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 C with smooth boundary. We prove, using the Lax-Milgram Lemma, that this MIND BVP has a unique solution in the classical weak setting H1(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 Hp(C), for s > 1/p, 1 < p < ∞. We apply a quasilocalization technique to the MIND BVP and obtain model Dirichlet-Neumann, Dirichletimpedance 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 < ∞ 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 H1(C) for arbitrary complex values of the nonzero constant in the impedance condition. 1. Formulation of the problem. Let S ⊂ R be some smooth, closed, orientable surface, bordering a compact inner Ω and outer Ω− := R \ Ω+ domains. By C we denote a subsurface of S, which has two faces C− and C and inherits the orientation from S: C 2010 AMS Mathematics subject classification. Primary 35J57; Secondary 45E10, 47B35.