Uniform estimates for the Penalized Boundary Obstacle Problem

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.

arXiv commentA section on decay rates for Higher holder norms has been added