Low regularity potentials in heterogeneous Cahn--Hilliard functionals

1. Problem Statement

The paper investigates the asymptotic behavior (as $\varepsilon \to 0$) of the Cahn–Hilliard functional in a highly irregular, heterogeneous setting. The functional models liquid-liquid phase separation and is given by:
$$F_\varepsilon(u) := \int_\Omega \left[ \frac{1}{\varepsilon} W(x, u(x)) + \varepsilon |\nabla u(x)|^2 \right] dx$$
where $u: \Omega \to \mathbb{R}^M$ is a phase parameter, and $W(x, u)$ is a potential energy density.

Key Challenges Addressed:

• Low Regularity of Wells: Previous literature typically assumed the "wells" (the stable phases $a(x)$ and $b(x)$ where $W=0$) were Lipschitz continuous. This paper considers wells that are merely of Bounded Variation (BV), allowing them to have jump discontinuities.
• Low Regularity of Potential: The potential $W$ is not assumed to be quadratic near the wells (a standard assumption in prior works). Instead, it satisfies mild growth conditions controlled by a function $f(t)$.
• Vector-Valued Case: The analysis covers the case where the phase variable is vector-valued ($M \ge 1$), which is significantly more complex than the scalar case due to the geometry of the transition paths (geodesics).

The goal is to establish Compactness and $\Gamma$-convergence of the family $\{F_\varepsilon\}$ to a limiting functional $F_0$ that describes the interfacial energy of the phase separation.

2. Methodology

The authors employ a combination of variational analysis, geometric measure theory, and Young measure techniques.

A. Structural Assumptions

The paper introduces a set of weak structural assumptions (H1–H8) on the potential $W$ and wells $a, b$:

• Carathéodory Condition: $W$ is measurable in $x$ and continuous in $u$.
• Well Separation: The distance between wells $|a(x) - b(x)|$ is bounded away from zero.
• Growth Conditions: $W$ is controlled by a function $f$ near the wells (non-degeneracy) and has at least linear growth at infinity. Crucially, $f$ satisfies a doubling condition ($f(2t) \le C f(t)$), which is much weaker than the quadratic growth required in previous works.
• Regularity of Wells: The wells $a, b$ are in $BV(\Omega) \cap L^\infty(\Omega)$. In the $\Gamma$-convergence result, they are piecewise continuous with jumps on a piecewise $C^2$ partition.
• Relative Continuity: A condition (H7) is imposed on the "adjusted" potential to allow freezing the spatial variable $x$ during blow-up arguments.

B. Key Technical Tools

1. Young Measures: Used to prove compactness. The authors transform the problem to a fixed-well setting using a change of variables $T_a(x)$, then analyze the generated Young measures to show the limit function takes values only in the wells $\{a(x), b(x)\}$.
2. Geodesic Distance Analysis: A central technical contribution is the study of the geodesic distance $d_W(x; p, q)$ associated with the degenerate metric $\sqrt{W}$.
   • They prove the existence of geodesics even when $W$ is discontinuous.
   • They establish a uniform bound on the Euclidean length of these geodesics under the weak doubling condition on $f$. This is critical because, in the vector-valued case, the transition path is not a straight line; without a length bound, the recovery sequence construction fails.
   • They provide a counter-example (Example 4.10) showing that if the growth conditions are too weak, geodesics can have infinite Euclidean length.
3. Approximation of Sets of Finite Perimeter: They use a refined approximation result (Proposition 4.15) to approximate the jump set of the limit function and the wells by $C^2$ surfaces, ensuring the energy converges correctly across the interfaces.
4. Blow-up Method: Used for the $\Gamma$-liminf inequality. By rescaling around a jump point and using the relative continuity of the potential, they "freeze" the spatial dependence to reduce the problem to a 1D optimal profile problem.

3. Key Contributions

A. Relaxation of Regularity Assumptions

The paper significantly weakens the hypotheses required for $\Gamma$-convergence compared to existing literature (e.g., Fonseca-Tartar, Bouchitté, Cristoferi-Ganedi-Fonseca):

• Wells: Allowed to be in $BV$ (jump discontinuities permitted) rather than Lipschitz.
• Potential Growth: The potential need not be quadratic near the wells. It only needs to satisfy a doubling condition (H8) and be controlled by a function $f$. This includes potentials that are concave near the wells, which were previously excluded.

B. Uniform Length Bound for Degenerate Geodesics

A major theoretical contribution is Theorem 4.8, which proves that under the doubling condition on $f$, the geodesics connecting the wells have a uniformly bounded Euclidean length.

• Significance: In the vector-valued case, the optimal transition profile is a curve solving a degenerate geodesic problem. If the curve's length is unbounded, the energy of the recovery sequence cannot be controlled. This result bridges the gap between low regularity potentials and the existence of valid recovery sequences.

C. Mass Constraint Handling

The authors extend their results to the case where a mass constraint ($\int_\Omega u = m$) is imposed. They adapt the recovery sequence construction by modifying the approximation of the phase set to strictly satisfy the volume constraint while preserving the energy convergence.

4. Main Results

Theorem 2.16 (Compactness)

Under assumptions (H1)–(H5), if a sequence $\{u_n\}$ has uniformly bounded energy $F_{\varepsilon_n}(u_n)$, then there exists a subsequence converging strongly in $L^1(\Omega; \mathbb{R}^M)$ to a function $u \in BV(\Omega; \{a, b\})$.

Theorem 2.17 ($\Gamma$-Convergence)

Under assumptions (H1)–(H8), the functionals $F_{\varepsilon_n}$ $\Gamma$-converge to the functional $F_\infty$ in the $L^1$ topology:
$$F_\infty(u) = \begin{cases} \int_{J_u \cap \Omega} d_W(x; u^-(x), u^+(x)) \, d\mathcal{H}^{N-1}(x) & \text{if } u \in BV(\Omega; \{a, b\}) \\ +\infty & \text{otherwise} \end{cases}$$
where $d_W(x; p, q)$ is the geodesic distance defined by:
$$d_W(x; p, q) = \inf \left\{ \int_{-1}^1 2\sqrt{W(x, \gamma(t))} |\gamma'(t)| \, dt : \gamma(-1)=p, \gamma(1)=q \right\}$$

Theorem 2.20 (Mass Constrained Case)

The compactness and $\Gamma$-convergence results hold even when a mass constraint is imposed, provided the growth function $f$ satisfies a slightly stronger condition ($f(t) \le |t|^\alpha$ with $\alpha > 1$) to ensure the differentiability of the potential near the wells for the mass correction.

5. Significance and Impact

• Mathematical Rigor: The paper resolves the technical difficulty of handling vector-valued phase transitions with discontinuous material properties (heterogeneous media). Previous results required Lipschitz continuity of the wells to ensure the transition paths were well-behaved.
• Physical Relevance: Many real-world materials (e.g., composites, polycrystals) have interfaces or inclusions where material properties jump. This model allows for a rigorous mathematical description of phase separation in such complex, non-smooth environments.
• Generalization of Potential: By removing the quadratic growth requirement, the theory applies to a broader class of physical potentials, including those with "flat" or concave bottoms near the wells, which are relevant in certain thermodynamic models.
• Technical Innovation: The proof of the uniform Euclidean length bound for geodesics in degenerate metrics is a standalone result of interest in geometric analysis and optimal transport theory.

In summary, this work provides a robust framework for analyzing phase separation in highly irregular heterogeneous media, extending the classical Cahn-Hilliard theory to settings with low regularity potentials and discontinuous material parameters.