The Neumann problem on Lipschitz domains in Hardy spaces of order less than one
Recently, B. E. J. Dahlberg and C. E. Kenig considered the Neumann problem, \(\Delta u= 0\) in \(D\), \(\partial u/\partial\nu= f\) on \(\partial D\), for Laplace's equation in a Lipschitz domain \(D\). One of their main results considers this problem when the data lies in the atomic Hardy space \(H^1(\partial D)\) and they show that the solution has gradient in \(L^1(\partial D)\). The aim of this paper is to establish an extension of their theorem for data in the Hardy space \(H^p(\partial D)\), \(1- \varepsilon< p< 1\), where \(0< \varepsilon< 1/n\) is a positive constant which depends only on \(m\), the maximum of the Lipschitz constants of the functions which define the boundary of the domain. We also extend G. Verchota's and Dahlberg and Kenig's theorem on the potential representation of solutions of the Neumann problem to the range \(1- \varepsilon< p< 1\). This has the interesting consequence that the double-layer potential is invertible on Hölder spaces \(C^\alpha(\partial D)\) for \(\alpha\) close to zero.
scientific article; zbMATH DE number 912151