Abstract
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.
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1 Introduction
The object of study in this paper is the energy functional appearing in the van der Waals–Cahn–Hillard theory [4, 7],
where \(u:\Omega \subset \mathbb {R}^n \rightarrow \mathbb {R}\) (\(n \ge 2\)) is the normalized density distribution of two phases of a material, \(|\nabla u|^2=\sum _{k=1}^n(\partial u/\partial x_k)^2\) and \(W:\mathbb {R} \rightarrow [0,\infty )\) is a double-well potential with two global minima at \(\pm 1\). 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 \(\varepsilon \) is small relative to the size of the domain \(\Omega \) and \(E_{\varepsilon }(u)\) is bounded, it is expected that u is close to \(+1\) or \(-1\) on most of \(\Omega \) while a spatial change between \(\pm 1\) occurs within a hypersurface-like region of \(O(\varepsilon )\) thickness which we may call the diffused interface of u. In this case, the quantity \(E_{\varepsilon }(u)\) 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 \(E_{\varepsilon }\).
In this direction, there have been a number of works studying the asymptotic behavior of \(E_{\varepsilon }\) as \(\varepsilon \rightarrow 0+\) under various assumptions. For the energy minimizers with appropriate side conditions, it is well-known that it \(\Gamma \)-converges to the area functional of the limit interface [10, 12,13,14, 20]. 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 \(E_{\varepsilon }\) different from the energy minimizers. For general critical points, Hutchinson and the first author [8] proved that the limit is an integral stationary varifold [1]. For general stable critical points, the first author and Wickramasekera [26] proved that the limit is an embedded real-analytic minimal hypersurface except for a closed singular set of codimension seven. More recently, Guaraco [6] showed that a uniform Morse index bound is sufficient to conclude the same regularity for \(n\ge 3\) and gave a new proof of Almgren–Pitts theorem [16] 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 [27].
While the investigations on the critical points of \(E_\varepsilon \) have direct links to the minimal surface theory as above, more generally, it turned out that suitable controls of the first variation of \(E_{\varepsilon }\) guarantee the analogous good asymptotic behaviors. For example, under the assumption that
with \(f_{\varepsilon }:=-\varepsilon \Delta u_\varepsilon +W'(u_{\varepsilon })/\varepsilon \) and \(p>n/2\), the first author [22, 25] 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 \(W^{1,p}(\Omega ):=\{u\in L^p(\Omega )\,:\, \nabla u\in L^p(\Omega )\}\). The mean curvature of the limit interface is characterized by the weak \(W^{1,p}\) limit of \(f_{\varepsilon }\) [18]. Another example concerns one of De Giorgi’s conjectures. Under the assumption that (with \(f_\varepsilon \) as above)
and \(n=2,3\), Röger–Schätzle [17] (independently [15] 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 \(u_{\varepsilon }\) satisfying
where \(v_{\varepsilon }\) 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 [11, 21], for example. It is also natural to investigate the effect of advection term as \(\varepsilon \rightarrow 0+\). We prove the analogous result Theorem 2.1 to [22, 25], 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 \(v_{\varepsilon }\). 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 existence problem for scalar constant or prescribed mean curvature using a min–max approach along the lines of Almgren–Pitts [16], we mention papers by Zhou and Zhu [28, 29].
As for the proof, just as in the case of [8, 22, 25], 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 \(\varepsilon v_\varepsilon \cdot \nabla u_\varepsilon \) as a perturbative term, and to do so, we need to control a certain “trace” norm of \(v_\varepsilon \) on diffused interface. If an \(\varepsilon \)-independent upper density ratio estimate of diffused surface measure is available, then we can control \(\varepsilon v_\varepsilon \cdot \nabla u_\varepsilon \) by the \(W^{1,p}\)-norm of \(v_\varepsilon \). 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 [25] with minor modifications.
The paper is organized as follows. In Sect. 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 Sect. 4, we prove the main theorem by modifying the proof in [22, 25], and in Sect. 5, we give some concluding remarks.
2 Assumptions and main results
We use the notation that \(U_r(a):=\{x\in \mathbb {R}^n\,:\, |x-a|<r\}\), \(B_r(a):=\{x\in \mathbb {R}^n\,:\, |x-a|\le r\}\), \(U_r:=U_r(0)\) and \(B_r:=B_r(0)\).
2.1 Assumptions
Throughout the paper, we assume that:
-
(a)
The function \(W: \mathbb {R} \rightarrow [0,\infty )\) is \(C^3\) and has two strict minima \(W(\pm 1)=W'(\pm 1)=0\).
-
(b)
For some \(\gamma \in (-1,1)\), \(W'>0\) on \((-1,\gamma )\) and \(W'<0\) on \((\gamma ,1)\).
-
(c)
For some \(\alpha \in (0,1)\) and \(\kappa >0\), \(W''(x)\ge \kappa \) for all \(|x| \ge \alpha \).
Let \(\Omega \subset \mathbb {R}^n\) be a bounded domain. We assume that we are given \(W^{1,2}(\Omega )\) functions \(\{u_i\}^\infty _{i=1}\), \(W^{1,p}(\Omega ; \mathbb {R}^n)\) vector fields \(\{v_i\}^\infty _{i=1}\) and positive constants \(\{\varepsilon _i\}^\infty _{i=1}\) satisfying
weakly on \(\Omega \) for each \(i\in {\mathbb {N}}\). In addition, assume that
and that there exist constants \(c_{0}\), \(E_0\) and \(\lambda _0\) such that, for all \(i\in {\mathbb {N}}\), 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 [30, 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 \(\Phi ^{-1}\) is the inverse function of \(\Phi \). It follows that \(u_i\) converges to u a.e. on \(\Omega \). By Fatou’s Lemma and (2.4), we have
This shows that \(u= \pm 1\) a.e. on \(\Omega \) and u is a BV function. For simplicity we write \(\partial ^* \{u=1\}\) as the reduced boundary [30] of \(\{u=1\}\) and \(\Vert \partial ^*\{u=1\}\Vert \) 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, 19] for a comprehensive treatment of varifolds.
Let \(\mathbf{G}(n,n-1)\) be the Grassmannian, i.e. the space of unoriented \((n-1)\)-dimensional subspaces in \(\mathbb {R}^n\). We also regard \(S \in \mathbf{G}(n,n-1)\) as the \(n\times n\) matrix representing the orthogonal projection of \(\mathbb {R}^n\) onto S. For two given square-matrices \(S_1\) and \(S_2\), we write \(S_1\cdot S_2:=\mathrm{trace}(S^t_1 \circ S_2)\), where the upper-script t indicates the transpose of the matrix and \(\circ \) is the matrix multiplication. We say that V is an \((n-1)\)-dimensional varifold in \(\Omega \subset \mathbb {R}^n\) if V is a Radon measure on \(\mathbf{G}_{n-1}(\Omega ):=\Omega \times \mathbf{G}(n,n-1)\). Let \(\mathbf{V}_{n-1}(\Omega )\) be the set of all \((n-1)\)-dimensional varifolds in \(\Omega \). Convergence in the varifold sense means convergence in the usual sense of measures. For \(V \in \mathbf{V}_{n-1}(\Omega )\), we let \(\Vert V\Vert \) be the weight measure of V. For \(V \in \mathbf{V}_{n-1}(\Omega )\), we define the first variation of V by
for any vector field \(g \in C^1_c(\Omega ;\mathbb {R}^n)\). We let \(\Vert \delta V\Vert \) be the total variation of \(\delta V\). If \(\Vert \delta V\Vert \) is absolutely continuous with respect to \(\Vert V\Vert \), then the Radon–Nikodym derivative \(\delta V/\Vert V\Vert \) exists as a vector-valued \(\Vert V\Vert \) measurable function. In this case, we define the generalized mean curvature vector of V by \(-\delta V/\Vert V\Vert \) 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 \(\mu _{i}\) on \(\Omega \) by
where \({\mathcal {L}}^n\) is the n-dimensional Lebesgue measure and \(\sigma :=\int _{-1}^1 \sqrt{2W(s)}\,ds\). Define \(V_i \in \mathbf{V}_{n-1}(\Omega )\) by
for \(\phi \in C_c (\mathbf{G}_{n-1}(\Omega ))\), where I is the \(n \times n\) identity matrix and \(\otimes \) is the tensor product of the two vectors. Note that \(I-\frac{\nabla u_i(x)}{|\nabla u_i(x)|}\otimes \frac{\nabla u_i(x)}{|\nabla u_i(x)|}\) represents the orthogonal projection to the \((n-1)\)-dimensional subspace \(\{ a\in {\mathbb {R}}^n\,:\, a\cdot \nabla u_i(x)=0\}\). By definition, we have
and by (2.6), we have
for each \(g\in C^1_c(\Omega ,\mathbb {R}^n)\).
2.3 Main Theorems
With the above assumptions and notation, we show:
Theorem 2.1
Suppose that \(u_i,v_i,\varepsilon _i\) satisfy (2.1)–(2.5) and let \(V_i\) be the varifold associated with \(u_i\) as in (2.8). On passing to a subsequence we can assume that
Then we have the following properties.
-
(1)
For each \(\phi \in C_c(\Omega )\),
$$\begin{aligned} \frac{1}{2}\Vert V\Vert (\phi )= & {} \lim _{i \rightarrow \infty } \frac{1}{\sigma }\int _{\Omega }\frac{\varepsilon _i}{2}|\nabla u_i|^2 \phi = \lim _{i \rightarrow \infty } \frac{1}{\sigma }\int _{\Omega } \frac{W(u_i)}{\varepsilon _i} \phi \\= & {} \lim _{i \rightarrow \infty } \frac{1}{\sigma }\int _\Omega |\nabla w_i| \phi . \end{aligned}$$ -
(2)
\(\mathrm{spt}\,\Vert \partial ^* \{u =1\}\Vert \subset \mathrm{spt}\,\Vert V\Vert \) and \(\{u_i\}\) converges locally uniformly to \(\pm 1\) on \(\Omega \setminus \mathrm{spt}\,\Vert V\Vert \).
-
(3)
For each \(0<b<1\), \(\{|u_i|\le 1-b \}\) locally converges to \(\mathrm{spt}\,\Vert V\Vert \) in the Hausdorff distance sense in \(\Omega \).
-
(4)
V is an integral varifold and the density \(\theta (x)\) of V satisfies
$$\begin{aligned} \theta (x)= {\left\{ \begin{array}{ll} \hbox {odd} &{} \mathcal {H}^{n-1}\,\, a.e. \ x\in \partial ^*\{u=1\}, \\ \hbox {even} &{} \mathcal {H}^{n-1}\,\, a.e. \ x\in \mathrm{spt}\,\Vert V\Vert \backslash \partial ^*\{u=1\}, \end{array}\right. } \end{aligned}$$ -
(5)
the generalized mean curvature vector \(H_V\) of V is given by
$$\begin{aligned} H_V(x)=(T_x\,\mathrm{spt}\,\Vert V\Vert )^{\perp }(v(x)), \end{aligned}$$for \(\Vert V\Vert \) a.e. \(x\in \Omega \).
-
(6)
For \({\tilde{\Omega }}\subset \subset \Omega \), there exists a constant \(\lambda _1\) depending only on \(c_0\),\(\lambda _0\),n,p,W, \(E_0\) and \(\mathrm{dist}({\tilde{\Omega }},\,\partial \Omega )\) such that
$$\begin{aligned} \int _{{\tilde{\Omega }}}|H_V(x)|^{\frac{p(n-1)}{n-p}}\,d\Vert V\Vert (x)\le \int _{{\tilde{\Omega }}}|v(x)|^{\frac{p(n-1)}{n-p}} \,d\Vert V\Vert (x)\le \lambda _1. \end{aligned}$$Note that \(\frac{p(n-1)}{n-p}>n-1\) due to (2.2).
Since V is integral and the generalized mean curvature vector is in the stated class, V satisfies various good properties described in [19, Section 17]. In particular, \(\mathrm{spt}\,\Vert V\Vert \) is a closed countably \((n-1)\)-rectifiable set (see [19, 17.9(1)]), and writing \(\Gamma := \mathrm{spt}\,\Vert V\Vert \), for any \(\phi \in C_c(\mathbf{G}_{n-1}(\Omega ))\),
Here, \(T_x\,\Gamma \in \mathbf{G}(n,n-1)\) is the approximate tangent space of \(\Gamma \) at x which exists \({\mathcal {H}}^{n-1}\) a.e. \(x\in \Gamma \). With this notation, (5) implies that \(H_V(x)=(T_x\,\Gamma )^{\perp }(v(x))\) for \({\mathcal {H}}^{n-1}\) a.e. \(x\in \Gamma \), i.e., the generalized mean curvature vector of V coincides with the projection of v to the orthogonal subspace \((T_x\,\Gamma )^{\perp }\) for \({\mathcal {H}}^{n-1}\) a.e. \(x\in \Gamma \). We emphasize the difference of characterization of the mean curvature vector from [25], where the similar equality holds only on the reduced boundary of \(\{u=1\}\), while the equality here holds on the whole support of \(\Vert V\Vert \) including on the “hidden boundary” \(\Gamma \setminus \partial ^*\{u=1\}\). If we additionally assume that \(\theta =1\) for \({\mathcal {H}}^{n-1}\) a.e. \(x\in \Gamma \), then because of the integrability of \(H_V\) and the Allard regularity theorem [1], except for a closed \({\mathcal {H}}^{n-1}\)-null set, \(\Gamma \) is locally a \(C^{1,2-\frac{n}{p}}\) hypersurface. Without the assumption \(\theta =1\), we can still conclude that \(\mathrm{spt}\,\Vert V\Vert \) is \(C^{1,2-\frac{n}{p}}\) hypersurface on a dense open set of \(\mathrm{spt}\,\Vert V\Vert \), even though we do not know if the complement is \({\mathcal {H}}^{n-1}\)-null or not.
2.4 A vectorial prescribed mean curvature problem
As an applicationFootnote 1 of Theorem 2.1 with suitable modifications, we prove the following:
Theorem 2.2
Let (M, g) be a smooth compact n-dimensional Riemannian manifold and let \(\rho \in W^{2,p}(M)\) be a given function, where \(p>\frac{n}{2}\). Then, there exists a non-zero integral varifold V in M such that
for \(\Vert V\Vert \) a.e. \(x\in M\).
Proof
We may assume \(p<n\). Consider the following functional for \(\varepsilon >0\) and \(u\in W^{1,2}(M)\):
By the Sobolev embedding, \(\rho \in C^{0,2-\frac{n}{p}}(M)\) and thus \(0<\exp (\min \,\rho )\le \exp (\rho )\le \exp (\max \,\rho )<\infty \). By considering the path space in \(W^{1,2}(M)\) connecting \(u\equiv 1\) and \(u\equiv -1\), the standard min–max method gives a non-trivial critical point \(u_\varepsilon \) for each \(\varepsilon >0\), with uniform strictly positive lower and upper bounds of \(F_{\varepsilon }(u_{\varepsilon })\). The critical point satisfies (2.1) with \(v=\nabla \rho \) and \(|u_\varepsilon |\le 1\), with an appropriate modification of the equation. Take a sequence \(\varepsilon _i\rightarrow 0+\) and a corresponding min–max critical points \(u_i\). Then the sequence \(u_i,\nabla \rho , \varepsilon _i\) satisfy all the assumptions of Theorem 2.1 with a small error terms coming from the metric of M (see Guaraco’s work [6] for an adaptation of the results to Riemannian manifolds in the case \(v_\varepsilon =0\)). The limit varifold V thus has the desired property. \(\square \)
For more remarks on the main results, see Sect. 5.
3 The estimate for the upper density ratio
In this section, we prove Theorem 3.8–3.10, which give \(\varepsilon \)-independent estimates of the upper and lower density ratios of the energy. Throughout this section, we drop the index i and set \(\Omega =U_1=\{|x|<1\}\) since the result is local. Assume \(u\in W^{1,2}(U_1)\) and \(v\in W^{1,p}(U_1;\mathbb {R}^n)\) satisfy (2.1) with a positive \(\varepsilon \) and (2.3)–(2.5) are satisfied for a given set of \(c_{0}, E_0, \lambda _0\). The exponent p satisfies (2.2). We first derive two preliminary properties for u, Lemmas 3.1 and 3.2.
Lemma 3.1
There exists \(c_{1}>0\) depending only on \(c_{0},\lambda _0,n,p\) and W such that
and
for \(0<\varepsilon <1/2\). If \(\varepsilon \ge 1/2\), then we have for any \(0<s<1\)
where \(c_{1}\) depends additionally on s. In both cases, we have \(u\in W_{loc}^{3,p}(U_1)\).
Proof
Consider the case \(0<\varepsilon <1/2\). Define \({\tilde{u}}(x):=u(\varepsilon x)\) and \(\tilde{v}(x):=\varepsilon v(\varepsilon x)\) for \(x\in U_{\varepsilon ^{-1}}\). After this change of variables, we obtain from (2.1) that
Under the change of variables, we obtain from (2.5)
For any \(U_2(x)\subset U_{\varepsilon ^{-1}}\), let \(\phi \in C^1_c(U_2(x))\) be a function such that \(0\le \phi \le 1\), \(\phi =1\) on \(B_1(x)\) and \(|\nabla \phi |\le 4\) on \(U_2(x)\). Use (3.4) with the test function \({\tilde{u}}\phi ^2\). Using also (2.3), we obtain
Since \(\frac{np}{n-p}>2\), (3.5) and (3.6) give
We next note that the function \({\tilde{u}}\phi \) weakly satisfies the following equation:
Using the standard \(L^p\) theory [5, 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 \(\nabla {\tilde{u}}\) is raised from q to \(q\cdot \frac{np}{np-q(2p-n)}\), 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 for \({\tilde{u}}\), and by the Sobolev inequality, the \(L^{\infty }_{loc}\) estimate for \(\nabla {\tilde{u}}\). Again by the \(L^p\) theory, we obtain the \(W^{2,\frac{np}{n-p}}_{loc}\) estimate of \({\tilde{u}}\). In particular, by the Sobolev inequality, we obtain (3.1) and (3.2). Since the right-hand side of (3.8) is in \(W^{1,p}_{loc}\) (note that \({\tilde{v}}\cdot \nabla ^2{\tilde{u}}\in L^{\frac{np}{2(n-p)}}_{loc}\) and \(\frac{np}{2(n-p)}>p\) by (2.2)), we have \({\tilde{u}}\in W^{3,p}_{loc}\) and the weak third-derivatives of \({\tilde{u}}\) exist. The case of \(\varepsilon \ge 1/2\) does not require the change of variables as above and the proof is omitted. \(\square \)
Lemma 3.2
Given \(0<s<1\), there exist constants \(0<\varepsilon _{1},\eta <1\) depending only on \(c_0,\lambda _0,W,n,p\) and s such that
for \(\varepsilon \le \varepsilon _{1}\).
Proof
Let \(q = \frac{np}{n-p}-1\) and \(\phi \in C^\infty _c\left( B_{\frac{s+1}{2}}\right) \) with \(\phi \ge 0\). Multiplying (2.1) by \([(u-1)_+]^q \phi ^2\), we have
By \(W'(u) \ge \kappa (u-1)\) for \(u\ge 1\) and (3.1), we obtain
which shows
By (2.3), (2.5) and iterating the computation above with suitable \(\phi \), we obtain
To derive a contradiction, assume that \(u(x_0)-1 \ge \varepsilon ^\eta \) for some \(x_0 \in B_{s}\). By (3.1), for \(y \in B_{\frac{\varepsilon ^{1+\eta }}{2c_{1}}}(x_0)\),
Then we have
which show by \(q = \frac{np}{n-p}-1\)
This is a contradiction if \(\eta \) and \(\varepsilon \) are sufficiently small. \(u \ge -1-\varepsilon ^\eta \) is proved similarly. \(\square \)
The next Lemma 3.3 is the starting point of the ultimate establishment of the monotonicity formula.
Lemma 3.3
For \(B_r(x) \subset U_1\), we have
Proof
Multiply both sides of (2.1) by \(\nabla u \cdot g\), where \(g = (g^1, \dots ,g^n) \in C^1_c (U_1; \mathbb {R}^n)\). By integration by parts, we obtain
We assume that \(x=0\) after a suitable translation and let \(g^j(y)=y_j \rho (|y|)\). Writing \(r=|y|\), (3.18) becomes
We choose \(\rho \) which is a smooth approximation of \(\chi _{B_r}\), the characteristic function of \(B_r\), and then we take a limit \(\rho \rightarrow \chi _{B_r}\). Then we have
By dividing the above equation by \(r^n\), the lemma follows. \(\square \)
We need the following lemma to control the negative contribution of the right-hand side of (3.17).
Lemma 3.4
Given \(0<s<1\), there exist constants \(0<\beta _{1}<1\) and \(0< \varepsilon _{2}<1\) which depend only on \(c_{0}\), \(\lambda _0\), W, n, p and s such that, if \(\varepsilon \le \varepsilon _ {2}\),
The proof of Lemma 3.4 is deferred to the end of this section. Next, for \(B_r(x)\subset U_1\) and \(0<r<\mathrm{dist}(x,\partial U_1)\), define
Using Lemmas 3.3 and 3.4, we prove:
Lemma 3.5
Given \(0<s<1\), there exist constants \(0<\varepsilon _{3},c_{4},c_{5}<1 \) which depend only on \(c_0,\lambda _0,W,n,p\) and s such that, if \(B_{\varepsilon ^{\beta _{1}}}(x) \subset U_s\), \(|u(x)|\le \alpha \) and \(\varepsilon \le \varepsilon _{3}\), then
Proof
By integrating (3.17) over [\(\varepsilon ,r\)], we have
By (3.19) and \(B_r(x)\subset U_s\), we have
Since \(|u(x)| \le \alpha \), using (3.1), we have \(|u(y)| \le \frac{\alpha +1}{2}\) for all \(y \in B_{\frac{(1-\alpha )\varepsilon }{2c_{1}}}(x)\). By choosing a larger \(c_{1}\) if necessary, we may assume \(\frac{(1-\alpha )}{2c_{1}} \le 1\). Define
With this choice, we obtain
Since a larger \(\beta _{1}\) satisfies (3.19) as well, we may assume \((3-\frac{n}{p})\beta _{1} -1 >0\) by choosing \(\beta _{1}<1\) sufficiently close to 1. Then we may show that the sum of (3.22) and (3.23) may be bounded from above by \(c_{4}\) for sufficiently small \(\varepsilon \) and \(c_{5}\) if \(r\le c_{5}\varepsilon ^{\beta _{1}}\). Then, (3.20) follows from (3.21)–(3.24). \(\square \)
Theorem 3.6
Given \(0<s<1\), there exist constants \(0<\varepsilon _{4},\beta _{2} <1\) and \(0<c_{6}\) which depend only on \(c_0\), \(\lambda _0\), W, n, p and s such that, if \(B_{r}(x) \subset U_s\), \(c_{5} \varepsilon ^{\beta _{1}}<r\) and \(\varepsilon \le \varepsilon _{{4}}\), then
Proof
The proof is similar to [25, Proposition 3.4] with a minor modification. Define \(\beta _{2}:=\frac{1-\beta _{1}}{2\beta _{1}}\) and \(\beta _{3}:=\frac{1+\beta _{1}}{2}\). \(\beta _{2}\) and \(\beta _{3}\) are chosen so that
We estimate the integral of (3.25) by separating \(B_r(x)\) into three disjoint sets. Define
Note that \(r> c_{5}\varepsilon ^{\beta _{1}}> \varepsilon ^{\beta _{3}}\) for small \(\varepsilon \).
The estimates of the integral over \(\mathcal {A}\) and \(\mathcal {B}\) are exactly the same as in [25]. Namely, for \({\mathcal {A}}\), we use \({\mathcal {L}}^n({\mathcal {A}})\le n\omega _n r^{n-1}\varepsilon ^{\beta _{3}}\) and (3.19) as well as \(r^{-1}\le (c_{5}\varepsilon ^{\beta _{1}})^{-1}\). For \({\mathcal {B}}\), we use (3.20) and prove that \({\mathcal {L}}^n({\mathcal {B}})\le c(n)\varepsilon ^{n\beta _{3}}N\), where N is an integer satisfying \({\mathcal {B}}\subset \cup _{i=1}^N B_{5\varepsilon ^{\beta _{3}}}(x_i)\). Here we only consider the estimate on \(\mathcal {C}\) and refer the reader to the proof of [25, Proposition 3.4]. Define
\(\phi \) is a Lipschitz function and is 0 on \(\{y\,:\,|y-x| > r \} \cup \{|u|< \alpha \}\), 1 on \(\mathcal {C}\) and \(|\nabla \phi |\le \varepsilon ^{-\beta _{3}}\). Differentiate (2.1) with respect to \(x_j\), multiply it by \(u_{x_j} \phi ^2\) 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 \(|u| \ge \alpha \) on the support of \(\phi \), we have \(W''\ge \kappa \). Thus
By \(|\nabla \phi | \le \varepsilon ^{-\beta _{3}}\) and (2.5), we have
where \(c_{8}=c_{7}+c(n,p)\lambda _0^2\). Since \(\phi =1\) on \(\mathcal {C}\), multiplying (3.31) by \(\frac{\varepsilon ^2}{\kappa r^n}\),
By the definitions of \(\beta _{1}\), \(\beta _{2}\), \(\beta _{3}\) and \(r \ge c_{5} \varepsilon ^{\beta _1}\), we have
and using \(\varepsilon \le r\),
where \(\frac{2n}{p}-3 <1\). Hence, we obtain
By re-defining \(\beta _{2}= \min \{\beta _{2}, 4-\frac{2n}{p} \}\) and the estimates of integrals over \(\mathcal {A}\), \(\mathcal {B}\) and \(\mathcal {C}\), we proved (3.25). \(\square \)
To proceed, we need the following theorem (see [30, Theorem 5.12.4]).
Theorem 3.7
Let \(\mu \) be a positive Radon measure on \(\mathbb {R}^n\) satisfying
Then there exists a constant c(n) such that
for all \(\phi \in C^1_c(\mathbb {R}^n)\).
Theorem 3.8
There exists a constant \(0<c_{9}\) which depends only on \(c_0\), \(\lambda _0\), W, \(E_0\), n and p such that, if \(0<\varepsilon <1/2\) and \(U_{2r}(x)\subset U_{1-\varepsilon }\), then
Proof
Define
By Lemma 3.1, we have \(\sup _{x\in U_{1-\varepsilon }}\varepsilon |\nabla u(x)|\le c_{1}\). Thus for any \(U_{2r}(x)\subset U_{1-\varepsilon }\), we have
and we have \(E_1<\infty \) for each \(\varepsilon \). In the following, we give an estimate of \(E_1\) depending only on \(c_0,\lambda _0,n,p,W\) and \(E_0\). Let \(U_{2r_0}(x_0)\subset U_{1-\varepsilon }\) be fixed such that
For simplicity, define
and change variables by \(\tilde{x}=(x-x_0)/l\), \(\tilde{r}=r/l\), \({\tilde{\varepsilon }}=\varepsilon /l\), \(\tilde{u}({\tilde{x}})=u(x)\) and \(\tilde{v}({\tilde{x}})=lv(x)\). Note that \(U_{l+\varepsilon }(x_0)\subset U_1\). In particular, we write
By (2.1), (2.4) and (2.5), we have
Define for \(B_{{\tilde{r}}}({\tilde{x}})\subset U_{1+{\tilde{\varepsilon }}}\)
Under the above change of variables, note that we have \(E(r,x)=\tilde{E}(\tilde{r},\tilde{x})\). Next, for any \(x\in B_{3l/4}(x_0)\), we have \(\mathrm{dist}\,(x,\partial U_{1-\varepsilon })\ge l/4\). Hence for any \(x\in B_{3l/4}(x_0)\) and \(r<l/8\le \mathrm{dist}\,(x,\partial U_{1-\varepsilon })/2\), by the definition of \(E_1\), we have
This shows (again using \(\mathrm{dist}(x,\partial U_{1-\varepsilon })\ge l/4\))
We next let \(c_{1}, c_{4},c_{5},c_{6},\varepsilon _{1},\varepsilon _{2},\varepsilon _{3},\varepsilon _{{4}},\beta _{1},\beta _{2}\) be constants obtained in Lemma 3.1–3.5 and Theorem 3.6 corresponding to the same \(c_{0},\lambda _0,n,p,W\) and \(s=3/4\). Then note that the estimates up to Theorem 3.6 hold for \({\tilde{u}}\) and \({\tilde{v}}\) for \(U_{3/4}\) and with respect to the new variables (\({\tilde{x}}\), \({\tilde{r}}\), \({\tilde{\varepsilon }}\) etc.) if
due to (3.38) and (3.40). It is important to note that \({\hat{\varepsilon }}\) depends only on \(c_0,\lambda _0,n,p\) and W. Note that (3.40) yields an upper bound for \(\Vert {\tilde{v}}\Vert _{L^{\frac{np}{n-p}} (U_{1+{\tilde{\varepsilon }}})}+\Vert \nabla {\tilde{v}}\Vert _{L^p(U_{1+{\tilde{\varepsilon }}})}\) independent of l, as \(l<1\) and \(2-\frac{n}{p}>0\). A priori, we do not know if (3.43) holds or not and we prove the desired estimate for \(E_1\) by exhausting all the possibilities.
First consider the case \({\tilde{\varepsilon }}\ge {\hat{\varepsilon }}\). We use (3.3) and (3.1), respectively, for \({\tilde{\varepsilon }}>1/2\) and \(1/2\ge {\tilde{\varepsilon }}\ge {\hat{\varepsilon }}\). Suppose that \({\tilde{\varepsilon }}>1/2\). By (3.37) and (3.3), we have
and since \(l{\tilde{\varepsilon }}=\varepsilon \le 1\), \(l<1\) and \({\tilde{r}}_0\le 1/2\), we obtain
If \(1/2\ge {\tilde{\varepsilon }}\ge {\hat{\varepsilon }}\), again by (3.37), (3.1) and \({\tilde{r}}_0\le 1/2\), we have
and we obtain
Thus by (3.44) and (3.45), if \({\tilde{\varepsilon }}\ge {\hat{\varepsilon }}\), \(E_1\) is bounded by a constant which depends only on \(c_0,\lambda _0,n,p\) and W.
For the rest of the proof, consider the case \({\tilde{\varepsilon }}<{\hat{\varepsilon }}\) and consider the following four cases (a)–(d) depending on how large \({\tilde{r}}_0=r_0/l\) is relative to \({\tilde{\varepsilon }}\) and \({\tilde{r}}_1\), where \({\tilde{r}}_1\) will be determined shortly depending only on \(c_0,\lambda _0,n,p,W\) and \(E_0\):
To control the term involving v in (3.17), define a Radon measure
By Theorem 3.7 and (3.42) (note that (3.42) has the restriction \({\tilde{r}}<1/8\) but this can be dropped easily by replacing \(4^{n-1}\) by a larger constant depending only on n), we have
for all \(\phi \in C^1_c(\mathbb {R}^n)\). By (3.17) and (3.25), if \(c_{5} \tilde{\varepsilon }^{\beta _1}<\tilde{r} \le \frac{1}{2}\), we have
Let \(\phi \in C^1_c(B_{\frac{3\tilde{r}}{2}})\) be such that \(0\le \phi \le 1\), \(\phi (y)=1\) in \(B_{\tilde{r}}\) and \(|\nabla \phi | \le \frac{4}{\tilde{r}}\). We use (3.46) with (3.40) by smoothly approximating \(|\tilde{v}|\) as
Hence, for \(c_{5}{\tilde{\varepsilon }}^{\beta _1}<{\tilde{r}}\le \frac{1}{2}\), (3.47) with (3.48) and (3.42) give
By integrating (3.49) over \({\tilde{r}}\in (\tilde{s}_1,\tilde{s}_2)\) with \(c_{5}{\tilde{\varepsilon }}^{\beta _1}<{\tilde{s}}_1<{\tilde{s}}_2\le \frac{1}{2}\), we obtain
where \(c_{10}\) depends only on \(c_0,\lambda _0,n,p\) and W. At this point, we choose \({\tilde{r}}_1<1/2\) depending only on \(c_0,\lambda _0,n,p\) and W so that
This in particular implies from (3.50) that if \(c_{5}{\tilde{\varepsilon }}^{\beta _1}<{\tilde{s}}_1<{\tilde{s}}_2\le {\tilde{r}}_1\), then
With this \({\tilde{r}}_1\) being fixed, we proceed to check that \(E_1\) is bounded in terms of \(c_0,\lambda _0,n,p,W,E_0\) in each case (a)–(d). Case (a): By (3.37), (3.39) and \({\tilde{r}}_1<{\tilde{r}}_0\), we have
Hence
and \(E_1\) is bounded by a constant depending only on \(c_0,\lambda _0,n,p,W\) and \(E_0\). Case (b): Since \(c_{5}{\tilde{\varepsilon }}^{\beta _1}<{\tilde{r}}_0\le {\tilde{r}}_1\), we may use (3.51) with \({\tilde{s}}_2=\tilde{r}_1\) and \({\tilde{s}}_1=\tilde{r}_0\). Then we obtain
Then, by (3.37) and (3.39), we obtain
which depends only on \(c_0,\lambda _0,n,p,W\) and \(E_0\). Case (c): By the same estimate used in the proof of Lemma 3.5, we have
We use (3.51) with \({\tilde{s}}_1=c_{5}{\tilde{\varepsilon }}^{\beta _1}\) and \({\tilde{s}}_2={\tilde{r}}_1\) to obtain
and (3.52) and (3.53) combined with (3.37) give
which depends only on \(c_0,\lambda _0,n,p,W\) and \(E_0\).
Case (d): Since \({\tilde{r}}_0\le {\tilde{\varepsilon }}\), we use (3.1) to obtain
This completes the estimate for \(E_1\). \(\square \)
Once we obtain the upper density estimate, we may obtain the following monotonicity formula.
Theorem 3.9
Given \(0<s<1\), there exist constants \(0<c_{11}\) and \(0<\varepsilon _5<1\) depending only on \(c_0,\lambda _0,n,p,W,E_0\) and s, such that, if \(c_{5} \varepsilon ^{\beta _{1}}\le s_1 <s_2\), \(B_{s_2}(x)\subset U_s\) and \(\varepsilon < \varepsilon _{5}\), then
Proof
Let \(\varepsilon _{5}=\min \{\varepsilon _{1},\varepsilon _{2},\varepsilon _{3},\varepsilon _{{4}},(1-s)/2\}\) corresponding to the given s and suppose that \(\varepsilon <\varepsilon _{5}\). For any \(x\in U_s\) and \(0<r<(1-s)/2\), by Theorem 3.8, \(E(r,x)\le c_{9} (1-s-\varepsilon )^{1-n}\), where the right-hand side is bounded by a constant depending only on \(c_0,\lambda _0,n,p,W,E_0\) and s. For \(B_{s_2}(x)\subset U_s\), we have (3.25) and (3.17). Arguing as (3.46)–(3.50) without change of variables (so \(l=1\)) and with \(\mu \) restricted to \(B_s\) in place of \(B_{3/4}\), we obtain (3.55). \(\square \)
Theorem 3.10
Given \(0<s<1\), there exist constants \(0<c_{12}\) depending only on \(c_0,\lambda _0,n,p,W,E_0\) and s such that, if \(\varepsilon <\varepsilon _{5}\), \(|u(x)|\le \alpha \), \(\varepsilon \le r\) and \(B_r(x)\subset U_s\), then
Proof
By Lemma 3.5, we may assume \(c_{5} \tilde{\varepsilon }^{\beta _{1}} \le r\) and \(E(c_{5}\varepsilon ^{\beta _{1}},x)\ge c_{4}\). In (3.55), let \(s_1=c_{5}\varepsilon ^{\beta _{1}}\) and \(s_2=r\). Fix \(r_1>0\) depending only on \(c_0,\lambda _0,n,p,W,E_0\) and s so that \(c_{{11}}(r_1^{2-\frac{n}{p}}+r_1^{\beta _{2}})\le c_{4}/2\). Then for \(c_{5}\varepsilon ^{\beta _{1}}\le r\le r_1\), (3.55) shows that \(E(r,x)\ge c_{4}/2\). For \(1>r>r_1\), \(E(r,x) \ge r_1^{n-1} E(r_1,x)\ge r_1^{n-1} c_{4}/2\). Thus, setting \(c_{{12}}=r_1^{n-1}c_{4}/2\), we have (3.56). \(\square \)
For the rest of the present section, we finish the proof of Lemma 3.4. We use the following result proved in [25, Lemma 3.9].
Lemma 3.11
Given \(0<\eta ,\beta _{4} <1\), \(\eta \le \beta _{4}\), \(0<c_{13}\), there exist \(\varepsilon _{6}>0\), \(c_{14}>0\) depending only on \(\eta \), \(\beta _{4}\), \(c_{{13}}\), n and W with the following properties: Suppose \(f\in C^3(U_{\varepsilon ^{-\beta _{4}}})\), \(g\in C^1(U_{\varepsilon ^{-\beta _{4}}})\) and \(0<\varepsilon \le \varepsilon _{6}\) satisfy
on \(U_{\varepsilon ^{-\beta _{4}}}\) and
Then
We remark that the assumptions on W are essentially used for the proof of Lemma 3.11.
Proof of Lemma 3.4
As in the proof of Lemma 3.1, define \(\tilde{u}(x):=u(\varepsilon x)\), \(\tilde{v}(x):=\varepsilon v(\varepsilon x)\), and subsequently drop \(\tilde{\cdot }\) for simplicity. We have
on \(U_{\varepsilon ^{-1}}\). With respect to the new variables, we need to prove
for some \(0<\beta _1<1\) for all sufficiently small \(\varepsilon \). Let \(\phi _\lambda \) be the standard mollifier, namely, define
where the constant \(C>0\) is selected so that \(\int _{\mathbb {R}^n} \phi =1\), and define \(\phi _\lambda (x) :=\frac{1}{\lambda ^n}\phi (\frac{x}{\lambda })\). For \(0<\beta _{5}<1\) to be chosen depending only on n and p later, define for \(x\in U_{\varepsilon ^{-1}-1}\)
We next define g to be
so that we have
To use Lemma 3.11, we next estimate the \(W^{1,n}\) norm of g on \(U_{\varepsilon ^{-\beta _{4}}}(x)\) with \(x\in U_{s\varepsilon ^{-1}}\), where \(0<\beta _4<1\) will be chosen depending only on n and p. In the following, let us write \(U_{\varepsilon ^{-\beta _{4}}}(x)\) as \(U_{\varepsilon ^{-\beta _{4}}}\) and \(U_{\frac{\varepsilon ^{-\beta _{4}}}{2}}(x)\) as \(U_{\frac{\varepsilon ^{-\beta _{4}}}{2}}\)for simplicity. For sufficiently small \(\varepsilon \) depending on s and \(\beta _4\) so that \(U_{2\varepsilon ^{-\beta _4}}(x) \subset U_{\varepsilon ^{-1}-1}\) (and so that we may use (3.1)), the first term of (3.62) can be estimated as
where \(c_{15}\) depends only on \(\phi \), n and \(c_{1}\). By (3.5), we obtain
where \(c_{16}\) depends only on \(\phi ,n,p,\lambda _0\) and \(c_{1}\). We next consider the second term of (3.62). By (3.60), (3.61) and
we compute
Here \(c_{17}\) depends only on \(\phi ,n,\lambda _0,c_{1}\) and W. Hence, (3.67)–(3.70) show
By (3.62), (3.66) and (3.71), we have
We use Lemma 3.11 to f and g. Due to Lemma 3.2, we have \(\sup |f|\le \sup |u|\le 1+\varepsilon ^{\eta }\) on \(U_{\varepsilon ^{-\beta _{4}}}\) and we may choose smaller \(\eta \) if necessary. Because of (3.1), we have \(c_{{13}}\ge \sup _{U_{\varepsilon ^{-\beta _{4}}}}(\frac{1}{2}|\nabla f|^2-W(f))\) for a constant depending only on \(c_{1}\) and W (here again we restrict \(\varepsilon \) small so that \(U_{\varepsilon ^{-\beta _4}}(x)\subset U_{\varepsilon ^{-1}-1}\cap U_{\varepsilon ^{-1}(1+s)/2}\)). Then we have all the assumptions for Lemma 3.11 and obtain
At this point, we fix sufficiently small \(0<\beta _{4},\beta _{5}<1\) depending only on n and p such that
This shows that we may choose a \(0<\beta _{1}<1\) such that
for all sufficiently small \(\varepsilon >0\). We may take the center of \(U_{\frac{\varepsilon ^{-\beta _{4}}}{2}}(=U_{\frac{\varepsilon ^{-\beta _{4}}}{2}}(x)\)) to be any \(x\in U_{s\varepsilon ^{-1}}\) so that we have the estimate on \(U_{s\varepsilon ^{-1}}\). By (3.61), (3.74) and
we may also replace f by u in (3.74) by choosing a larger \(0<\beta _{1}<1\) if necessary. This proves the desired estimate. \(\square \)
4 Rectifiability and integrality of the limit varifold
In this section, we recover the index i and assume that \(\{u_i\}_{i=1}^{\infty }\), \(\{v_i\}_{i=1}^{\infty }\) and \(\{\varepsilon _i\}_{i=1}^n\) satisfy (2.1)–(2.5). Define \(\mu _i\) and \(V_i\) as in (2.7) and (2.8). By the standard weak compactness theorem of Radon measures, there exists a subsequence (denoted by the same index) and a Radon measure \(\mu \) and a varifold V such that
Lemma 4.1
For \(x \in \mathrm{spt}\,\mu \), there exists a subsequence \(x_i \in U_1\) such that \(u_i(x_i) \in [-\alpha , \alpha ]\) and \(\lim _{i\rightarrow \infty }x_i =x\).
Proof
We prove this by contradiction and assume that there exists some \(r>0\) such that \(|u_i|\ge \alpha \) on \(U_r(x)\) for all large i. Without loss of generality, assume \(u_i \ge \alpha \) on \(U_r(x)\). Then we repeat the same argument leading to (3.30) with \(\phi \) there replaced by \(C^1_c(U_r(x))\). The argument shows that \(\lim _{i \rightarrow \infty } \int \frac{\varepsilon _i}{2}|\nabla u_i|^2 \phi ^2=0\). Next, multiplying \(u_i-1\) to the equation (2.1) and using \(W'(u_i)(u_i-1)\ge \frac{\kappa }{2}(u_i-1)^2\), we obtain
By integration by parts and (2.5), the right-hand side of (4.1) converges to 0. This shows that \(\mu (U_r(x))=0\) and contradicts \(x\in \mathrm{spt}\, \mu \). \(\square \)
Theorem 4.2
There exist constants \(0<D_1 \le D_2 < \infty \) which depend only on \(c_0,\lambda _0, n,p,W,E_0\) and s such that, for \(x \in \mathrm{spt}\, \mu \cap U_{s}\) and \(B_r(x)\subset U_{s}\), we have
Proof
This follows immediately from Theorem 3.8, 3.10 and Lemma 4.1. \(\square \)
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 [22, Proposition 4.3]:
Theorem 4.3
\(\xi _i\), \(\frac{\varepsilon _i}{2}|\nabla u_i|^2 - |\nabla w_i|\) and \(\frac{W(u_i)}{\varepsilon _i}- |\nabla w_i|\) all converge to zero in \(L^1_{loc}(U_1)\).
Proof of Theorem 2.1
Recall that . Since
by Theorem 4.3, converges to \(\mu \). We also know that \(\Vert V_i\Vert \) converges to \(\Vert V\Vert \) by definition, thus we have \(\Vert V\Vert =\mu \). This proves (1). The claims (2) and (3) follows from Theorem 3.8, 3.10 and Lemma 4.1 (see also [22, Proposition 4.2]). Next, by (2.9), (3.18) and Theorem 4.3,
for \(g \in C^1_c (U_1;\mathbb {R}^n)\), where \(\lim _{i\rightarrow \infty }o(1)=0\). By Theorem 3.8 and \(\mathrm{spt}\,g \subset U_{s}\) for some \(0<s<1\), we have a uniform bound on E(r, x) (corresponding to \(u_i\)) for \(B_r(x)\subset U_{s}\). Hence, by Theorem 3.7, we have
where the integrations are over \(\mathrm{spt}\,g\). 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 \(\epsilon >0\) be arbitrary and let \({\tilde{v}}\) be a smooth vector field such that \(\Vert v-{\tilde{v}}\Vert _{W^{1,p}(U_{s})}<\epsilon \). By the varifold convergence \(V_i \rightarrow V\), we have
We may arbitrarily approximate the quantities in (4.5) by v by the same argument in (4.4), hence by (4.3)–(4.5), we obtain
Hence, \(\Vert \delta V\Vert \) is a Radon measure on \(U_1\). By (4.2) and Allard’s rectifiability theorem [1, 5.5.(1)], V is rectifiable. Since V is rectifiable, there exist an \({\mathcal {H}}^{n-1}\) measurable and countably \(n-1\)-rectifiable set \(\Gamma \) such that
The set \(\Gamma \) is the measure-theoretic support of \(\Vert V\Vert \) and \(T_x\,\Gamma \) is the approximate tangent space of \(\Gamma \) which exists for \({\mathcal {H}}^{n-1}\) a.e. on \(\Gamma \). Next from (4.6), \(\Vert \delta V\Vert \) is absolutely continuous with respect to \(\Vert V\Vert \) and the generalized mean curvature \(H_V\) exists. By (4.6) and (4.7), we have \(H_V(x)=(T_x\,\Gamma )^{\perp }(v(x))\) holds for \(\Vert V\Vert \) a.e. for x. This proves (5), except that we do not yet take \(\Gamma ={\mathrm {spt}}\,\Vert V\Vert \). The proof of (4) is the same as [25] for the following reason. We may set \(f=\varepsilon \nabla u\cdot v\) in [25] and we have \(\Vert \varepsilon \nabla u\cdot v\Vert _{L^{\frac{np}{n-p}}(U_{s})}\le c_{1}\lambda _0\) 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 \(L^{\frac{np}{n-p}}\) norm as well as the estimate (3.2). See the comment in the proof of [22, Proposition 4.8] where it is explained that (3.2) is necessary. We should also point out that the \(L^n\) control of \(\nabla f\) is not needed in the proof. Finally, by arguing as in (4.4) and the Hölder inequality, we have
for any function \(\phi \in C^1_c(U_{s};\mathbb {R}^+)\) and we have the same inequality for \(v\in W^{1,p}(U_1)\) by the density argument. Thus we have (6). By the well-known property of varifold having the generalized mean curvature in \(L^q\) with \(q>n-1\) (see [19, 17.9(1)]), \(\mathrm{spt}\,\Vert V\Vert \) coincide with the measure-theoretic support \(\Gamma \), so that we have (5) with \(\Gamma =\mathrm{spt}\,\Vert V\Vert \). This concludes the proof of Theorem 2.1. \(\square \)
5 Concluding remarks
In [11, 21], we studied the singular perturbation problem for
and proved that the time-parametrized family of limit varifolds satisfies the motion law of “normal velocity \(=\) mean curvavture vector + \(v^{\perp }\)” in a weak formulation (see [9, 23] for the case of \(v_\varepsilon =0\)). 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 \(\varepsilon \). 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 [11, 21].
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 \(H\cdot \nu =f\), where \(\nu \) 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 \(\rho \), 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 \(F_{\varepsilon }\) in the proof of Theorem 2.2, since we have a very successful analogy in [24, 26]. In this direction, we mention that a construction of prescribed scalar mean curvature hypersurfaces along the lines suggested here has been announced recently [3] (see also [2]).
Notes
The authors thank Nick Edelen for a discussion which inspired this application.
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Tonegawa, Y., Tsukamoto, Y. A diffused interface with the advection term in a Sobolev space. Calc. Var. 59, 184 (2020). https://doi.org/10.1007/s00526-020-01860-z
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DOI: https://doi.org/10.1007/s00526-020-01860-z