Mixed impedance boundary value problems for the Laplace-Beltrami equation. (English) Zbl 1454.35128

Castro, Luis; Duduchava, Roland; Speck, Frank-Olme

J. Integral Equations Appl. 32, No. 3, 275-292 (2020).

The paper under review is concerned with the analysis of the mixed impedance-Neumann-Dirichlet boundary value problem for the Laplace-Beltrami equation on a compact smooth surface with smooth boundary. In the first part of this paper, by using the Lax-Milgram Lemma, the authors establish that this problem has a unique solution in the classical weak setting. Next, the authors consider the mixed impedance-Neumann-Dirichlet boundary value problem in a nonclassical setting of the Bessel potential space. The basic tool for this analysis is a quasilocalization technique. This allows to obtain a necessary and sufficient condition for the Fredholmness of the mixed impedance-Neumann-Dirichlet boundary value problem and to indicate a large set of the space parameters for which the initial BVP is uniquely solvable in the nonclassical setting. As a consequence, the authors prove that the mixed impedance-Neumann-Dirichlet boundary value problem has a unique solution in the classical weak setting for arbitrary complex values of the nonzero constant in the impedance condition.

Reviewer: Vicenţiu D. Rădulescu (Craiova)

MSC:

35J57	Boundary value problems for second-order elliptic systems
45E10	Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators

Keywords:

Laplace-Beltrami equation; impedance-Neumann-Dirichlet condition; Fredholm property; unique solvability; Bessel potential space

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