A Dirichlet problem for harmonic forms in \(C^1\)-domains
type: Item
instance of: scholarly article
title: A Dirichlet problem for harmonic forms in \(C^1\)-domains
zbMATH Open document ID: 0893.31006
DOI: 10.1007/BF01294158
author: Renata Selvaggi
author: Irene Sisto
published in: Integral Equations and Operator Theory
publication date: 1998-08-16T00:00:00Z
review text: The object of the paper is the solvability of Dirichlet's problem for harmonic \(s \)-forms in a \(C^1\)-domain in \(\mathbb{R}^n\) with boundary data of class \(L^p\). This problem can be considered as an extension of a similar one, solved by C. Miranda in 1953, who studied the existence of a soution to Dirichlet's problem for harmonic differential \(s\)-forms in a \(C^{2, \alpha}\) domain with Hölder continuous data. The authors use the same mathematical technique, the layer potential representation, and obtain a system \(S_s\) of differential forms which is studied in the space \(W_s^{1,p} (\partial \Omega)\) of the \(s\)-forms of class \(L^p\) on \(\partial \Omega\). The paper is devoted to the study of the internal Dirichlet problem. The authors show that the linear operator associated with \(S_s\) is compact on \(W_s^{1,p} (\partial \Omega)\) and that the \(L^p\)-solutions of the homogeneous system are closed forms in \(W_s^{1,p} (\partial \Omega)\). Using some results of their own obtained in 1978 und 1996, the authors give an extension of the existence results proved by C. Miranda in 1953. The last part of the paper deals with the external Dirichlet problem.
Mathematics Subject Classification ID: 31B20
Mathematics Subject Classification ID: 35J55
Mathematics Subject Classification ID: 31B10
zbMATH DE Number: 1113569
zbMATH Keywords: harmonic forms in \(C^1\)-domains
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: differential forms
zbMATH Keywords: existence
reviewed by: Vasile Ionescu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1113569