A Remark on Hausdorff Measure in Obstacle Problems

Abstract

In this paper, we consider the identical zero obstacle problem for the second order elliptic equation

where $\Omega$ is an open bounded domain of ${\mathbb{R}}^N,N\ge 2$. We prove that the free boundary has finite $(N-1)$-Hausdorff measure, which extends the previous works by Caffarelli, Lee and Shahgholian for $p$-Laplacian equations with $p=2,p>2$ respectively and contains the singular case of $1<p<2$.