# A Remark on Hausdorff Measure in Obstacle Problems

### Jun Zheng

Lanzhou University, Lanzhou (Gansu), China### Peihao Zhao

Lanzhou University, Lanzhou (Gansu), China

## Abstract

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

$-\text { div}\ a(\nabla u)=-1\quad \text{in}\ \mathcal {D}'(\Omega),$

where $\Omega$ is an open bounded domain of $\mathbb{R}^{N},N\geq 2$. 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 $1< p <2$.

## Cite this article

Jun Zheng, Peihao Zhao, A Remark on Hausdorff Measure in Obstacle Problems. Z. Anal. Anwend. 31 (2012), no. 4, pp. 427–439

DOI 10.4171/ZAA/1467