A limiting free boundary problem for a degenerate operator in Orlicz-Sobolev spaces
type: Item
instance of: scholarly article
title: A limiting free boundary problem for a degenerate operator in Orlicz-Sobolev spaces
zbMATH Open document ID: 1460.35145
author: Jefferson Abrantes Santos
author: Sérgio H. M. Soares
published in: Revista Matemática Iberoamericana
publication date: 2021-03-07T00:00:00Z
review text: Summary: A free boundary optimization problem involving the \(\Phi \)-Laplacian in Orlicz-Sobolev spaces is considered for the case where \(\Phi\) does not satisfy the natural conditions introduced by Lieberman. A minimizer \(u_\Phi\) having non-degeneracy at the free boundary is proved to exist and some important consequences are established, namely, the Lipschitz regularity of \(u_\Phi\) along the free boundary, that the positivity set of \(u_\Phi\) has locally uniform positive density, and that the free boundary is porous with porosity \(\delta > 0\) and has finite \((N - \delta)\)-Hausdorff measure. The method is based on a truncated minimization problem in terms of the Taylor polynomial of \(\Phi\) of order \(2k\). The proof demands to revisit the Lieberman proof of a Harnack inequality and verify that the associated constant with this inequality is independent of \(k\) provided that \(k\) is sufficiently large.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 7318498
zbMATH Keywords: free boundary problems
zbMATH Keywords: degenerate elliptic equations
zbMATH Keywords: minimization problem
zbMATH Keywords: Orlicz-Sobolev spaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/rmi/1180
OpenAlex ID: W3008820208
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DOI: 10.4171/RMI/1180
description: scientific article; zbMATH DE number 7318498