A diffused interface with the advection term in a Sobolev space

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.

1 Introduction

The object of study in this paper is the energy functional appearing in the van der Waals–Cahn–Hillard theory [4, 7],

$$\begin{aligned} E_{\varepsilon }(u)= \int _\Omega \frac{\varepsilon |\nabla u|^2}{2}+\frac{W(u)}{\varepsilon }, \end{aligned}$$

(1.1)

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

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0+} \big (E_{\varepsilon }(u_{\varepsilon })+\Vert f_{\varepsilon }\Vert _{W^{1,p}(\Omega )}\big )<\infty \end{aligned}$$

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)

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0+}\big (E_{\varepsilon }(u_{\varepsilon })+\varepsilon ^{-1}\Vert f_{\varepsilon }\Vert _{L^2(\Omega )}^2\big )<\infty \end{aligned}$$

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

$$\begin{aligned} - \varepsilon \Delta u_{\varepsilon } + \frac{W'(u_{\varepsilon })}{\varepsilon }= \varepsilon v_\varepsilon \cdot \nabla u_{\varepsilon }, \end{aligned}$$

(1.2)

where $v_{\varepsilon }$ is considered here as a given vector field and we assume that

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0+}\big (E_{\varepsilon }(u_{\varepsilon })+\Vert v_{\varepsilon }\Vert _{W^{1,p}(\Omega )}\big )<\infty \end{aligned}$$

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

$$\begin{aligned} -\varepsilon _{i} \Delta u_{i} + \frac{W'(u_{i})}{\varepsilon _{i}} = \varepsilon _{i} v_{i} \cdot \nabla u_{i} \end{aligned}$$

(2.1)

weakly on $\Omega$ for each $i\in {\mathbb {N}}$. In addition, assume that

$$\begin{aligned} \lim _{i \rightarrow \infty }\varepsilon _i=0, \quad \frac{n}{2}< p<n \end{aligned}$$

(2.2)

and that there exist constants $c_{0}$, $E_0$ and $\lambda _0$ such that, for all $i\in {\mathbb {N}}$, we have:

$$\begin{aligned}&\Vert u_i\Vert _{L^{\infty }(\Omega )}\le c_{0}, \end{aligned}$$

(2.3)

$$\begin{aligned}&\int _{\Omega } \left( \frac{\varepsilon _i |\nabla u_i|^2}{2}+\frac{W(u_i)}{\varepsilon _{i}}\right) \le E_0, \end{aligned}$$

(2.4)

$$\begin{aligned}&\Vert v_i\Vert _{L^{\frac{np}{n-p}}(\Omega )}+\Vert \nabla v_i\Vert _{L^{p}(\Omega )} \le \lambda _0. \end{aligned}$$

(2.5)

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

$$\begin{aligned} \Phi (s) := \int ^s_{-1} \sqrt{W(t)/2} \ dt ,\quad w_i(x):= \Phi (u_i(x)). \end{aligned}$$

By the Cauchy–Schwarz inequality and (2.4), we obtain

$$\begin{aligned} \int _{\Omega } |\nabla w_i| \le \frac{1}{2}\int _{\Omega } \left( \frac{\varepsilon _i |\nabla u_i|^2}{2}+\frac{W(u_i)}{\varepsilon _i}\right) \le \frac{1}{2}E_0 . \end{aligned}$$

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

$$\begin{aligned} u (x): = \Phi ^{-1}(w (x)). \end{aligned}$$

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

$$\begin{aligned} \int _{\Omega } W(u) = \int _{\Omega } \lim _{i\rightarrow \infty } W(u_i) \le \liminf _{i \rightarrow \infty } \int _{\Omega } W(u_i) =0. \end{aligned}$$

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

$$\begin{aligned} \delta V(g) := \int _{\mathbf{G}_{n-1}(\Omega )} \nabla g(x) \cdot S \ dV(x,S) \end{aligned}$$

(2.6)

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

$$\begin{aligned} d \mu _i:= \frac{1}{\sigma }\Big (\frac{\varepsilon _i |\nabla u_i|^2}{2}+\frac{W(u_i)}{\varepsilon _i}\Big )d \mathcal {L}^n, \end{aligned}$$

(2.7)

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

$$\begin{aligned} V_i (\phi ) := \int _{\{ |\nabla u_i| \ne 0 \}} \phi \Big ( x,I-\frac{\nabla u_i(x)}{|\nabla u_i(x)|} \otimes \frac{\nabla u_i(x)}{|\nabla u_i(x)|} \Big ) d\mu _i(x) \end{aligned}$$

(2.8)

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

$$\begin{aligned} \delta V_i (g) = \int _{\{ |\nabla u_i| \ne 0 \}} \nabla g \cdot \Big ( I-\frac{\nabla u_i}{|\nabla u_i|} \otimes \frac{\nabla u_i}{|\nabla u_i|} \Big ) d\mu _i \end{aligned}$$

(2.9)

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

$$\begin{aligned} v_i \rightarrow v\ \text{ weakly } \text{ in } \ W^{1,p}, \ \ u_i \rightarrow u \ a.e., \ \ V_i \rightarrow V \text{ in } \text{ the } \text{ varifold } \text{ sense. } \end{aligned}$$

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 $\phi \in C_c(\mathbf{G}_{n-1}(\Omega ))$,

$$\begin{aligned} \int _{\mathbf{G}_{n-1}(\Omega )} \phi (x,S)\,dV(x,S)=\int _{\Gamma } \phi (x,T_x\,\Gamma )\theta (x)\,d{\mathcal {H}}^{n-1}(x). \end{aligned}$$

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 application¹ 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

$$\begin{aligned} H_V(x)=(T_x \,\mathrm{spt}\,\Vert V\Vert )^{\perp }(\nabla \rho (x)) \end{aligned}$$

for $\Vert V\Vert$ a.e. $x\in M$.

Proof

We may assume $p<n$. Consider the following functional for $\varepsilon >0$ and $u\in W^{1,2}(M)$:

$$\begin{aligned} F_{\varepsilon }(u):=\int _{M}\Big (\frac{\varepsilon |\nabla u|^2}{2}+\frac{W(u)}{\varepsilon }\Big )\exp (\rho )\,d{\omega }_g. \end{aligned}$$

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

$$\begin{aligned} \sup _{x\in U_{1-\varepsilon }}\varepsilon |\nabla u(x)|\le c_{1} \end{aligned}$$

(3.1)

and

$$\begin{aligned} \sup _{x,x'\in U_{1-\varepsilon }} \varepsilon ^{3-\frac{n}{p}}\frac{|\nabla u(x)-\nabla u(x')|}{|x-x'|^{2-\frac{n}{p}}}\le c_{1} \end{aligned}$$

(3.2)

for $0<\varepsilon <1/2$. If $\varepsilon \ge 1/2$, then we have for any $0<s<1$

$$\begin{aligned} \sup _{x\in U_{s}} |\nabla u(x)|\le c_{1} \end{aligned}$$

(3.3)

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

$$\begin{aligned} -\Delta \tilde{u}+W'(\tilde{u})= \tilde{v} \cdot \nabla \tilde{u} \ \ \ \text{ weakly } \text{ on } U_{\varepsilon ^{-1}}. \end{aligned}$$

(3.4)

Under the change of variables, we obtain from (2.5)

$$\begin{aligned} \Vert {\tilde{v}}\Vert _{L^{\frac{np}{n-p}}(U_{\varepsilon ^{-1}})} + \Vert \nabla {\tilde{v}}\Vert _{L^{p}(U_{\varepsilon ^{-1}})} \le \lambda _0 \varepsilon ^{2-\frac{n}{p}}. \end{aligned}$$

(3.5)

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

$$\begin{aligned} \begin{aligned} \int |\nabla {\tilde{u}}|^2\phi ^2&\le c_{0}\int (2\phi |\nabla \phi \Vert \nabla {\tilde{u}}| +|W'|\phi ^2+|{\tilde{v}}\Vert \nabla {\tilde{u}}|\phi ^2)\\&\le \frac{1}{2}\int |\nabla {\tilde{u}}|^2\phi ^2+\int \left( 4c_{0}^2|\nabla \phi |^2+c_{0}|W'|\phi ^2 +c_{0}^2|{\tilde{v}}|^2\phi ^2\right) . \end{aligned} \end{aligned}$$

(3.6)

Since $\frac{np}{n-p}>2$, (3.5) and (3.6) give

$$\begin{aligned} \sup _{B_2(x)\subset U_{\varepsilon ^{-1}}}\int _{B_1(x)}|\nabla {\tilde{u}}|^2\le c(c_{0},\lambda _0,n,p,W). \end{aligned}$$

(3.7)

We next note that the function ${\tilde{u}}\phi$ weakly satisfies the following equation:

$$\begin{aligned} -\Delta ({\tilde{u}}\phi )=-{\tilde{u}}\Delta \phi -2\nabla \phi \cdot \nabla {\tilde{u}}+( {\tilde{v}}\cdot \nabla {\tilde{u}}-W'({\tilde{u}}))\phi . \end{aligned}$$

(3.8)

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

$$\begin{aligned} \begin{aligned}&\nabla {\tilde{u}}\in L^q_{loc}\,\Longrightarrow \, {\tilde{v}}\cdot \nabla {\tilde{u}} \in L_{loc}^{\frac{npq}{np+q(n-p)}}\,\Longrightarrow \, {\tilde{u}}\in W^{2,\frac{npq}{np+q(n-p)}}_{loc}\\&\quad \Longrightarrow \,\nabla {\tilde{u}}\in L_{loc}^{\frac{npq}{np-q(2p-n)}} \end{aligned} \end{aligned}$$

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

$$\begin{aligned} \sup _{x\in B_s}|u(x)| \le 1+ \varepsilon ^{\eta } \end{aligned}$$

(3.9)

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

$$\begin{aligned}&-\varepsilon \int q [(u-1)_+]^{q-1} |\nabla u|^2 \phi ^2 +2 [(u-1)_+]^q \phi \nabla \phi \cdot \nabla u \nonumber \\&\quad = \int \frac{W'}{\varepsilon }[(u-1)_+]^q \phi ^2 - \int \varepsilon \nabla u\cdot v [(u-1)_+]^q \phi ^2. \end{aligned}$$

(3.10)

By $W'(u) \ge \kappa (u-1)$ for $u\ge 1$ and (3.1), we obtain

$$\begin{aligned}&\frac{\kappa }{\varepsilon } \int [(u-1)_+]^{q+1} \phi ^2 + \int \varepsilon q [(u-1)_+]^{q-1} |\nabla u|^2 \phi ^2 \nonumber \\&\quad \le 2\varepsilon \int [(u-1)_+]^q \phi |\nabla \phi | |\nabla u| + c_{1} \int |v| [(u-1)_+]^q \phi ^2 \nonumber \\&\quad \le \frac{q \varepsilon }{2} \int [(u-1)_+]^{q-1} |\nabla u|^2 \phi ^2 + \frac{8\varepsilon }{q} \int [(u-1)_+]^{q+1} |\nabla \phi |^2 \nonumber \\&\quad \quad + \,\frac{\kappa }{2\varepsilon } \int [(u-1)_+]^{q+1} \phi ^2 + \frac{\varepsilon ^q c(q, c_{1})}{\kappa ^q} \int |v|^{q+1} \phi ^2, \end{aligned}$$

(3.11)

which shows

$$\begin{aligned} \frac{\kappa }{2\varepsilon } \int [(u-1)_+]^{q+1} \phi ^2 \le \frac{8\varepsilon }{q} \int [(u-1)_+]^{q+1} |\nabla \phi |^2 + \frac{\varepsilon ^q c(q,c_{1})}{\kappa ^q} \int |v|^{q+1} \phi ^2.\nonumber \\ \end{aligned}$$

(3.12)

By (2.3), (2.5) and iterating the computation above with suitable $\phi$, we obtain

$$\begin{aligned} \int _{B_{s}} [(u-1)_+]^{q+1} \le c_{2}(s,q,\lambda _0,n,p,W ,c_{0},c_{1}) \varepsilon ^{q+1}. \end{aligned}$$

(3.13)

To derive a contradiction, assume that $u(x_0)-1 \ge \varepsilon ^\eta$ for some $x_0 \in B_{s}$. By (3.1), for $y \in B_{\frac{\varepsilon ^{1+\eta }}{2c_{1}}}(x_0)$,

$$\begin{aligned} u(y)-1 \ge u(x_0)-1-\sup |\nabla u|\frac{\varepsilon ^{1+\eta }}{2c_{1}} \ge \frac{\varepsilon ^\eta }{2}. \end{aligned}$$

(3.14)

Then we have

$$\begin{aligned} c_{2} \varepsilon ^{q+1} \ge \int _{B_{\frac{\varepsilon ^{1+\eta }}{2c_{1}}}(x_0)} [(u-1)_+]^{q+1} \ge \left( \frac{\varepsilon ^\eta }{2}\right) ^{q+1} \omega _n \left( \frac{\varepsilon ^{1+\eta }}{2c_{1}}\right) ^n, \end{aligned}$$

(3.15)

which show by $q = \frac{np}{n-p}-1$

$$\begin{aligned} \varepsilon ^{\eta \frac{np}{n-p} - \frac{np}{n-p} + n+n\eta } \le c_{3}(s,q,\lambda _0,n,p,W,c_{0},c_{1}). \end{aligned}$$

(3.16)

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

$$\begin{aligned} \begin{aligned}&\frac{d}{dr}\left\{ \frac{1}{r^{n-1}} \int _{B_r(x)} \left( \frac{\varepsilon |\nabla u|^2}{2}+\frac{W(u)}{\varepsilon }\right) \right\} = \frac{1}{r^n} \int _{B_r(x)} \left( \frac{W(u)}{\varepsilon }-\frac{\varepsilon |\nabla u|^2}{2} \right) \\&\quad +\frac{\varepsilon }{r^{n+1}} \int _{\partial B_r(x)}\left( (y-x) \cdot \nabla u \right) ^2 + \frac{\varepsilon }{r^n} \int _{B_r(x)} (v \cdot \nabla u)((y-x) \cdot \nabla u). \end{aligned} \end{aligned}$$

(3.17)

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

$$\begin{aligned} \int \Big ( \Big (\frac{\varepsilon |\nabla u|^2}{2}+\frac{W}{\varepsilon }\Big ) \mathrm {div}g -\varepsilon \sum _{i,j} u_{y_i} u_{y_j} g^i_{y_j} + \varepsilon (v \cdot \nabla u)(\nabla u \cdot g) \Big ) =0. \end{aligned}$$

(3.18)

We assume that $x=0$ after a suitable translation and let $g^j(y)=y_j \rho (|y|)$. Writing $r=|y|$, (3.18) becomes

$$\begin{aligned} \begin{aligned}&\int \Big ( \Big (\frac{\varepsilon |\nabla u|^2}{2}+ \frac{W}{\varepsilon } \Big ) \left( r \rho ' + n \rho \right) - \varepsilon \frac{\rho '}{r} (y \cdot \nabla u)^2 \\&\quad - \varepsilon |\nabla u|^2 \rho + \varepsilon (\nabla u \cdot v) (\nabla u \cdot y) \rho \Big ) = 0. \end{aligned} \end{aligned}$$

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

$$\begin{aligned} \begin{aligned}&-(n-1)\int _{B_r} \left( \frac{\varepsilon |\nabla u|^2}{2}+\frac{W}{\varepsilon } \right) + r \int _{ \partial B_r} \left( \frac{\varepsilon |\nabla u|^2}{2}+\frac{W}{\varepsilon } \right) \\&\quad = \int _{B_r} \left( \frac{W}{\varepsilon }-\frac{\varepsilon |\nabla u|^2}{2}\right) +\frac{\varepsilon }{r} \int _{\partial B_r} (y \cdot \nabla u)^2 + \varepsilon \int _{B_r} (\nabla u \cdot v)( \nabla u \cdot y). \end{aligned} \end{aligned}$$

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 $\varepsilon \le \varepsilon _ {2}$,

$$\begin{aligned} \sup _{B_{s}} \left( \frac{\varepsilon }{2} |\nabla u|^2 - \frac{W(u)}{\varepsilon }\right) \le \varepsilon ^{-\beta _{1}} . \end{aligned}$$

(3.19)

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

$$\begin{aligned} E(r,x):= \frac{1}{r^{n-1}} \int _{B_r(x)} \left( \frac{\varepsilon |\nabla u|^2}{2}+\frac{W(u)}{\varepsilon }\right) . \end{aligned}$$

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

$$\begin{aligned} E(r,x)\ge c_{4} \ \ \ \text{ for } \text{ all } \ \varepsilon \le r \le c_{5} \varepsilon ^{\beta _{1}}. \end{aligned}$$

(3.20)

Proof

By integrating (3.17) over [$\varepsilon ,r$], we have

$$\begin{aligned} E(r,x)-E(\varepsilon ,x)\ge & {} -\int ^r_\varepsilon \frac{d\tau }{\tau ^n} \int _{B_\tau (x)}\left( \frac{\varepsilon }{2} |\nabla u|^2 - \frac{W(u)}{\varepsilon }\right) _+ \nonumber \\&\quad +\int ^r_\varepsilon \frac{d\tau }{\tau ^n} \int _{B_\tau (x)} \varepsilon (\nabla u \cdot v)( \nabla u \cdot (y-x)). \end{aligned}$$

(3.21)

By (3.19) and $B_r(x)\subset U_s$, we have

$$\begin{aligned} \int ^r_\varepsilon \frac{d\tau }{\tau ^n} \int _{B_\tau (x)}\left( \frac{\varepsilon }{2} |\nabla u|^2 - \frac{W(u)}{\varepsilon }\right) _+ \le \omega _n r \varepsilon ^{-\beta _1} . \end{aligned}$$

(3.22)

By (3.1) and (2.5), we have

$$\begin{aligned} \begin{aligned} \Big |\int ^r_\varepsilon \frac{d\tau }{\tau ^n} \int _{B_\tau (x)} \varepsilon (\nabla u \cdot v)( \nabla u \cdot (y-x))\Big |&\le \int ^r_0 \frac{d\tau }{\tau ^{n-1}} \int _{B_\tau (x)} c_{1}^2 \varepsilon ^{-1} |v| \\&\le c(\lambda _0,n,p,c_{1}) r^{3-\frac{n}{p}} \varepsilon ^{-1}. \end{aligned} \end{aligned}$$

(3.23)

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

$$\begin{aligned} c_{4}:=\frac{\omega _n}{2} \frac{(1-\alpha )^n}{(2c_{1})^n} \min _{|t| \le \frac{1+\alpha }{2}} W(t)>0. \end{aligned}$$

With this choice, we obtain

$$\begin{aligned} E(\varepsilon ,x)\ge & {} \frac{1}{\varepsilon ^{n-1}} \int _{B_{\frac{(1-\alpha )\varepsilon }{2c_{1}}}(x)} \frac{W(u)}{\varepsilon } \nonumber \\\ge & {} \omega _n \frac{(1-\alpha )^n}{(2c_{1})^n} \min _{|t| \le \frac{1+\alpha }{2}} W(t) = 2c_{4}. \end{aligned}$$

(3.24)

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

$$\begin{aligned} \frac{1}{r^n} \int _{B_r(x)} \left( \frac{\varepsilon }{2} |\nabla u|^2 - \frac{W(u)}{\varepsilon }\right) _+ \le \frac{c_{6}}{r^{1-\beta _{2}}}(E(r,x)+1) . \end{aligned}$$

(3.25)

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

$$\begin{aligned}&\beta _{1} \beta _{2}= \beta _{3} - \beta _{1},\end{aligned}$$

(3.26)

$$\begin{aligned}&0<\beta _{2}<1, \ \ \ \ 0<\beta _{1}<\beta _{3}<1. \end{aligned}$$

(3.27)

We estimate the integral of (3.25) by separating $B_r(x)$ into three disjoint sets. Define

$$\begin{aligned} \mathcal {A} := & {} B_r(x) \backslash B_{r-\varepsilon ^{\beta _{3}}}(x) , \\ \mathcal {B}:= & {} \{y\in B_{r-\varepsilon ^{\beta _{3}}}(x)\, :\, \mathrm{dist}(\{|u| \le \alpha \},y)< \varepsilon ^{\beta _{3}} \} , \\ \mathcal {C}:= & {} \{y\in B_{r-\varepsilon ^{\beta _{3}}}(x) \,:\, \mathrm{dist}(\{|u| \le \alpha \},y) \ge \varepsilon ^{\beta _{3}} \} . \end{aligned}$$

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

$$\begin{aligned} \phi (z) := \min \left\{ 1, \varepsilon ^{-\beta _{3}} \mathrm{dist}(\{y\,:\,|y-x| \ge r \} \cup \{|u|\le \alpha \},z )\right\} . \end{aligned}$$

$\phi$ 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

$$\begin{aligned} \int \sum _{j} \varepsilon u_{x_j} \Delta u_{x_j} \phi ^2= \int \frac{W''}{\varepsilon }|\nabla u|^2\phi ^2 - \varepsilon \sum _j\int (v\cdot \nabla u)_{x_j}\phi ^2u_{x_j}. \end{aligned}$$

(3.28)

By integration by parts, the Cauchy–Schwarz inequality and (3.1), we obtain

$$\begin{aligned}&\int \varepsilon |\nabla ^2 u |^2 \phi ^2 +\frac{W''}{\varepsilon }|\nabla u |^2 \phi ^2 \nonumber \\&\quad = \int -\sum _{i,j} 2\varepsilon u_{x_j}u_{x_i x_j} \phi \phi _{x_i} - \varepsilon \nabla u \cdot v (\Delta u \phi ^2 +2 \phi \nabla u \cdot \nabla \phi ) \nonumber \\&\quad \le \frac{1}{2} \int \varepsilon |\nabla ^2 u |^2 \phi ^2 +c_{7}\int (\varepsilon |\nabla u|^2|\nabla \phi |^2 + |v|^2 \phi ^2 \varepsilon ^{-1}), \end{aligned}$$

(3.29)

where $c_{7}$ depends on $c_{1}$. Since $|u| \ge \alpha$ on the support of $\phi$, we have $W''\ge \kappa$. Thus

$$\begin{aligned} \int \frac{\varepsilon }{2} |\nabla ^2 u|^2 \phi ^2 + \frac{\kappa }{\varepsilon }|\nabla u|^2 \phi ^2 \le c_{7} \int (\varepsilon |\nabla u|^2|\nabla \phi |^2 + |v|^2 \phi ^2 \varepsilon ^{-1}). \end{aligned}$$

(3.30)

By $|\nabla \phi | \le \varepsilon ^{-\beta _{3}}$ and (2.5), we have

$$\begin{aligned} \int \frac{\kappa }{\varepsilon }|\nabla u|^2 \phi ^2\le & {} c_{7} \left( \varepsilon ^{-2\beta _{3}} \int _{B_r} \varepsilon |\nabla u|^2 +\varepsilon ^{-1} \Vert v\Vert ^2_{L^{\frac{np}{n-p}}} (\omega _n r^n)^{\frac{np-2(n-p)}{np}} \right) \nonumber \\\le & {} c_{8} \left( \varepsilon ^{-2\beta _{3}} \int _{B_r} \varepsilon |\nabla u|^2 + \varepsilon ^{-1} r^{n-\frac{2(n-p)}{p}} \right) , \end{aligned}$$

(3.31)

where $c_{8}=c_{7}+c(n,p)\lambda _0^2$. Since $\phi =1$ on $\mathcal {C}$, multiplying (3.31) by $\frac{\varepsilon ^2}{\kappa r^n}$,

$$\begin{aligned} \frac{1}{r^n} \int _{\mathcal {C}} \frac{\varepsilon }{2} |\nabla u|^2 \le \frac{c_{8}}{\kappa } \left( \frac{\varepsilon ^{2-2\beta _{3}}}{r} E(r,x) + \varepsilon r^{2-\frac{2n}{p}}\right) . \end{aligned}$$

(3.32)

By the definitions of $\beta _{1}$, $\beta _{2}$, $\beta _{3}$ and $r \ge c_{5} \varepsilon ^{\beta _1}$, we have

$$\begin{aligned} \frac{\varepsilon ^{2-2\beta _{3}}}{r} \le \frac{\varepsilon ^{\beta _{1} \beta _{2}}}{r^{1-\beta _{2}}c_{5}^{\beta _{2}}}, \end{aligned}$$

(3.33)

and using $\varepsilon \le r$,

$$\begin{aligned} \varepsilon r^{2-\frac{2n}{p}} \le \frac{1}{r^{\frac{2n}{p}-3}}, \end{aligned}$$

(3.34)

where $\frac{2n}{p}-3 <1$. Hence, we obtain

$$\begin{aligned} \frac{1}{r^n} \int _{\mathcal {C}}\frac{\varepsilon }{2}|\nabla u|^2 \le \frac{c_{8}}{\kappa } \left( \frac{\varepsilon ^{\beta _{1} \beta _{2}}}{r^{1-\beta _{2}}c_{5}^{\beta _2}}E(r,x)+ \frac{1}{r^{\frac{2n}{p}-3}} \right) . \end{aligned}$$

(3.35)

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

$$\begin{aligned} K(\mu ):=\sup _{B_r(x)\subset \mathbb {R}^n} \frac{1}{r^{n-1}} \mu (B_r(x))< \infty . \end{aligned}$$

Then there exists a constant c(n) such that

$$\begin{aligned} \left| \int _{\mathbb {R}^n} \phi \,d\mu \right| \le c(n) K(\mu ) \int _{\mathbb {R}^n} |\nabla \phi | \,d\mathcal {L}^n \end{aligned}$$

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

$$\begin{aligned} \mathrm{dist}(x, \partial U_{1-\varepsilon })^{n-1}\,E(r,x) \le c_{9}. \end{aligned}$$

(3.36)

Proof

Define

$$\begin{aligned} E_1:=\sup _{U_{2r}(x)\subset U_{1-\varepsilon }} \mathrm{dist}(x, \partial U_{1-\varepsilon })^{n-1} E(r,x). \end{aligned}$$

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

$$\begin{aligned} E(r,x)\le \omega _n r\left( \frac{c_{1}^2}{2\varepsilon }+\sup _{|t|\le c_{0}}\frac{W(t)}{\varepsilon }\right) \le \frac{c(n,c_{1},W)}{\varepsilon } \end{aligned}$$

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

$$\begin{aligned} \mathrm{dist}(x_0, \partial U_{1-\varepsilon })^{n-1}E(r_0,x_0) > \frac{3}{4} E_1. \end{aligned}$$

(3.37)

For simplicity, define

$$\begin{aligned} l:=\mathrm{dist}\,(x_0,\partial U_{1-\varepsilon })=1-\varepsilon -|x_0| \end{aligned}$$

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

$$\begin{aligned} {\tilde{r}}_0:=r_0/l\le 1/2. \end{aligned}$$

By (2.1), (2.4) and (2.5), we have

$$\begin{aligned}&-\tilde{\varepsilon } \Delta \tilde{u} + \frac{W'(\tilde{u})}{\tilde{\varepsilon }} = \tilde{\varepsilon } \tilde{v} \cdot \nabla \tilde{u} \,\, \text{ for } {\tilde{x}}\in U_{1+{\tilde{\varepsilon }}}, \end{aligned}$$

(3.38)

$$\begin{aligned}&\int _{U_{1+{\tilde{\varepsilon }}}} \left( \frac{\tilde{\varepsilon } |\nabla \tilde{u}|^2}{2}+\frac{W(\tilde{u})}{\tilde{\varepsilon }}\right) \le l^{1-n} E_0, \end{aligned}$$

(3.39)

$$\begin{aligned}&\Vert \tilde{v}\Vert _{L^{\frac{np}{n-p}}(U_{1+{\tilde{\varepsilon }}})} +\Vert \nabla \tilde{v}\Vert _{L^{p}(U_{1+{\tilde{\varepsilon }}})} \le l^{2-\frac{n}{p}}\lambda _0. \end{aligned}$$

(3.40)

Define for $B_{{\tilde{r}}}({\tilde{x}})\subset U_{1+{\tilde{\varepsilon }}}$

$$\begin{aligned} \tilde{E}(\tilde{r},\tilde{x}):= \frac{1}{\tilde{r}^{n-1}} \int _{B_{\tilde{r}}(\tilde{x})} \left( \frac{\tilde{\varepsilon } |\nabla \tilde{u}|^2}{2}+\frac{W(\tilde{u})}{\tilde{\varepsilon }}\right) . \end{aligned}$$

(3.41)

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

$$\begin{aligned} \mathrm{dist}(x,\partial U_{1-\varepsilon })^{n-1}E(r,x)\le E_1. \end{aligned}$$

This shows (again using $\mathrm{dist}(x,\partial U_{1-\varepsilon })\ge l/4$)

$$\begin{aligned} \sup _{\tilde{x}\in B_{\frac{3}{4}},\, 0<\tilde{r}<\frac{1}{8}} \tilde{E}(\tilde{r},\tilde{x}) \le 4^{n-1}l^{1-n}E_1. \end{aligned}$$

(3.42)

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

$$\begin{aligned} {\tilde{\varepsilon }}\le {\hat{\varepsilon }}:=\min \{\varepsilon _{1},\varepsilon _{2},\varepsilon _{3},\varepsilon _{{4}},1/2\} \end{aligned}$$

(3.43)

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

$$\begin{aligned} \frac{3}{4l^{n-1}} E_1<{\tilde{E}}({\tilde{r}}_0,0)\le \omega _n{\tilde{r}}_0({\tilde{\varepsilon }} c_{1}^2+2 \sup _{|x|\le c_0}W(x)) \end{aligned}$$

and since $l{\tilde{\varepsilon }}=\varepsilon \le 1$, $l<1$ and ${\tilde{r}}_0\le 1/2$, we obtain

$$\begin{aligned} E_1< \frac{4}{3} \omega _n(c_{1}^2+2\sup _{|x|\le c_0} W(x)). \end{aligned}$$

(3.44)

If $1/2\ge {\tilde{\varepsilon }}\ge {\hat{\varepsilon }}$, again by (3.37), (3.1) and ${\tilde{r}}_0\le 1/2$, we have

$$\begin{aligned} \frac{3}{4l^{n-1}} E_1< & {} \tilde{E}(\tilde{r}_0,0)\le \omega _n (c_{1}^2 +\sup _{|x|\le c_0}W(x)) \frac{\tilde{r}_0}{\tilde{\varepsilon }} \\\le & {} \omega _n (c_{1}^2 +\sup _{|x|\le c_0}W(x)) \frac{1}{2\hat{\varepsilon }} \end{aligned}$$

and we obtain

$$\begin{aligned} E_1<\omega _n (c_{1}^2 +\sup _{|x|\le c_0}W(x)) \frac{2}{3\hat{\varepsilon }}. \end{aligned}$$

(3.45)

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 $E_0$:

$$\begin{aligned} (a)\,\tilde{r}_1< \tilde{r}_0 \le \frac{1}{2},\,\, (b)\,c_{5} \tilde{\varepsilon }^{\beta _{1}}< \tilde{r}_0 \le \tilde{r}_1, \,\, (c)\,\tilde{\varepsilon }< \tilde{r}_0 \le c_{5} \tilde{\varepsilon }^{\beta _{1}},\,\, (d)\, 0<\tilde{r}_0\le \tilde{\varepsilon }. \end{aligned}$$

To control the term involving v in (3.17), define a Radon measure

$$\begin{aligned} \mu (A) := \int _{A \cap B_{\frac{3}{4}}} \left( \frac{\tilde{\varepsilon }|\nabla \tilde{u}|^2}{2}+\frac{W(\tilde{u})}{\tilde{\varepsilon }}\right) . \end{aligned}$$

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

$$\begin{aligned} \Big |\int _{B_{\frac{3}{4}}} \phi \, d \mu \Big | \le c(n) l^{1-n} E_1 \int _{\mathbb {R}^n} |\nabla \phi |\, d \mathcal {L}^n \end{aligned}$$

(3.46)

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

$$\begin{aligned} \begin{aligned} \frac{d}{d\tilde{r}} \tilde{E}(\tilde{r},0) \ge&-\frac{c_{6}}{\tilde{r}^{1-\beta _2}}(\tilde{E}(\tilde{r},0)+1) - \frac{1}{\tilde{r}^{n-1}} \int _{B_{\tilde{r}}} {\tilde{\varepsilon }}|\tilde{v}| |\nabla \tilde{u}|^2 \\&+ \frac{1}{\tilde{r}^n} \int _{B_{\tilde{r}}} \left( \frac{W(\tilde{u})}{\tilde{\varepsilon }} - \frac{\tilde{\varepsilon }}{2} |\nabla \tilde{u}|^2 \right) _+ . \end{aligned} \end{aligned}$$

(3.47)

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

$$\begin{aligned} \begin{aligned}&\int _{B_{\tilde{r}}} {\tilde{\varepsilon }}|\tilde{v}| |\nabla \tilde{u}|^2 \le \int _{B_{\frac{3\tilde{r}}{2}}} {\tilde{\varepsilon }}\phi |\tilde{v}| |\nabla \tilde{u}|^2 \le c(n)l^{1-n}E_1 \int _{B_{\frac{3\tilde{r}}{2}}} | \nabla (\phi |\tilde{v}|)| \\&\quad \le c(n)l^{1-n}E_1 \int _{B_{\frac{3\tilde{r}}{2}}} \frac{4}{\tilde{r}}|\tilde{v}| + |\nabla \tilde{v}| \le c(n) l^{3-n-\frac{n}{p}} \lambda _0 \tilde{r}^{n-\frac{n}{p}}E_1. \end{aligned} \end{aligned}$$

(3.48)

Hence, for $c_{5}{\tilde{\varepsilon }}^{\beta _1}<{\tilde{r}}\le \frac{1}{2}$, (3.47) with (3.48) and (3.42) give

$$\begin{aligned} \begin{aligned} \frac{d}{d\tilde{r}} \tilde{E}(\tilde{r},0)&\ge -c_{6}\tilde{r}^{\beta _2-1}(c(n)l^{1-n}E_1+1)- c(n) l^{3-n-\frac{n}{p}} \lambda _0 {\tilde{r}}^{1-\frac{n}{p}} E_1\\&\quad +\frac{1}{\tilde{r}^n} \int _{B_{\tilde{r}}} \left( \frac{W(\tilde{u})}{\tilde{\varepsilon }} - \frac{\tilde{\varepsilon }}{2} |\nabla \tilde{u}|^2 \right) _+ . \end{aligned} \end{aligned}$$

(3.49)

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

$$\begin{aligned} \begin{aligned} \tilde{E}({\tilde{s}}_2,0) - \tilde{E}(\tilde{s}_1,0)&\ge -c_{10}({\tilde{s}}_2^{\beta _2}+l^{2-\frac{n}{p}}{\tilde{s}}_2^{2-\frac{n}{p}})l^{1-n}E_1 -c_{10} {\tilde{s}}_2^{\beta _2}\\&\quad +\int _{{\tilde{s}}_1}^{{\tilde{s}}_2}\frac{d{\tilde{r}}}{\tilde{r}^n} \int _{B_{\tilde{r}}} \left( \frac{W(\tilde{u})}{\tilde{\varepsilon }} - \frac{\tilde{\varepsilon }}{2} |\nabla \tilde{u}|^2 \right) _+ , \end{aligned} \end{aligned}$$

(3.50)

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

$$\begin{aligned} c_{{10}}\left( {\tilde{r}}^{\beta _2}_1+{\tilde{r}}_1^{2-\frac{n}{p}}\right) <\frac{1}{4}. \end{aligned}$$

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

$$\begin{aligned} {\tilde{E}}({\tilde{s}}_2,0)-{\tilde{E}}({\tilde{s}}_1,0)\ge -c_{{10}}-\frac{1}{4} l^{1-n}E_1. \end{aligned}$$

(3.51)

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

$$\begin{aligned} \frac{3}{4} l^{1-n}E_1\le \tilde{E}(\tilde{r}_0,0)\le \tilde{r}_0^{1-n} l^{1-n} E_0 \le \tilde{r}_1^{1-n} l^{1-n} E_0. \end{aligned}$$

Hence

$$\begin{aligned} E_1\le \frac{4}{3} {\tilde{r}}_1^{1-n}E_0 \end{aligned}$$

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

$$\begin{aligned} \tilde{E}(\tilde{r}_1,0) - \tilde{E}(\tilde{r}_0,0) \ge -c_{{10}} - \frac{1}{4} l^{1-n}E_1. \end{aligned}$$

Then, by (3.37) and (3.39), we obtain

$$\begin{aligned} E_1\le 4\left( {\tilde{E}}({\tilde{r}}_1,0)+c_{{10}})l^{n-1}\le 4({\tilde{r}}_1^{1-n}E_0+c_{{10}}\right) , \end{aligned}$$

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

$$\begin{aligned} \tilde{E}(c_{5} \tilde{\varepsilon }^{\beta _1},0) - \tilde{E} (\tilde{r}_0,0) \ge -c_{4}. \end{aligned}$$

(3.52)

We use (3.51) with ${\tilde{s}}_1=c_{5}{\tilde{\varepsilon }}^{\beta _1}$ and ${\tilde{s}}_2={\tilde{r}}_1$ to obtain

$$\begin{aligned} \tilde{E}(\tilde{r}_1,0) - \tilde{E}(c_{5} \tilde{\varepsilon }^{\beta _1},0) \ge -c_{{10}} - \frac{1}{4} l^{1-n}E_1, \end{aligned}$$

(3.53)

and (3.52) and (3.53) combined with (3.37) give

$$\begin{aligned} E_1\le 4l^{n-1} ({\tilde{E}}({\tilde{r}}_1,0)+c_{4}+c_{{10}})\le 4{\tilde{r}}_1^{1-n}E_0+4(c_{4}+c_{{10}}), \end{aligned}$$

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

$$\begin{aligned} \tilde{E}(\tilde{r}_0,0) \le \omega _n\left( c_{1}^2 +\sup _{|x|\le c_0}W(x)\right) \frac{\tilde{r}_0}{\tilde{\varepsilon }} \le \omega _n \left( c_{1}^2 +\sup _{|x|\le c_0}W(x)\right) . \end{aligned}$$

(3.54)

Then (3.54) and (3.37) gives

$$\begin{aligned} E_1<\frac{4}{3}\omega _n\left( c_{1}^2 +\sup _{|x|\le c_0}W(x)\right) l^{n-1}\le \omega _n \left( c_{1}^2 +\sup _{|x|\le c_0}W(x)\right) . \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} E(s_2,x)-E(s_1,x) \ge&-c_{{11}} \left( s_2^{2-\frac{n}{p}}+s_2^{\beta _{2}}\right) \\&+ \int ^{s_2}_{s_1} \frac{d \tau }{\tau ^n} \int _{B_{\tau }(x)} \Big ( \frac{W(u)}{\varepsilon } - \frac{\varepsilon }{2} |\nabla u|^2 \Big )_+. \end{aligned} \end{aligned}$$

(3.55)

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

$$\begin{aligned} E(r,x) \ge c_{{12}}. \end{aligned}$$

(3.56)

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

$$\begin{aligned} -\Delta f+W'(f) = g \end{aligned}$$

on $U_{\varepsilon ^{-\beta _{4}}}$ and

$$\begin{aligned} \sup _{U_{\varepsilon ^{-\beta _{4}}}} |f| \le 1+\varepsilon ^\eta , \ \ \ \sup _{U_{\varepsilon ^{-\beta _{4}}}} \left( \frac{1}{2}|\nabla f|^2 -W(f) \right) \le c_{{13}}. \end{aligned}$$

Then

$$\begin{aligned} \sup _{B_{\frac{\varepsilon ^{-\beta _{4}}}{2}}} \left( \frac{1}{2}|\nabla f|^2 -W(f) \right) \le c_{13} (\varepsilon ^{-\beta _{4}}\Vert g\Vert _{W^{1,n}(B_{\varepsilon ^{-\beta _{4}}})}+\varepsilon ^\eta ). \end{aligned}$$

(3.57)

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

$$\begin{aligned} -\Delta u + W'(u)= \nabla u \cdot v \end{aligned}$$

on $U_{\varepsilon ^{-1}}$. With respect to the new variables, we need to prove

$$\begin{aligned} \sup _{U_{s\varepsilon ^{-1}}} \left( \frac{1}{2}|\nabla u|^2 -W(u) \right) \le \varepsilon ^{1-\beta _1} \end{aligned}$$

(3.58)

for some $0<\beta _1<1$ for all sufficiently small $\varepsilon$. Let $\phi _\lambda$ be the standard mollifier, namely, define

$$\begin{aligned} \phi (x) := {\left\{ \begin{array}{ll} C \exp \left( \frac{1}{|x|^2-1} \right) &{} \text{ for } |x|<1 \\ 0 &{} \text{ for } |x| \ge 1, \end{array}\right. } \end{aligned}$$

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 $x\in U_{\varepsilon ^{-1}-1}$

$$\begin{aligned} f(x):=(u*\phi _{\varepsilon ^{\beta _{5}}})(x)=\int u(x-y)\phi _{\varepsilon ^{\beta _{5}}}(y) \,dy. \end{aligned}$$

(3.59)

By (3.1) and (3.2), we have

$$\begin{aligned} \sup _{U_{\varepsilon ^{-1}-1}}|f-u|\le & {} 2c_{1} \varepsilon ^{\beta _{5}}, \end{aligned}$$

(3.60)

$$\begin{aligned} \sup _{U_{\varepsilon ^{-1}-1}}|\nabla f -\nabla u|\le & {} 2c_{1} \varepsilon ^{\beta _{5}}\left( 2-\frac{n}{p}\right) . \end{aligned}$$

(3.61)

We next define g to be

$$\begin{aligned} g:= (\nabla u \cdot v)*\phi _{\varepsilon ^{\beta _{5}}}+( W'(f)-W'(u)*\phi _{\varepsilon ^{\beta _{5}}}), \end{aligned}$$

(3.62)

so that we have

$$\begin{aligned} -\Delta f + W'(f)= g. \end{aligned}$$

(3.63)

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

$$\begin{aligned} \Vert (\nabla u \cdot v)*\phi _{\varepsilon ^{\beta _{5}}}\Vert _{W^{1,n}(U_{\varepsilon ^{-\beta _4}})} \le c_{15}(1+\varepsilon ^{-\beta _{5}})\Vert v\Vert _{L^n(U_{2\varepsilon ^{-\beta _{4}}})} \end{aligned}$$

(3.64)

where $c_{15}$ depends only on $\phi$, n and $c_{1}$. By (3.5), we obtain

$$\begin{aligned} \Vert v\Vert _{L^n(U_{2\varepsilon ^{-\beta _{4}}})}\le & {} \Vert v\Vert _{L^{\frac{np}{n-p}}(U_{2\varepsilon ^{-\beta _{4}}})} \{\omega _n(2\varepsilon ^{-\beta _{4}})^n \}^{\frac{2p-n}{np}} \nonumber \\\le & {} \lambda _0 \varepsilon ^{(2-\frac{n}{p})(1-\beta _{4}) } (2^n \omega _n)^{\frac{2p-n}{np}} . \end{aligned}$$

(3.65)

Thus (3.64) and (3.65) show

$$\begin{aligned} \Vert (\nabla u \cdot v)*\phi _{\varepsilon ^{\beta _{5}}}\Vert _{W^{1,n}(U_{\varepsilon ^{-\beta _{4}}})} \le c_{16} \varepsilon ^{(2-\frac{n}{p})(1-\beta _{4})-\beta _{5}} \end{aligned}$$

(3.66)

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

$$\begin{aligned} W'(f)-W'(u)*\phi _{\varepsilon ^{\beta _{5}}} = (W'(f)-W'(u))+(W'(u)-W'(u)*\phi _{\varepsilon ^{\beta _{5}}}), \end{aligned}$$

we compute

$$\begin{aligned}&\sup |W'(f)-W'(u)| \le \sup |W''|\,\sup |u-f|\,\le c_{17}\varepsilon ^{\beta _{5}}, \ \end{aligned}$$

(3.67)

$$\begin{aligned}&\sup |\nabla (W'(f)-W'(u))| \le \sup |W''| \sup |\nabla f -\nabla u| \nonumber \\&\qquad \,\,\,+ \sup |\nabla u| \sup |W'''| \sup |u-f|\nonumber \\&\qquad \le c_{17}\varepsilon ^{\beta _{5}}(2-\frac{n}{p}), \end{aligned}$$

(3.68)

$$\begin{aligned}&\sup |W'(u)-W'(u)*\phi _{\varepsilon ^{\beta _{5}}} | \le c_{17}\varepsilon ^{\beta _{5}}, \end{aligned}$$

(3.69)

$$\begin{aligned}&\sup |\nabla (W'(u)-W'(u)*\phi _{\varepsilon ^{\beta _{5}}}) | \le c_{17}\varepsilon ^{\beta _{5}}(2-\frac{n}{p}). \end{aligned}$$

(3.70)

Here $c_{17}$ depends only on $\phi ,n,\lambda _0,c_{1}$ and W. Hence, (3.67)–(3.70) show

$$\begin{aligned} \Vert W'(f)-W'(u)*\phi _{\varepsilon ^{\beta _{5}}}\Vert _{W^{1,n}(U_{\varepsilon ^{-\beta _{4}}}) \le 4c_{17} \varepsilon ^{\beta _{5}}(2-\frac{n}{p})-\beta _{4}}. \end{aligned}$$

(3.71)

By (3.62), (3.66) and (3.71), we have

$$\begin{aligned} \Vert g\Vert _{W^{1,n}(U_{\varepsilon ^{-\beta _{4}}})} \le c_{16} \varepsilon ^{(2-\frac{n}{p})(1-\beta _{4})-\beta _{5}} +4c_{17} \varepsilon ^{\beta _{5}}(2-\frac{n}{p})-\beta _{4}. \end{aligned}$$

(3.72)

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

$$\begin{aligned} \begin{aligned} \sup _{U_{\frac{\varepsilon ^{-\beta _{4}}}{2}}}&\Big ( \frac{1}{2}|\nabla f|^2 -W(f) \Big ) \\&\le c_{14}(\varepsilon ^{(2-\frac{n}{p})(1-\beta _{4})-\beta _{4}-\beta _{5}}+\varepsilon ^{\beta _{5}}(2-\frac{n}{p})-2\beta _{4}+\varepsilon ^\eta ). \end{aligned} \end{aligned}$$

(3.73)

At this point, we fix sufficiently small $0<\beta _{4},\beta _{5}<1$ depending only on n and p such that

$$\begin{aligned} \left( 2-\frac{n}{p}\right) (1-\beta _{4})-\beta _{4}-\beta _{5}>0, \ \ \ \beta _{5}\left( 2-\frac{n}{p}\right) -2\beta _{4}>0. \end{aligned}$$

This shows that we may choose a $0<\beta _{1}<1$ such that

$$\begin{aligned} \sup _{U_{\frac{\varepsilon ^{-\beta _{4}}}{2}}} \left( \frac{1}{2}|\nabla f|^2 -W(f) \right) \le \varepsilon ^{1-\beta _{1}} \end{aligned}$$

(3.74)

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

$$\begin{aligned} \sup |W(f)-W(u)| \le \sup |W'| \sup |u-f| \le c_{18}\varepsilon ^{\beta _{5}}, \end{aligned}$$

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

$$\begin{aligned} \mu _i\rightarrow \mu ,\,\, \,V_i\rightarrow V. \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} \int \frac{W}{\varepsilon _i}\phi ^2&\le c(W) \int \frac{(u_i-1)^2}{\varepsilon _i}\phi ^2 \le \frac{2c(W)}{\kappa }\int \frac{W'(u_i)(u_i-1)}{\varepsilon _i}\phi ^2 \\&=\frac{2c(W)}{\kappa }\int (\varepsilon _i \Delta u_i (u_i -1)+\varepsilon _i (v_i \cdot \nabla u_i)(u_i-1))\phi ^2. \end{aligned} \end{aligned}$$

(4.1)

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

$$\begin{aligned} D_1 r^{n-1} \le \mu (B_r(x)) \le D_2 r^{n-1}. \end{aligned}$$

(4.2)

Proof

This follows immediately from Theorem 3.8, 3.10 and Lemma 4.1. $\square$

For the subsequent use, define

$$\begin{aligned} \xi _i:=\frac{\varepsilon _i|\nabla u_i|^2}{2}-\frac{W(u_i)}{\varepsilon _i}. \end{aligned}$$

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,

$$\begin{aligned} \begin{aligned} \delta V_i(g)=&\int _{\{|\nabla u_i|\ne 0\}}\mathrm{div}\,g\,d\mu _i \\&- \frac{1}{\sigma }\int _{\{|\nabla u_i|\ne 0\}}\nabla g\cdot \Big (\frac{\nabla u_i}{|\nabla u_i|}\otimes \frac{\nabla u_i}{|\nabla u_i|}\Big )( \varepsilon _i|\nabla u_i|^2-\xi _i) \\ =&- \frac{1}{\sigma }\int \varepsilon _i(v_i\cdot \nabla u_i)(\nabla u_i\cdot g)\, +o(1) \end{aligned} \end{aligned}$$

(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

$$\begin{aligned} \begin{aligned} \int&\varepsilon _i ((v_i-v) \cdot \nabla u_i)(\nabla u_i \cdot g) \le c\left( \int |v_i-v|^p |g|^p \varepsilon _i |\nabla u_i|^2 \right) ^{\frac{1}{p}} \\&\le c\left( \int |\nabla (|v_i-v|^p |g|^p)| \right) ^{\frac{1}{p}} \\&\le c \left( \Vert \nabla v_i - \nabla v\Vert _{L^p} \Vert v_i-v\Vert _{L^p}^{p-1} +\Vert v_i-v\Vert _{L^p}^p \right) ^{\frac{1}{p}} \end{aligned} \end{aligned}$$

(4.4)

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

$$\begin{aligned} \begin{aligned} \frac{1}{\sigma }\int \varepsilon _i ({\tilde{v}} \cdot \nabla u_i)(\nabla u_i \cdot g)&= \int S^{\perp }({\tilde{v}})\cdot g\, dV_i(x,S) +o(1) \\&\rightarrow \int S^{\perp }({\tilde{v}})\cdot g\, dV(x,S). \end{aligned} \end{aligned}$$

(4.5)

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

$$\begin{aligned} \delta V(g) = -\int S^{\perp }(v)\cdot g \, dV(x,S). \end{aligned}$$

(4.6)

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

$$\begin{aligned} \int S^{\perp }(v)\cdot g\,dV(x,S)=-\int (T_x\,\Gamma )^{\perp }(v(x))\cdot g(x)\,d\Vert V\Vert (x). \end{aligned}$$

(4.7)

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

$$\begin{aligned} \int _{U_{s}} \phi ^{\frac{p(n-1)}{n-p}} d\Vert V\Vert \le c \Vert \nabla \phi \Vert _{L^p} \Vert \phi \Vert _{L^{np/(n-p)}}^{n(p-1)/(n-p)} \end{aligned}$$

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

$$\begin{aligned} \partial _t u_{\varepsilon } +v_\varepsilon \cdot \nabla u_\varepsilon = \Delta u_\varepsilon - \frac{W'(u_\varepsilon )}{\varepsilon ^2} \end{aligned}$$

(5.1)

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]).