An H-convergence-based implicit function theorem for homogenization of nonlinear non-smooth elliptic systems

Technical Summary: An H-convergence-based implicit function theorem for homogenization of nonlinear non-smooth elliptic systems

Problem Statement
The paper addresses the homogenization of boundary value problems for semilinear second-order elliptic systems of the form:
$$\partial_{x_i} \left( a_{ij}^{\alpha\beta}(\varepsilon, x)\partial_{x_j}u^\beta(x) + f_i^\alpha(x, u(x)) \right) = 0 \quad \text{in } \Omega,$$
subject to homogeneous Dirichlet boundary conditions. Here, $\varepsilon > 0$ is a small homogenization parameter, $\Omega \subset \mathbb{R}^N$ is a bounded Lipschitz domain, and the diffusion tensors $a_{ij}^{\alpha\beta}(\varepsilon, \cdot)$ satisfy the Legendre ellipticity condition. The nonlinearity $f_i^\alpha$ is assumed to be $C^1$-smooth as a map from $C(\Omega; \mathbb{R}^n)$ to $L^{p_0}(\Omega)$ with $p_0 > N$.

The diffusion tensors are assumed to $H$-converge to a homogenized tensor $\hat{a}_{ij}^{\alpha\beta}$ as $\varepsilon \to 0$. The central problem is to establish the existence and local uniqueness of weak solutions $u_\varepsilon$ to the oscillating problem (1.1) near a non-degenerate weak solution $u_0$ of the homogenized problem (1.9), specifically proving strong convergence $\|u_\varepsilon - u_0\|_\infty \to 0$.

Methodology
The author employs a generalized implicit function theorem approach, adapted for singular perturbations and homogenization, rather than the classical implicit function theorem. The core difficulty addressed is that the linearized operators for the oscillating problem converge only in the weak sense of $H$-convergence, not with respect to a uniform operator norm, which precludes the direct application of classical theorems.

The methodology proceeds through the following steps:

1. Abstract Setting: The boundary value problems are reformulated as operator equations $A_\varepsilon u + DF(u) = 0$ in appropriate Banach spaces.
2. Approximate Solutions: Instead of using the homogenized solution $u_0$ as a starting point for iteration (which fails due to lack of regularity or convergence properties in the specific norms required), the author constructs a family of approximate solutions $\bar{u}_\varepsilon$ defined by the linear problem $A_\varepsilon \bar{u}_\varepsilon + DF(u_0) = 0$.
3. Fixed Point Argument: A generalized implicit function theorem (based on R.J. Magnus) is applied. The proof utilizes Banach's fixed point theorem on a sequence of iterates defined by the approximate solutions.
4. Function Space Selection: The choice of Banach spaces $U$ and $V$ is critical and depends on the dimension $N$ and the structure of the diffusion tensor:
   • Case $N=2$: The proof utilizes Sobolev spaces $W^{1,p}_0(\Omega; \mathbb{R}^n)$ with $p > 2$ (Meyers type estimates).
   • Case Triangular Systems ($a_{ij}^{\alpha\beta} = 0$ for $\alpha > \beta$): The proof utilizes Sobolev-Morrey spaces $W^{1,2,\lambda}_0(\Omega; \mathbb{R}^n)$ with $\lambda > N-2$. This allows handling non-smooth data where standard Sobolev embeddings might fail or where the homogenized tensor loses the triangular structure.
5. Regularity Tools: The proofs rely heavily on maximal regularity estimates for linear elliptic systems with non-smooth data (Meyers and Morrey estimates) to ensure the necessary compactness and continuity properties in the chosen function spaces.

Key Contributions and Results
The main result is Theorem 1.1, which states that if $u_0$ is a non-degenerate weak solution to the homogenized problem (i.e., the linearized homogenized operator is invertible), then for sufficiently small $\varepsilon$, there exists a unique weak solution $u_\varepsilon$ to the original problem such that $\|u_\varepsilon - u_0\|_\infty < \delta$ and $\|u_\varepsilon - u_0\|_\infty \to 0$ as $\varepsilon \to 0$.

Specific technical contributions include:

• Relaxation of Smoothness Assumptions: The result holds without requiring additional smoothness of the domain boundary $\partial\Omega$, the diffusion coefficients $a_{ij}^{\alpha\beta}$, the nonlinearity $f$, or the solution $u_0$. It accommodates non-smooth data ($L^\infty$ coefficients).
• No Global Uniqueness Required: The theorem establishes local uniqueness near $u_0$ without assuming global uniqueness of solutions for the nonlinear problem.
• Handling Non-Triangular Homogenization: In the case of triangular systems (condition 1.8), the author notes that the homogenized tensor $\hat{a}_{ij}^{\alpha\beta}$ is generally not triangular. The use of Sobolev-Morrey spaces allows the proof to proceed despite this loss of structure in the limit.
• Strong $L^\infty$ Convergence: The result guarantees convergence in the $L^\infty$ norm, which is stronger than the standard weak $H^1$ convergence typical in homogenization theory.
• Hölder Continuity: As a byproduct (Remark 1.7), the solutions $u_\varepsilon$ are shown to be Hölder continuous.

Significance and Claims
The paper claims that this result is new even in the linear case (where $f$ is independent of $u$). In the linear case, while the Lax-Milgram lemma guarantees unique weak solvability in $W^{1,2}_0$, it is not immediately clear that solutions belong to $L^\infty$ or that they converge strongly in $L^\infty$ under $H$-convergence with non-smooth data.

The author emphasizes that the approach bridges the gap between singular perturbation theory and homogenization theory by utilizing approximate solutions constructed via linearization rather than asymptotic expansions involving correctors. This avoids the need for smoothing operators and additional regularity assumptions on $u_0$.

The paper explicitly states that it does not provide a rate of convergence for $\|u_\varepsilon - u_0\|_\infty \to 0$ due to the non-smoothness of the data, contrasting this with cases where smooth data allows for rates like $O(\varepsilon)$ or $O(\varepsilon^\omega)$. The work is presented as a foundational existence and local uniqueness result for non-smooth, non-periodic homogenization of nonlinear systems.