Maximal \(L^{p}\)-regularity for the Laplacian on Lipschitz domains
type: Item
instance of: scholarly article
title: Maximal \(L^{p}\)-regularity for the Laplacian on Lipschitz domains
zbMATH Open document ID: 1122.35028
author: J. Martínez
published in: Mathematische Zeitschrift
publication date: 2007-09-11T00:00:00Z
review text: The author considers the Laplacian with Dirichlet or Neumann boundary conditions on bounded Lipschitz domains \(\Omega,\) both with the following two domains of definition: \[ D_1(\Delta)=\{u\in W^{1,p}(\Omega): \;\Delta u\in L^p(\Omega),\;Bu=0\} \] or \[ D_2(\Delta)=\{u\in W^{2,p}(\Omega): \;Bu=0\} \] where \(B\) is the boundary operator. It is proved that, under certain restrictions on the range of \(p,\) these operators generate positive analytic contraction semigroups on \(L^p(\Omega)\) which implies maximal regularity for the corresponding Cauchy problems. In particular, if \(\Omega\) is bounded and convex and \(1<p\leq 2\) the Laplacian with domain \(D_2(\Delta)\) has the maximal regularity property, as in the case of smooth domains. Moreover, it is constructed an example that proves that, in general, the Dirichlet-Laplacian with domain \(D_1(\Delta)\) is not even a closed operator.
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 5189585
zbMATH Keywords: Laplacian
zbMATH Keywords: boundary conditions
zbMATH Keywords: Lipschitz domain
zbMATH Keywords: analytic semigroups
reviewed by: Lubomira Softova
MaRDI profile type: Publication
full work available at URL: s00209-006-0055-6
OpenAlex ID: W2055870358
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DOI: 10.1007/S00209-006-0055-6
description: scientific article; zbMATH DE number 5189585