A Dirichlet problem for harmonic forms in \(C^1\)-domains
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.
scientific article; zbMATH DE number 1113569