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

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.


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
Full Text: DOI Euclid


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