 1, we can still conclude that spt V is C 1,2− n p hypersurface on a dense open set of spt V , even though we do not know if the complement is H n−1 -null or not.

2.4.
A vectorial prescribed mean curvature problem. As an application 1 of Theorem 2.1, we prove the following: Let Ω ⊂ R n be a bounded domain with Lipschitz boundary and let ρ ∈ W 2,p (Ω) be a given function, where p > n 2 . Then, there Proof. We may assume p < n. Consider the following functional for ε > 0 and u ∈ W 1,2 (Ω): By the Sobolev embedding, ρ ∈ C 0,2− n p (Ω) and thus 0 < exp(min ρ) ≤ exp(ρ) ≤ exp(max ρ) < ∞. By considering the path space in W 1,2 (Ω) connecting u ≡ 1 and u ≡ −1, the standard min-max method gives a nontrivial critical point u ε for each ε > 0, with uniform strictly positive lower and upper bounds of F ε (u ε ) (see for example [4] for the detail). The critical point satisfies (2.1) with v = ∇ρ and |u ε | ≤ 1. Take a sequence ε i → 0+ and a corresponding min-max critical points u i . Then the sequence u i , ∇ρ, ε i satisfy all the assumptions of Theorem 2.1. The limit varifold V thus has the desired property.
For more remarks on the main results, see Section 5.

The estimate for the upper density ratio
In this section, we prove Theorem 3.8-3.10, which give ε-independent estimates of the upper and lower density ratios of the energy. Throughout this section, we drop the index i and set Ω = U 1 = {|x| < 1} since the result is local. Assume u ∈ W 1,2 (U 1 ) and v ∈ W 1,p (U 1 ; R n ) satisfy (2.1) with a positive ε and (2.3)-(2.5) are satisfied for a given set of c 0 , E 0 , λ 0 . The exponent p satisfies (2.2). We first derive two preliminary properties for u, Lemma 3.1 and 3.2.
Lemma 3.1. There exists c 1 > 0 depending only on c 0 , λ 0 , n, p and W such that sup and sup x, The authors thank Nick Edelen for a discussion which inspired this application.
for 0 < ε < 1/2. If ε ≥ 1/2, then we have for any 0 < s < 1 where c 1 depends additionally on s. In both cases, we have Under the change of variables, we obtain from (2.5) Since np n−p > 2, (3.5) and (3.6) give We next note that the functionũφ weakly satisfies the following equation: Using the standard L p theory [3, Theorem 9.11] to (3.8), we may start a bootstrapping argument as follows. Staring with q = 2, we have with the corresponding estimates relating these norms. Note that the exponent of integrability of ∇ũ is raised from q to q · np np−q(2p−n) , with the factor strictly larger than one. Thus, in a finite number of bootstrapping, we obtain the W 2,s loc (with s > n) estimate forũ, and by the Sobolev inequality, the L ∞ loc estimate for ∇ũ. Again by the L p theory, we obtain the W 2, np n−p loc estimate ofũ. In particular, by the Sobolev inequality, we obtain (3.1) and (3.2). Since the right-hand side of (3.8) is in W loc and the weak third-derivatives ofũ exist. The case of ε ≥ 1/2 does not require the change of variables as above and the proof is omitted. Lemma 3.2. Given 0 < s < 1, there exist constants 0 < ε 1 , η < 1 depending only on c 0 , λ 0 , W, n, p and s such that To derive a contradiction, assume that u(x 0 ) − 1 ≥ ε η for some x 0 ∈ B s . By (3.1), for y ∈ B ε 1+η 2c 1 (3.14) Then we have This is a contradiction if η and ε are sufficiently small. u ≥ −1−ε η is proved similarly.
The next Lemma 3.3 is the starting point of the ultimate establishment of the monotonicity formula. (3.17) Proof. Multiply both sides of (2.1) by ∇u · g, where g = (g 1 , · · · , g n ) ∈ C 1 c (U 1 ; R n ). By integration by parts, we obtain We assume that x = 0 after a suitable translation and let g j (y) = y j ρ(|y|).
Writing r = |y|, (3.18) becomes We choose ρ which is a smooth approximation of χ Br , the characteristic function of B r , and then we take a limit ρ → χ Br . Then we have By dividing the above equation by r n , the lemma follows.
We need the following lemma to control the negative contribution of the right-hand side of (3.17).
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−1 ε β 3 and (3.19) as well as Here we only consider the estimate on C and refer the reader to the proof of [23,Proposition 3.4]. Define φ is a Lipschitz function and is 0 on {|y| > r} ∪ {|u| < α}, 1 on C and |∇φ| ≤ ε −β 3 . Differentiate (2.1) with respect to x j , multiply it by u x j φ 2 and sum over j. Then we have By integration by parts, the Cauchy-Schwarz inequality and (3.1), we obtain where c 7 depends on c 1 . Since |u| ≥ α on the support of φ, we have W ≥ κ.
Proof of Lemma 3.4. As in the proof of Lemma 3.1, defineũ(x) := u(εx), v(x) := εv(εx), and subsequently drop· for simplicity. We have −∆u + W (u) = ∇u · v on U ε −1 . With respect to the new variables, we need to prove for some 0 < β 1 < 1 for all sufficiently small ε. Let φ λ be the standard mollifier, namely, define where the constant C > 0 is selected so that R n φ = 1, and define φ λ (x) := 1 λ n φ( x λ ). For 0 < β 5 < 1 to be chosen depending only on n and p later, (3.59) By (3.1) and (3.2), we have We next define g to be To use Lemma 3.11, we next estimate the W 1,n norm of g on U ε −β 4 (x) with x ∈ U sε −1 , where 0 < β 4 < 1 will be chosen depending only on n and p.
In the following, let us write U ε −β 4 (x) as U ε −β 4 and U ε −β 4 2 (x) as U ε −β 4 2 for simplicity. The first term of (3.62) can be estimated as where c 15 depends only on φ, n and c 1 . By (3.5), we obtain where c 16 depends only on φ, n, p, λ 0 and c 1 . We next consider the second term of (3.62). By (3.60), (3.61) and we compute

67)
Proof. We prove this by contradiction and assume that there exists some r > 0 such that |u i | ≥ α on U r (x) for all large i. Without loss of generality, assume u i ≥ α on U r (x). Then we repeat the same argument leading to (3.30) with φ there replaced by C 1 c (U r (x)). The argument shows that lim i→∞ ε i 2 |∇u i | 2 φ 2 = 0. Next, multiplying u i − 1 to the equation (2.1) and (4.1) By integration by parts and (2.5), the right-hand side of (4.1) converges to 0. This shows that µ(U r (x)) = 0 and contradicts x ∈ spt µ.
Theorem 4.2. There exist constants 0 < D 1 ≤ D 2 < ∞ which depend only on c 0 , λ 0 , n, p, W, E 0 and s such that, for x ∈ spt µ ∩ U s and B r (x) ⊂ U s , we have Proof. This follows immediately from Theorem 3.8, 3.10 and Lemma 4.1.
For the subsequent use, define 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]: We also know that V i converges to V by definition, thus we have V = µ. This proves (1). The claims (2) and (3) for g ∈ C 1 c (U 1 ; 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 u i ) for B r (x) ⊂ U s . Hence, by Theorem 3.7, we have where the integrations are over spt g. The above converges to 0 since we may choose a further subsequence of v i which converges to v strongly in L p loc . Thus in the right-hand side of (4.3), we may replace v i by v. Let > 0 be arbitrary and letṽ be a smooth vector field such that v −ṽ W 1,p (Us) < .  Hence, δV is a Radon measure on U 1 . By (4.2) and Allard's rectifiability theorem [1, 5.5. (1)], V is rectifiable. Next from (4.6), δV is absolutely continuous with respect to V 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 following reason. We may set f = ε∇u·v in [23] and we have ε∇u·v ≤ c 1 λ 0 due to Lemma 3.1. In the proof, as long as we have the monotonicity formula (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 of integrality and we only need the control of L np n−p norm as well as the estimate (3.2). Finally, by arguing as in (4.4) and the Hölder inequality, we have for any function φ ∈ C 1 c (U s ; R + ) and we have the same inequality for v ∈ W 1,p (U 1 ) by the density argument. Thus we have (6).

Concluding remarks
In [9,19], we studied the singular perturbation problem for ∂ t u ε + v ε · ∇u ε = ∆u ε − W (u ε ) ε 2 (5.1) and proved that the time-parametrized family of limit varifolds satisfies the motion law of "normal velocity = mean curvavture vector + v ⊥ " in a weak formulation (see [7,21] for the case of v ε = 0). In these works, we assumed that the prescribed initial data satisfies a boundedness of the upper density ratio. Part of the difficulty was to show that the upper density ratio bound can be controlled locally in time and uniformly with respect to ε. For the equilibrium problem, it is certainly not natural to assume such an upper density ratio estimate. It is interesting to see if one can drop the upper density 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 the scalar version, i.e. given a scalar function (or constant) f , one looks for a hypersurface satisfying H · ν = f , where ν is the normal unit vector. The vectorial version is physically natural from the view point of force balance, in that the problem seeks the equality between the surface tension force and an external force acting on the surface. It must be said that the prescribed vector field in Theorem 2.2 is the gradient of a potential ρ, and not a general vector field. This is rather restrictive for applications and it is interesting to know if there can be a remedy for generalizations. If there may not exist a 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 since the functional is well-behaved functional-analytically. As a further question, it is also interesting to investigate the asymptotic behavior of stable critical points of F ε in the proof of Theorem 2.2, since we have a very successful analogy in [22,24].
Though we did not attempt to do so in this paper, we expect that all the analysis and main results can be transplanted to the setting of general Riemannian manifold with smooth metric, since the analysis is local in nature.
One would need to treat extra error terms coming from the metric which can be controlled.