Mixed impedance boundary value problems for the Laplace–Beltrami equation

Abstract

This work is devoted to the analysis of the mixed impedance-Neumann–Dirichlet boundary value problem (MIND BVP) for the Laplace–Beltrami equation on a compact smooth surface $\mathcal{𝒞}$ with smooth boundary. We prove, using the Lax–Milgram Lemma, that this MIND BVP has a unique solution in the classical weak setting ${ℍ}^1\left(\mathcal{𝒞}\right)$ when considering positive constants in the impedance condition. The main purpose is to consider the MIND BVP in a nonclassical setting of the Bessel potential space ${ℍ}_p^s\left(\mathcal{𝒞}\right)$, for $s>1∕p$, $1<p<\infty$. We apply a quasilocalization technique to the MIND BVP and obtain model Dirichlet–Neumann, Dirichlet-impedance and Neumann-impedance BVPs for the Laplacian in the half-plane. The model mixed Dirichlet–Neumann BVP was investigated by R. Duduchava and M. Tsaava (2018). The other two are investigated in the present paper. This allows to write a necessary and sufficient condition for the Fredholmness of the MIND BVP and to indicate a large set of the space parameters $s>1∕p$ and $1<p<\infty$ for which the initial BVP is uniquely solvable in the nonclassical setting. As a consequence, we prove that the MIND BVP has a unique solution in the classical weak setting ${ℍ}^1\left(\mathcal{𝒞}\right)$ for arbitrary complex values of the nonzero constant in the impedance condition.