Very weak solutions of parabolic systems of \(p\)-Laplacian type
type: Item
instance of: scholarly article
title: Very weak solutions of parabolic systems of \(p\)-Laplacian type
zbMATH Open document ID: 1011.35039
DOI: 10.1007/BF02384505
author: Juha Kinnunen
author: John L. Lewis
published in: Arkiv för Matematik
publication date: 2003-03-25T00:00:00Z
review text: The authors study very weak solutions of a class of second-order parabolic systems \[ \frac{\partial u_i}{\partial t} = \operatorname {div} A_i(x,t,\nabla u) + B_i(x,t,\nabla u) \] including systems of \(p\)-Laplacian type. The term ``very weak solutions'' denotes functions which lie in a Sobolev space with exponent less than the natural Sobolev exponent and which satisfy the equation in the sense of distributions. The main result states that very weak solutions which are defined using a Sobolev space with appropriate exponent \(r\) are in fact weak solutions. This allows to get the usual regularity results under weaker assumptions. The proof of this result involves the construction of suitable test functions and the application of reverse Hölder inequalities.
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 35K40
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 1887463
zbMATH Keywords: weak solutions
zbMATH Keywords: regularity
zbMATH Keywords: reverse Hölder inequality
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1887463