Mixed boundary value problems for the Laplace–Beltrami equation

Abstract

We investigate the mixed Dirichlet–Neumann boundary value problems for the Laplace–Beltrami equation on a smooth bounded surface with a smooth boundary in non-classical setting in the Bessel potential space for , . To the initial BVP we apply a quasi-localization and obtain a model BVP for the Laplacian. The model mixed BVP on the half plane is investigated by the potential method and is reduced to an equivalent system of Mellin convolution equations in Sobolev–Slobodečkii space. Boundary integral equations are ivestigated in both Bessel potential and Sobolev–Slobodečkii spaces. The symbol of the obtained system is written explicitly and is responsible for the Fredholm properties and the index of the system. An explicit criterion for the unique solvability of the initial BVP in the non-classical setting is derived as well.