A Remark on Hausdorff Measure in Obstacle Problems

  • Jun Zheng

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

    Lanzhou University, Lanzhou (Gansu), China


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

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

where Ω\Omega is an open bounded domain of RN,N2\mathbb{R}^{N},N\geq 2. We prove that the free boundary has finite (N1)(N-1)-Hausdorff measure, which extends the previous works by Caffarelli, Lee and Shahgholian for pp-Laplacian equations with p=2,p>2p=2,p>2 respectively and contains the singular case of 1<p<21< 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