Boundary regularity under generalized growth conditions
type: Item
instance of: scholarly article
title: Boundary regularity under generalized growth conditions
zbMATH Open document ID: 1420.49041
DOI: 10.4171/ZAA/1628
author: Petteri Harjulehto
author: Peter Hästö
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2019-03-21T00:00:00Z
review text: In this paper, the authors consider minimizers of the integral function \[ \int_{\Omega}\varphi\big(x,|Du|\big)\ dx, \] where the function $\varphi$ is quite flexible inasmuch as its growth is concerned. For example, as part of the authors' results, the case \[ \varphi(x,t)=t^p, \] which is the standard Dirichlet $p$-energy, the case \[ \varphi(x,t)=t^{p(x)}, \] which is the variable growth setting, and the case \[ \varphi(x,t)=t^p+a(x)t^q, \] which is the double phase setting, are included as special cases of the main result. It is, therefore, interesting that the authors' results possess this degree of generality. The main result of the paper states that if the complement of the set $\Omega$ is (in the sense of capacity) locally fat at a given boundary point $x_0\in\partial\Omega$, then that point must be regular, which means that the boundary value is attained in a pointwise sense, not only in a Sobolev sense. The paper seems to be clearly written, and so, it would be of interest to researchers working in the general area of regularity theory in the calculus of variations or PDEs.
Mathematics Subject Classification ID: 49N60
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 7039932
zbMATH Keywords: Dirichlet energy integral
zbMATH Keywords: regular boundary point
zbMATH Keywords: minimizer
zbMATH Keywords: superminimizer
zbMATH Keywords: generalized Orlicz space
zbMATH Keywords: Musielak-Orlicz spaces
zbMATH Keywords: weak Harnack inequality
zbMATH Keywords: nonstandard growth
zbMATH Keywords: variable exponent
zbMATH Keywords: double phase
reviewed by: Christopher Goodrich
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/zaa/1628
OpenAlex ID: W2910790477
Wikidata QID: Q128634555
description: scientific article; zbMATH DE number 7039932