\(L_p\)-theory of boundary integral equations on a contour with inward peak
The unique solvability of the boundary integral equations for the Dirichlet problem in \(\mathbb{R}^2\): \[ \Delta u=0 \quad\text{in }\Omega, \qquad u|_\Gamma= \varphi \] and for the Neumann problem: \[ \Delta u=0 \quad\text{in }\Omega, \qquad (\partial u/\partial n)_\Gamma=\psi \] are studied in the paper. \(\Omega\) is a simply connected domain in the complex plane and \(\Gamma\) its boundary with an inward peak in \(z=0\). The problem was studied by the authors more than ten years ago, and they published their results in several papers quoted in the bibliography. In the present paper they analyse the integral operators corresponding to the two boundary value problems to which a pair of function spaces are associated. The integral operators map one space isomorphically onto another. The principal results of the paper concern their continuity, their range, their character (Fredholm or not) and their injectivity. In the final part the exterior Dirichlet problem and Neumann problem are studied, with outward peak.
scientific article; zbMATH DE number 1211545