Homogenization of fully nonlinear elliptic equations with oscillating dirichlet boundary data

TL;DR

This paper studies the homogenization process for fully nonlinear elliptic equations with oscillating Dirichlet boundary data, showing convergence of solutions to a limit that solves a related boundary value problem.

Contribution

It establishes conditions under which solutions to oscillating boundary problems converge to a homogenized solution, extending understanding of boundary effects in nonlinear elliptic equations.

Findings

Solution $u_$ converges uniformly on compact sets.

Limit function $ar u$ solves a specific boundary value problem.

Convergence requires no flat boundary spots and large elliptic constant ratio.

Abstract

This paper deals with the homogenization of fully nonlinear second order equation with an oscillating Dirichlet boundary data when the operator and boundary data are $\e$-periodic. We will show that the solution $u_\e$ converges to some function $\bar u(x)$ uniformly on every compact subset $K$ of the domain $D$. Moreover, $\bar u$ is a solution to some boundary value problem. For this result, we assume that the boundary of the domain has no (rational) flat spots and the ratio of elliptic constants $\Lambda / \lambda$ is sufficiently large.