The EMS Publishing House is now EMS Press and has its new home at

Please find all EMS Press journals and articles on the new platform.

Zeitschrift für Analysis und ihre Anwendungen

Full-Text PDF (137 KB) | Metadata | Table of Contents | ZAA summary
Volume 31, Issue 4, 2012, pp. 427–439
DOI: 10.4171/ZAA/1467

Published online: 2012-10-10

A Remark on Hausdorff Measure in Obstacle Problems

Jun Zheng[1] and Peihao Zhao[2]

(1) Lanzhou University, Lanzhou (Gansu), China
(2) Lanzhou University, Lanzhou (Gansu), China

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 $1< p <2$.

Keywords: Elliptic equation, obstacle problem, free boundary, Hausdorff measure

Zheng Jun, Zhao Peihao: A Remark on Hausdorff Measure in Obstacle Problems. Z. Anal. Anwend. 31 (2012), 427-439. doi: 10.4171/ZAA/1467