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 $\mathbb{H}^1(\mathcal{C})$ 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 $\mathbb{H}^s_p(\mathcal{C})$, 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 $\mathbb{H}^1(\mathcal{C})$ for arbitrary complex values of the nonzero constant in the impedance condition.