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 H 1 (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 H s p (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 H 1 (C) for arbitrary complex values of the nonzero constant in the impedance condition.

On Some Elliptic Boundary Value Problems in Conic Domains

“…The theory of pseudodifferential equations is widely used in numerous branches of mathematics and physics. Specifically, such equations appear in problems of electromagnetic wave scattering (see, e.g., [1][2][3]), where the factorization method is widely applied. Below, the multidimensional variant of this method is used to derive integral representations of solutions to the considered boundary value problems.…”