Riesz potential estimates for a class of double phase problems
type: Item
instance of: scholarly article
title: Riesz potential estimates for a class of double phase problems
zbMATH Open document ID: 1387.35276
author: Sun-Sig Byun
author: Yeonghun Youn
published in: Journal of Differential Equations
publication date: 2017-11-16T00:00:00Z
review text: This paper deals with the equation \[ -\text{div}(\gamma(x) A(x,D)u)= \mu,\tag{1} \] where \(\mu\) is a finite Borel measure defined on the bounded domain \(\Omega\subset\mathbb{R}^n\). The mapping \(A(x,z)\) is \(C^1\) regular in \(z\), \(A_z(\cdot)\) is Carathéodory regular and satisfies several non-standard growth, ellipticity and continuity assumptions. The coefficient \(\gamma\) is Dini-continuous. The authors prove in Theorem 1.1 an optimal pointwise gradient estimate for such a measure data problem via Riesz pontentials of \(\mu\) and log-Dini continuity assumptions imposed on the modulating coeffient \(a(x)\). As a consequence in Theorem 1.2 an optimized \(C^1\) regularity criterion for the weak solution \(u\) of (1) is found. \(u\) is supposed to belong to some appropriate Musiełak-Orlicz spaces.
Mathematics Subject Classification ID: 35J62
zbMATH DE Number: 6807498
zbMATH Keywords: measure data problem
zbMATH Keywords: gradient estimate
zbMATH Keywords: Musielak-Orlicz space
reviewed by: Peter R. Popivanov
MaRDI profile type: Publication
full work available at URL: j.jde.2017.09.038
OpenAlex ID: W2763439237
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DOI: 10.1016/J.JDE.2017.09.038
description: scientific article; zbMATH DE number 6807498