The role of interplay between coefficients in the $G$-convergence of some elliptic equations

TL;DR

This paper investigates how solutions to certain elliptic equations behave under changes in coefficients, focusing on the interplay between matrix convergence and coefficient perturbations.

Contribution

It provides new insights into the combined effects of matrix $G$-convergence and coefficient perturbations on elliptic equation solutions.

Findings

Characterization of solution stability under $G$-convergence.

Analysis of the influence of coefficient perturbations on solution behavior.

Conditions ensuring continuous dependence of solutions on coefficients.

Abstract

We study the behavior of the solutions $u$ of the linear Dirichlet problems $- \mathrm{div} (M(x) \nabla u) + a(x) u = f(x)$ with respect to perturbations of the matrix $M(x)$ (with respect to the $G$-convergence) and with respect to perturbations of the nonnegative coefficient $a(x)$ and of the right hand side $f(x)$ satisfying the condition $|f (x)| \leq Q \, a (x)$.