Surface Tension and $\varGamma$-Convergence of Van der Waals–Cahn–Hilliard Phase Transitions in Stationary Ergodic Media

Abstract

We study the large scale equilibrium behavior of Van der Waals–Cahn–Hilliard phase transitions in stationary ergodic media. Specifically, we are interested in free energy functionals of the following form

$$\begin{aligned} \mathcal {F}^{\omega }(u) = \int _{\mathbb {R}^{d}} \left( \frac{1}{2} \varphi ^{\omega }(x,Du(x))^{2} + W(u(x)) \right) \, dx, \end{aligned}$$

where W is a double-well potential and $\varphi ^{\omega }(x,\cdot )$ is a stationary ergodic Finsler metric. We show that, at large scales, the random energy $\mathcal {F}^{\omega }$ can be approximated by the anisotropic perimeter associated with a deterministic Finsler norm $\tilde{\varphi }$. To find $\tilde{\varphi }$, we build on existing work of Alberti, Bellettini, and Presutti, showing, in particular, that there is a natural sub-additive quantity in this context.

Appendices

Appendix A: The Fundamental Estimate of $\varGamma$-Convergence

We state and prove the version of the fundamental estimate of $\varGamma$-convergence used throughout the paper. For problems of this type, the fundamental estimate was originally established in [5].

Theorem 6

Assume that the families $(u_{\epsilon })_{\epsilon> 0}, (v_{\epsilon })_{\epsilon > 0} \subseteq H^{1}_{\text {loc}}(\mathbb {R}^{d}; [-1,1])$ and bounded open sets $(U_{\epsilon })_{\epsilon> 0}, (V_{\epsilon })_{\epsilon > 0}, U' \subseteq \mathbb {R}^{d}$ satisfy the following conditions:

• (i) $C_{0} := \sup \left\{ \mathcal {F}^{\omega }_{\epsilon }(u_{\epsilon }; U') + \mathcal {F}^{\omega }_{\epsilon }(v_{\epsilon }; V_{\epsilon }) \, \mid \, \epsilon > 0\right\} < \infty$

• (ii) $U_{\epsilon } \subseteq U$ and $V_{\epsilon } \subseteq V$ for all $\epsilon > 0$, and

  $$\begin{aligned} D&:= \inf \left\{ \text {dist}(U_{\epsilon },\partial U') \, \mid \, \epsilon> 0\right\}> 0 \\ R&:= \sup \left\{ \text {diam}(V_{\epsilon }) \, \mid \, \epsilon > 0\right\} < \infty . \end{aligned}$$

• (iii) There is a $v \in L^{1}_{\text {loc}}(\mathbb {R}^{d}; \{-1,1\})$ such that $\lim _{\epsilon \rightarrow 0^{+}} \Vert v_{\epsilon } - v\Vert _{L^{1}(V_{\epsilon })} = 0$.

If there is a measurable set $E \subseteq \mathbb {R}^{d}$ such that $\mathcal {L}^{d}(E) \le \zeta$ and $\Vert u_{\epsilon } - v_{\epsilon }\Vert _{L^{1}(V_{\epsilon } \setminus E)} \rightarrow 0$ as $\epsilon \rightarrow 0^{+}$, then there is a constant $C > 0$ depending only on D, $\lambda$, $\varLambda$, and W and a family of cut-off functions $(\psi _{\epsilon })_{\epsilon > 0} \subseteq C^{\infty }_{c}(U';[0,1])$ satisfying $\psi _{\epsilon } \equiv 1$ on $U_{\epsilon }$ such that

$$\begin{aligned} \mathcal {F}^{\omega }_{\epsilon }(\psi _{\epsilon } u_{\epsilon } + (1 - \psi _{\epsilon }) v_{\epsilon }; U_{\epsilon } \cup V_{\epsilon }) \le \mathcal {F}^{\omega }_{\epsilon }(u_{\epsilon }; U') + \mathcal {F}^{\omega }_{\epsilon }(v_{\epsilon }; V_{\epsilon }) + C \zeta + o(1). \end{aligned}$$

We remark that we sometimes apply the fundamental estimate with $\epsilon$-independent open sets U, $U'$, and V.

Proof

Fix $\epsilon > 0$. Let $N_{\epsilon } = \lceil \epsilon ^{-1} \rceil$ and pick open sets $U_{1}^{\epsilon },U_{2}^{\epsilon },\dots ,U_{N_{\epsilon }}^{\epsilon }$ such that

• (1) $U_{1}^{\epsilon } \Subset U_{2}^{\epsilon } \Subset \dots \Subset U_{N_{\epsilon }}^{\epsilon }$

• (2) $U_{1}^{\epsilon } = U_{\epsilon }$ and $U_{N_{\epsilon }}^{\epsilon } = U'$

• (3) $\text {dist}(U_{i}^{\epsilon }, \partial U_{i + 1}^{\epsilon }) \ge \frac{D}{N_{\epsilon }}$ for each $i \in \{1,2,\dots ,N_{\epsilon } - 1\}$.

For each $i \in \{1,2,\dots ,N_{\epsilon } - 1\}$, pick $\psi _{i} \in C^{\infty }_{c}(U_{i + 1}^{\epsilon };[0,1])$ such that $\psi ^{\epsilon }_{i} \equiv 1$ in $U_{i}^{\epsilon }$ and

$$\begin{aligned} \Vert D\psi _{i}\Vert _{L^{\infty }(U_{i + 1}^{\epsilon })} \le \frac{2 N_{\epsilon }}{D}. \end{aligned}$$

For convenience, write $w^{\epsilon }_{i} = \psi ^{\epsilon }_{i} u_{\epsilon } + (1 - \psi ^{\epsilon }_{i})v_{\epsilon }$.

For a fixed i, we can write

$$\begin{aligned} \mathcal {F}^{\omega }_{\epsilon }(w_{i}^{\epsilon }; U_{\epsilon } \cup V_{\epsilon })&\le \mathcal {F}^{\omega }_{\epsilon }(u_{\epsilon }; U') + \mathcal {F}^{\omega }_{\epsilon }(v_{\epsilon }; V_{\epsilon }) + \mathcal {F}^{\omega }_{\epsilon }(w^{\epsilon }_{i}; (U_{i + 1}^{\epsilon } \setminus U_{i}^{\epsilon }) \cap V_{\epsilon }). \end{aligned}$$

Appealing to the definitions, we estimate the error term as

$$\begin{aligned} \mathcal {F}^{\omega }_{\epsilon }(w^{\epsilon }_{i}; (U_{i + 1}^{\epsilon } \setminus U_{i}^{\epsilon }) \cap V_{\epsilon }) \le e_{1}(i,\epsilon ) + e_{2}(i,\epsilon ) + e_{3}(i,\epsilon ) \end{aligned}$$

where

$$\begin{aligned} e_{1}(i,\epsilon )&= \epsilon \varLambda \int _{(U_{i + 1}^{\epsilon } \setminus U_{i}^{\epsilon }) \cap V_{\epsilon }} \left( |Du_{\epsilon }(x)|^{2} + |Dv_{\epsilon }(x)|^{2}\right) \, dx \\ e_{2}(i,\epsilon )&= \epsilon ^{-1} \int _{(U_{i + 1}^{\epsilon } \setminus U_{i}^{\epsilon }) \cap V_{\epsilon }} W(w^{\epsilon }_{i}(x)) \, dx \\ e_{3}(i,\epsilon )&= \epsilon \varLambda \left( \frac{2 N_{\epsilon }}{D} \right) ^{2} \int _{(U_{i + 1}^{\epsilon } \setminus U_{i}^{\epsilon }) \cap V_{\epsilon }} (u_{\epsilon }(x) - v_{\epsilon }(x))^{2} \, dx. \end{aligned}$$

Summing over i, we find

$$\begin{aligned} \sum _{i = 1}^{N_{\epsilon } - 1} \mathcal {F}^{\omega }_{\epsilon }(w^{\epsilon }_{i}; (U_{i + 1}^{\epsilon } \setminus U_{i}^{\epsilon }) \cap V_{\epsilon })&\le \frac{\varLambda }{\lambda } C_{0} + \epsilon \varLambda \left( \frac{2 N_{\epsilon }}{D} \right) ^{2} \int _{U' \cap V_{\epsilon }} (u_{\epsilon }(x) - v_{\epsilon }(x))^{2} \ dx \\&\quad + \epsilon ^{-1} \sum _{i = 1}^{N_{\epsilon } - 1} \int _{(U_{i + 1}^{\epsilon } \setminus U_{i}^{\epsilon }) \cap V_{\epsilon }} W(w_{i}^{\epsilon }(x)) \, dx. \end{aligned}$$

Thus, there is a $j_{\epsilon } \in \{1,2,\dots ,N_{\epsilon } - 1\}$ such that

$$\begin{aligned} \mathcal {F}^{\omega }_{\epsilon }(w^{\epsilon }_{j_{\epsilon }}; (U_{j_{\epsilon } + 1}^{\epsilon } \setminus U_{j_{\epsilon }}^{\epsilon }) \cap V_{\epsilon })&\le \epsilon \varLambda (N_{\epsilon } - 1)^{-1} \left( \frac{2 N_{\epsilon }}{D}\right) ^{2} \int _{V'} (u_{\epsilon }(x) - v_{\epsilon }(x))^{2} \, dx \\&\quad + \epsilon ^{-1} (N_{\epsilon } - 1)^{-1} \sum _{i = 1}^{N_{\epsilon } - 1} \int _{(U^{\epsilon }_{i + 1} \setminus U^{\epsilon }_{i}) \cap V_{\epsilon }} W(w_{i}^{\epsilon }(x)) \, dx \\&\quad + \frac{\varLambda }{\lambda } C_{0} (N_{\epsilon } - 1)^{-1}. \end{aligned}$$

Observe that there is a constant $C_{1} > 0$ depending only on D, $\lambda$, and $\varLambda$ such that

$$\begin{aligned} \max \left\{ \epsilon \varLambda (N_{\epsilon } - 1)^{-1} \left( \frac{2 N_{\epsilon }}{D}\right) ^{2}, \epsilon ^{-1} (N_{\epsilon } - 1)^{-1} \right\} \le C_{1}. \end{aligned}$$

Moreover, by assumption,

$$\begin{aligned} \limsup _{\epsilon \rightarrow 0^{+}} \int _{V_{\epsilon }} \left( u_{\epsilon }(x) - v_{\epsilon }(x)\right) ^{2} \, dx \le 4\zeta . \end{aligned}$$

Similarly, since W is continuous on $[-1,1]$ and $\mathcal {L}^{d}(V_{\epsilon }) \le \omega _{d} R^{d}$ independently of $\epsilon$, we find

$$\begin{aligned} \limsup _{\epsilon \rightarrow 0^{+}} \left[ \sum _{i = 1}^{N_{\epsilon } - 1} \int _{(U^{\epsilon }_{i + 1} \setminus U^{\epsilon }_{i}) \cap V_{\epsilon }} W(w_{i}^{\epsilon }(x)) \, dx \right] \le \Vert W\Vert _{L^{\infty }([-1,1])} \zeta . \end{aligned}$$

Therefore, as $\epsilon \rightarrow 0^{+}$,

$$\begin{aligned} \mathcal {F}^{\omega }_{\epsilon }(w_{j_{\epsilon }}^{\epsilon }; U_{\epsilon } \cup V_{\epsilon }) \le \mathcal {F}^{\omega }_{\epsilon }(u_{\epsilon }; U') + \mathcal {F}^{\omega }_{\epsilon }(v_{\epsilon }; V_{\epsilon }) + C_{1}(4 + \Vert W\Vert _{L^{\infty }([-1,1])}) \zeta + o(1). \end{aligned}$$

The theorem follows by setting $\psi _{\epsilon } = \psi ^{\epsilon }_{j_{\epsilon }}$ and $C = C_{1}(4 + \Vert W\Vert _{L^{\infty }([-1,1])})$. $\square$

Appendix B: Elements of Ergodic Theory

1.1 B.1 Conditional Expectation and Translations

In the proof of the thermodynamic limit, the following lemma was used. In the sequel, if X is a random variable on $(\varOmega ,\mathscr {B},\mathbb {P})$ and $\mathcal {G} \subseteq \mathcal {F}$ is a sub-$\sigma$-algebra, we let $\mathbb {E}(X \mid \mathcal {G})$ denote some fixed representative of the conditional expectation of X with respect to $\mathcal {G}$. Recall that $\mathbb {E}(X \mid \mathcal {G})$ is $\mathcal {G}$-measurable by definition, and it is unique up to almost sure equivalence.

Lemma 3

If X is a random variable on $(\varOmega ,\mathscr {B},\mathbb {P})$, $e \in S^{d -1}$, and $x \in \langle e \rangle ^{\perp }$, then

$$\begin{aligned} \mathbb {E}(X \circ \tau _{x} \, \mid \, \varSigma _{e}) = \mathbb {E}(X \, \mid \, \varSigma _{e}) \quad \mathbb {P}\text {-almost surely.} \end{aligned}$$

Proof

Recall that $E \in \varSigma _{e}$ if and only if $1_{E} \circ \tau _{x} = 1_{E}$ no matter the choice of $x \in \langle e \rangle ^{\perp }$. Thus, for such an E and x, we find

$$\begin{aligned} \mathbb {E}(X \circ \tau _{x} : E) = \mathbb {E}((X \circ \tau _{x}) 1_{E}) =\mathbb {E}((X 1_{E})\circ \tau _{x}) = \mathbb {E}(X : E). \end{aligned}$$

Since E was arbitrary, uniqueness implies $\mathbb {E}(X \circ \tau _{x} \, \mid \, \varSigma _{e}) = \mathbb {E}(X \, \mid \, \varSigma _{e})$ almost surely. $\square$

1.2 B.2 Sub-additive ergodic theorem

Since we will be applying this theorem in a somewhat unconventional “non-ergodic” form, we will state it carefully and give most of the details of the proof.

We follow Dal Maso and Modica [13] in the following definition:

Definition 5

Given $e \in S^{d - 1}$, a random function $\psi ^{\omega } : \mathcal {U}_{0}^{d - 1} \rightarrow \mathbb {R}$ is a sub-additive process over e if it satisfies the following conditions:

• (i) If $G, G_{1},\dots ,G_{N} \in \mathcal {U}_{0}^{d - 1}$, $G_{i} \subseteq G$ for each i, $G_{i} \cap G_{j} = \phi$ for distinct i, j, and $\mathcal {L}^{d - 1}(G \setminus (\cup _{j = 1}^{N} G_{j})) = 0$, then

  $$\begin{aligned} \psi ^{\omega } (G) \le \sum _{i = 1}^{N} \psi ^{\omega }(G_{i}). \end{aligned}$$

• (ii) If $G \in \mathcal {U}_{0}^{d - 1}$ and $x \in \langle e \rangle ^{\perp }$, then

  $$\begin{aligned} \psi ^{\omega }(G + O_{e}^{-1}(x)) = \psi ^{\tau _{x}\omega }(G). \end{aligned}$$

Here is the version of the sub-additive ergodic theorem we will use.

Theorem 7

Suppose $\psi ^{\omega }$ is a sub-additive process over e and there is a constant $C_{\psi } > 0$ such that

$$\begin{aligned} 0 \le \psi ^{\omega }(G) \le C_{\psi } \mathcal {L}^{d - 1}(G) \quad \text {if} \, \, G \in \mathcal {U}_{0}^{d - 1}. \end{aligned}$$

(31)

Define $\overline{\psi }^{\omega }$ and $\underline{\psi }^{\omega }$ by

$$\begin{aligned} \overline{\psi }^{\omega }&= \limsup _{R \rightarrow \infty } R^{1 - d} \psi ^{\omega }(Q(0,R)) \\ \underline{\psi }^{\omega }&= \liminf _{R \rightarrow \infty } R^{1 - d} \psi ^{\omega }(Q(0,R)). \end{aligned}$$

Then $\overline{\psi }^{\omega },\underline{\psi }^{\omega }$ are $\varSigma _{e}$-measurable random variables and $\overline{\psi }^{\omega } = \underline{\psi }^{\omega }$ almost surely. Moreover, there is an event $\varOmega _{\psi } \in \varSigma _{e}$ such that $\mathbb {P}(\varOmega _{\psi }) = 1$ and if $\omega \in \varOmega _{\psi }$ and Q is a cube in $\mathbb {R}^{d - 1}$, then

$$\begin{aligned} \underline{\psi }^{\omega } \mathcal {L}^{d - 1}(Q) = \lim _{R \rightarrow \infty } R^{1- d} \psi ^{\omega }(RQ). \end{aligned}$$

(32)

The fact that $\overline{\psi }$, $\underline{\psi }$, and $\varOmega _{\psi }$ are $\varSigma _{e}$-measurable is used in Lemma 1 and Theorem 2.

Proof

First, we claim that

$$\begin{aligned} \overline{\psi }^{\omega }&= \limsup _{\mathbb {N} \ni N \rightarrow \infty } N^{1 - d} \psi ^{\omega }(Q(0,N)) \\ \underline{\psi }^{\omega }&= \liminf _{\mathbb {N} \ni N \rightarrow \infty } N^{1 - d} \psi ^{\omega }(Q(0,N)). \end{aligned}$$

Indeed, this follows immediately from (31) and condition (i) in the definition of sub-additive process. If $N = \lfloor R \rfloor$, then

$$\begin{aligned}&(N + 1)^{1- d} \psi ^{\omega }(Q(0,N + 1)) - C \left\{ 1 - \left( \frac{R}{N + 1}\right) ^{d - 1}\right\} \le R^{1 - d} \psi ^{\omega }(Q(0,R)) \\&R^{1 - d} \psi ^{\omega }(Q(0,R)) \le N^{1 - d} \psi ^{\omega }(Q(0,N)) + C \left\{ 1 - \left( \frac{N}{R}\right) ^{d - 1}\right\} . \end{aligned}$$

Thus, the limit inferior and limit superior do not change if we restrict R to $\mathbb {N}$.

Now we claim that $\overline{\psi }^{\omega }$ and $\underline{\psi }^{\omega }$ are $\varSigma _{e}$-measurable. This follows from condition (ii) in the definition of sub-additive process. Suppose $x \in \langle e \rangle ^{\perp }$. Notice that $Q(0,N) \subseteq Q(x,N + |x|_{\infty })$. Moreover, $\frac{N + |x|_{\infty }}{N} \rightarrow 1$ as $N \rightarrow \infty$. Using (31), we find

$$\begin{aligned} \psi ^{\omega }(Q(O_{e}^{-1}(x),N + |x|_{\infty })) \le \psi ^{\omega }(Q(0,N)) + C_{\psi } ((N + |x|_{\infty })^{d - 1} - N^{d - 1}). \end{aligned}$$

Therefore, from the equality $\psi ^{\tau _{x}\omega }(Q(0,N + |x|_{\infty })) = \psi ^{\omega }(Q(O_{e}^{-1}(x),N + |x|_{\infty }))$, we deduce that $\overline{\psi }^{\tau _{x} \omega } \le \overline{\psi }^{\omega }$ and $\underline{\psi }^{\tau _{x}\omega } \le \underline{\psi }^{\omega }$. Replacing x with $-x$ yields the complementary inequality. Thus, $\overline{\psi }^{\omega }$ and $\underline{\psi }^{\omega }$ are $\varSigma _{e}$-measurable.

Let $\varOmega ^{(1)} = \{\omega \in \varOmega \, \mid \, \overline{\psi }^{\omega } = \underline{\psi }^{\omega }\}$. By the (multi-parameter) sub-additive ergodic theorem (cf. [19]), $\mathbb {P}(\varOmega ^{(1)}) = 1$. In other words, $\overline{\psi }^{\omega } = \underline{\psi }^{\omega }$ almost surely.

Next, let $\mathcal {D}_{\mathbb {Q}}$ be the family of all open cubes in $\mathbb {R}^{d -1}$ with rational endpoints, that is, $\mathcal {D}_{\mathbb {Q}} = \{Q(y,\rho ) \, \mid \, y \in \mathbb {Q}^{d - 1}, \, \, \rho \in \mathbb {Q} \cap (0,\infty )\}$. Define the event $\varOmega ^{(2)}$ by

$$\begin{aligned} \varOmega ^{(2)} = \bigcap _{Q \in \mathcal {D}_{\mathbb {Q}}} \left\{ \omega \in \varOmega \, \mid \, \underline{\psi }^{\omega }\mathcal {L}^{d -1}(Q) = \lim _{\mathbb {N} \ni N \rightarrow \infty } N^{1- d} \psi ^{\omega }(NQ) \right\} . \end{aligned}$$

The sub-additive ergodic theorem implies $\varOmega ^{(2)}$ is a countable intersection of probability one events. Therefore, $\mathbb {P}(\varOmega ^{(2)}) = 1$. Arguing as before, we see that $\varOmega ^{(2)}$ can alternatively be characterized as follows:

$$\begin{aligned} \varOmega ^{(2)} = \bigcap _{Q \in \mathcal {D}_{\mathbb {Q}}} \left\{ \omega \in \varOmega \, \mid \, \underline{\psi }^{\omega }\mathcal {L}^{d -1}(Q) = \lim _{R \rightarrow \infty } R^{1- d} \psi ^{\omega }(RQ) \right\} . \end{aligned}$$

Similarly, the arguments used to show $\underline{\psi }^{\omega }$ and $\overline{\psi }^{\omega }$ are $\varSigma _{e}$-measurable can be adapted to prove $\varOmega ^{(2)} \in \varSigma _{e}$.

Finally, let $\mathcal {D} = \{Q(y,\rho ) \, \mid \, y \in \mathbb {R}^{d - 1}, \, \rho > 0\}$. We claim that

$$\begin{aligned} \varOmega ^{(2)} = \bigcap _{Q \in \mathcal {D}} \left\{ \omega \in \varOmega \, \mid \, \underline{\psi }^{\omega }\mathcal {L}^{d -1}(Q) = \lim _{R \rightarrow \infty } R^{1- d} \psi ^{\omega }(RQ) \right\} . \end{aligned}$$

This follows since we can approximate any cube in $\mathcal {D}$ by a cube in $\mathcal {D}_{\mathbb {Q}}$. As the arguments are similar to the ones already exposed, we omit the details.

The proposition follows with $\varOmega _{\psi } = \varOmega ^{(1)} \cap \varOmega ^{(2)}$. $\square$

In Sect. 4.4, we will use the following version of Birkhoff’s ergodic theorem. Since we use the fact that the event on which it holds is in $\varSigma$, we provide a complete statement and sketch the proof:

Theorem 8

If $E \in \mathscr {B}$, then there is an $\varOmega _{E} \in \varSigma$ satisfying $\mathbb {P}(\varOmega _{E}) = 1$ such that if Q is any open cube in $\mathbb {R}^{d}$, then

$$\begin{aligned} \mathbb {P}(E) \mathcal {L}^{d}(Q) = \lim _{R \rightarrow \infty } R^{-d} \mathcal {L}^{d}(\{x \in RQ \, \mid \, \tau _{x} \omega \in E\}) \quad \text {if} \, \, \omega \in \varOmega _{E}. \end{aligned}$$

The $\varSigma$-measurability of $\varOmega _{E}$ is used in the proof of Proposition 9 to show that the event $\hat{\varOmega }$ in Theorem 1 is itself $\varSigma$-measurable.

Proof

Define a sub-additive process $\psi ^{\omega } : \mathcal {U}_{0}^{d} \rightarrow \mathbb {R}$ by

$$\begin{aligned} \psi ^{\omega }(A) = \mathcal {L}^{d}(\{x \in A \, \mid \, \tau _{x} \omega \in E\}). \end{aligned}$$

It is straightforward to verify that $\psi ^{\omega }$ satisfies Definition 5 with $d - 1$ replaced by d and $\langle e \rangle ^{\perp }$ replaced by $\mathbb {R}^{d}$. These changes have no bearing on the proof of Theorem 7 other than switching $\varSigma$ for $\varSigma _{e}$.

As for the value of the limit, we can see that it should be $\mathbb {P}(E)$ by appealing to the dominated convergence theorem and ergodicity. $\square$