On homogenization problems for fully nonlinear equations with oscillating Dirichlet boundary conditions

TL;DR

This paper investigates how to recover Dirichlet boundary conditions in the limit for fully nonlinear equations with oscillating boundary data, using blow-up techniques for homogenization in elliptic and parabolic problems.

Contribution

It introduces a method to determine the limiting Dirichlet boundary condition for fully nonlinear equations with oscillating boundary data, via a blow-up analysis near the boundary.

Findings

Established a blow-up approach for boundary analysis

Derived the limiting Dirichlet condition for homogenized equations

Applied techniques to both elliptic and parabolic problems

Abstract

We study two types of asymptotic problems whose common feature - and difficulty- is to exhibit oscillating Dirichlet boundary conditions : the main contribution of this article is to show how to recover the Dirichlet boundary condition for the limiting equation. These two types of problems are (i) periodic homogenization problems for fully nonlinear, second-order elliptic partial differential equations set in a half-space and (ii) parabolic problems with an oscillating in time Dirichlet boundary condition. In order to obtain the Dirichlet boundary condition for the limiting problem, the key step is a blow-up argument near the boundary which leads to the study of Dirichlet problems set on half space type domains and of the asymptotic behavior of the solutions when the distance to the boundary tends to infinity.