Mixed Boundary Value Problem on Hypersurfaces

Abstract

The purpose of the present paper is to investigate the mixed Dirichlet-Neumann boundary value problems for the anisotropic Laplace-Beltrami equation divC(A∇Cφ)=f on a smooth hypersurface C with the boundary Γ=∂C in Rn. A(x) is an n×n bounded measurable positive definite matrix function. The boundary is decomposed into two nonintersecting connected parts Γ=ΓD∪ΓN and on ΓD the Dirichlet boundary conditions are prescribed, while on ΓN the Neumann conditions. The unique solvability of the mixed BVP is proved, based upon the Green formulae and Lax-Milgram Lemma. Further, the existence of the fundamental solution to divS(A∇S) is proved, which is interpreted as the invertibility of this operator in the setting Hp,#s(S)→Hp,#s-2(S), where Hp,#s(S) is a subspace of the Bessel potential space and consists of functions with mean value zero.