A remark on the regularity of the solutions of the \(p\)-Laplacian and its application to their finite element approximation
type: Item
instance of: scholarly article
title: A remark on the regularity of the solutions of the \(p\)-Laplacian and its application to their finite element approximation
zbMATH Open document ID: 0799.35085
author: John W. Barrett
author: Wenbin Liu
published in: Journal of Mathematical Analysis and Applications
publication date: 1994-11-14T00:00:00Z
review text: The authors show that solutions of the inhomogeneous, two-dimensional \(p\)-Laplace equation with (nonzero) boundary data have higher regularity than previously known. Specifically, if \(\Omega\) is a domain in \(\mathbb{R}^ 2\), if \(f\in L^ q\) for some \(q>2\), and if \(1<p\leq 2\), then any weak solution of \(-\text{div} (|\nabla u|^{p-2} \nabla u)=f\) in \(\Omega\), \(u=g\) on \(\partial\Omega\) has weak second derivatives in \(L^ 2\) in either of two cases: (1) \(\partial\Omega \in C^ 2\) and \(g\in H^ 2\), or (2) \(\Omega\) is convex and \(g\equiv 0\). An estimate for these derivatives is given, and this estimate is used to obtain error bounds in a finite element approximation to the problem. Reviewer's comment: The author mentions regularity results of \textit{E. DiBenedetto} [Nonlinear Anal., Theory Methods Appl. 7, 827-850 (1983; Zbl 0539.35027)] and the reviewer [Nonlinear Anal., Theory Methods Appl. 12, No. 11, 1203-1219 (1988; Zbl 0675.35042)] in the introduction, but does not discuss these results in Remark 2.1, where they are particularly relevant. For example, if \(g\in C^{1,\alpha}\cap H^ 2\) for some \(\alpha\in (0,1)\), the reviewer's results ensure that the gradient of the solution is globally bounded and this fact simplifies the proofs of Theorems 2.1 and 2.2.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 65N30
zbMATH DE Number: 484029
zbMATH Keywords: inhomogeneous, two-dimensional \(p\)-Laplace equation
zbMATH Keywords: error bounds in a finite element approximation
reviewed by: Gary M. Lieberman
MaRDI profile type: Publication
full work available at URL: jmaa.1993.1319
OpenAlex ID: W4212972812
DOI: 10.1006/JMAA.1993.1319
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description: scientific article; zbMATH DE number 484029