Quantitative Homogenization of Elliptic PDE with Random Oscillatory Boundary Data

TL;DR

This paper investigates how solutions to nonlinear elliptic PDEs with rapidly oscillating random boundary data behave on average, establishing homogenization results with probabilistic convergence rates.

Contribution

It provides the first almost sure homogenization results for nonlinear elliptic PDEs with random oscillatory boundary data, including convergence rates.

Findings

Proves almost sure homogenization under certain conditions.

Establishes convergence rates in probability.

Demonstrates concentration of measure effects.

Abstract

We study the averaging behavior of nonlinear uniformly elliptic partial differential equations with random Dirichlet or Neumann boundary data oscillating on a small scale. Under conditions on the operator, the data and the random media leading to concentration of measure, we prove an almost sure and local uniform homogenization result with a rate of convergence in probability.