Uniform estimates for the Penalized Boundary Obstacle Problem

TL;DR

This paper establishes uniform estimates for the fractional penalized obstacle problem, especially for the boundary case with s=1/2, extending previous results to a more general setting motivated by homogenization theory.

Contribution

It introduces uniform estimates for the fractional penalized obstacle problem, generalizing earlier results to the boundary case and independent of the penalization parameter.

Findings

Obtained sharp estimates for solutions independent of epsilon.

Extended classical results to fractional and boundary obstacle problems.

Provided tools relevant for homogenization theory applications.

Abstract

In this paper, motivated by a problem arising in random homogenization theory, we initiate the study of uniform estimates for the fractional penalized obstacle problem, $ \Delta^{s}u^{\epsilon} = \beta_{\epsilon} (u^{\epsilon})$. In particular we consider the penalized boundary obstacle problem, $s = \frac{1}{2}$, and obtain sharp estimates for the solution independent of the penalizing parameter $\epsilon$. This is a generalization of a result due to H. Brezis and D. Kinderlehrer.