Mixed boundary value problems for the Laplace-Beltrami equations

TL;DR

This paper studies mixed boundary value problems for the Laplace-Beltrami equation on smooth hypersurfaces using advanced functional analysis, establishing solvability criteria in non-classical Bessel potential spaces.

Contribution

It introduces a novel approach applying quasilocalization and Mellin convolution equations to analyze mixed BVPs in Bessel potential spaces, providing explicit solvability conditions.

Findings

Derived Fredholm properties and index of the system.

Established unique solvability criteria in non-classical settings.

Reduced model BVP to Mellin convolution equations with explicit symbols.

Abstract

We investigate the mixed Dirichlet-Neumann boundary value problems for the Laplace-Beltrami equation on a smooth hypersurface $\mathcal{C}$ with the smooth boundary in non-classical setting in the Bessel potential spaces $\mathbb{H}^1_p(\mathcal{C})$ for $1<p<\infty$. To the initial BVP we apply quasilocalization and obtain model BVPs for the Laplacian. The model mixed BVP on the half plane is investigated by potential method and is reduced to an equivalent system of Mellin convolution equations in Bessel potential and Besov spaces. The symbol of the obtained system is written explicitly, which provides Fredholm properties and the index of the system. The unique solvability criteria for the initial mixed BVP in the non-classical setting is derived.