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 ${\text{div}}_{\mathcal{C}}(A{\nabla }_{\mathcal{C}}\phi )=f$ on a smooth hypersurface $\mathcal{C}$ with the boundary $\mathrm{\Gamma }=\partial \mathcal{C}$ in ${\mathbb{R}}^n$. $A(x)$ is an $n\times n$ bounded measurable positive definite matrix function. The boundary is decomposed into two nonintersecting connected parts $\mathrm{\Gamma }={\mathrm{\Gamma }}_D\cup {\mathrm{\Gamma }}_N$ and on ${\mathrm{\Gamma }}_D$ the Dirichlet boundary conditions are prescribed, while on ${\mathrm{\Gamma }}_N$ 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 ${\text{div}}_{\mathcal{S}}(A{\nabla }_{\mathcal{S}})$ is proved, which is interpreted as the invertibility of this operator in the setting ${\mathbb{H}}_{p,\#}^s(\mathcal{S})\to {\mathbb{H}}_{p,\#}^{s-2}(\mathcal{S})$, where ${\mathbb{H}}_{p,\#}^s(\mathcal{S})$ is a subspace of the Bessel potential space and consists of functions with mean value zero.