Uniform estimates for the Penalized Boundary Obstacle Problem

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.