On the A-obstacle problem and the Hausdorff measure of its free boundary

S. Challal; A. Lyaghfouri; J. F. Rodrigues

doi:10.1007/s10231-010-0177-7

On the A-obstacle problem and the Hausdorff measure of its free boundary

S. Challal, A. Lyaghfouri, J. F. Rodrigues

Research output: Contribution to journal › Article › peer-review

34 Citations (Scopus)

Abstract

In this paper, we prove existence and uniqueness of an entropy solution to the A-obstacle problem, for L ¹-data. We also extend the Lewy-Stampacchia inequalities to the general framework of L ¹-data and show convergence and stability results. We then prove that the free boundary has finite (N - 1)-Hausdorff measure, which completes previous works on this subject by Caffarelli for the Laplace operator and by Lee and Shahgholian for the p-Laplace operator when p > 2.

Original language	English
Pages (from-to)	113-165
Number of pages	53
Journal	Annali di Matematica Pura ed Applicata
Volume	191
Issue number	1
DOIs	https://doi.org/10.1007/s10231-010-0177-7
Publication status	Published - Jan 2012
Externally published	Yes

Keywords

A-Laplace operator
Entropy solution
Free boundary
Hausdorff measure
Lewy-Stampacchia inequalities
Obstacle problem
Stability

ASJC Scopus subject areas

Applied Mathematics

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AU - Lyaghfouri, A.

AU - Rodrigues, J. F.

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N2 - In this paper, we prove existence and uniqueness of an entropy solution to the A-obstacle problem, for L 1-data. We also extend the Lewy-Stampacchia inequalities to the general framework of L 1-data and show convergence and stability results. We then prove that the free boundary has finite (N - 1)-Hausdorff measure, which completes previous works on this subject by Caffarelli for the Laplace operator and by Lee and Shahgholian for the p-Laplace operator when p > 2.

AB - In this paper, we prove existence and uniqueness of an entropy solution to the A-obstacle problem, for L 1-data. We also extend the Lewy-Stampacchia inequalities to the general framework of L 1-data and show convergence and stability results. We then prove that the free boundary has finite (N - 1)-Hausdorff measure, which completes previous works on this subject by Caffarelli for the Laplace operator and by Lee and Shahgholian for the p-Laplace operator when p > 2.

KW - A-Laplace operator

KW - Entropy solution

KW - Free boundary

KW - Hausdorff measure

KW - Lewy-Stampacchia inequalities

KW - Obstacle problem

KW - Stability

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On the A-obstacle problem and the Hausdorff measure of its free boundary

Abstract

Keywords

ASJC Scopus subject areas

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