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

TL;DR

This paper establishes an implicit function theorem for homogenization of nonlinear elliptic systems with non-smooth data, using H-convergence and gradient estimates to prove the existence and uniqueness of solutions.

Contribution

It introduces a new implicit function theorem framework for homogenization of nonlinear non-smooth elliptic systems based on H-convergence.

Findings

Existence of a unique weak solution near the homogenized solution for small parameters.

Application of Meyers and Morrey gradient estimates in the proof.

Demonstration of the implicit function theorem approach in homogenization context.

Abstract

We consider homogenization of Dirichlet problems for semilinear elliptic systems with non-smooth data. We suppose that the diffusion tensors H-converge if the homogenization parameter tends to zero. Our result is of implicit function theorem type: For small homogenization parameter there exists exactly one weak solution close to a given non-degenerate weak solution to the homogenized problem. For the proofs we use gradient estimates of Meyers (if the space dimension is two) or Morrey (if the diffusion tensors are triangular) type for solutions to linear elliptic systems.