On the regularity of the free boundary in the \(p\)-Laplacian obstacle problem
type: Item
instance of: scholarly article
title: On the regularity of the free boundary in the \(p\)-Laplacian obstacle problem
zbMATH Open document ID: 1387.35631
author: Alessio Figalli
author: Brian Krummel
author: Xavier Ros-Oton
published in: Journal of Differential Equations
publication date: 2017-05-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1701.05262
review text: The paper is focused on the obstacle problem \(\min\{-\Delta_pu,u-\varphi\}=0\) in \(\Omega\subset \mathbb{R}^n\) for the \(p\)-Laplacian operator \(\Delta_pu=\operatorname{div} (|\nabla u|^{p-2}\nabla u)\), \(1<p<\infty\), where \(\Omega\) is a bounded smooth domain, and \(\varphi\) and \(g\) are given smooth functions. The authors study the regularity of the free boundary near those free boundary points \(x_0\in\partial\{u>\varphi\}\) at which \(\nabla \varphi(x_0)=0\). For \(p\not=2\) and \(\varphi(x)=-|x|^2\) in \(\mathbb{R}^2\), they construct global \(2\)-homogeneous solutions for the problem above with free boundaries having a corner at the origin. Then, under a concavity assumption for \(\varphi\), they prove that the free boundary is countably \((n-1)\) rectifiable, and present a nondegeneracy property for \(u\) at all free boundary points.
Mathematics Subject Classification ID: 35R35
zbMATH DE Number: 6712479
zbMATH Keywords: obstacle problem
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: free boundary
reviewed by: Rodica Luca
MaRDI profile type: Publication
OpenAlex ID: W2580644902
arXiv ID: 1701.05262
cites work: Optimal regularity for the obstacle problem for the \(p\)-Laplacian
cites work: The regularity of free boundaries in higher dimensions
cites work: The obstacle problem revisited
cites work: On the \(A\)-obstacle problem and the Hausdorff measure of its free boundary
cites work: On the regularity of the free boundary for quasilinear obstacle problems
cites work: On the porosity of free boundaries in degenerate variational inequalities
cites work: Hausdorff measure and stability for the \(p\)-obstacle problem \((2<p<\infty)\).
cites work: Regularity of free boundaries in obstacle-type problems
cites work: A strong maximum principle for some quasilinear elliptic equations
Wikidata QID: Q112631922
DOI: 10.1016/J.JDE.2017.03.035
description: scientific article