Fall 2020 Mixed impedance boundary value problems for the Laplace–Beltrami equation
Luis Castro, Roland Duduchava, Frank-Olme Speck
J. Integral Equations Applications 32(3): 275-292 (Fall 2020). DOI: 10.1216/jie.2020.32.275


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 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 ps(𝒞), for s>1p, 1<p<. 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>1p 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 1(𝒞) for arbitrary complex values of the nonzero constant in the impedance condition.


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Luis Castro. Roland Duduchava. Frank-Olme Speck. "Mixed impedance boundary value problems for the Laplace–Beltrami equation." J. Integral Equations Applications 32 (3) 275 - 292, Fall 2020. https://doi.org/10.1216/jie.2020.32.275


Received: 11 February 2019; Revised: 1 November 2019; Accepted: 4 November 2019; Published: Fall 2020
First available in Project Euclid: 17 September 2020

zbMATH: 07283058
MathSciNet: MR4150701
Digital Object Identifier: 10.1216/jie.2020.32.275

Primary: 35J57
Secondary: 45E10 , 47B35

Keywords: Bessel potential space , boundary integral equation , boundary value problem , Fredholm criteria, symbol , Fredholm property , impedance-Neumann–Dirichlet condition , Laplace–Beltrami equation , potential method , unique solvability

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium


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Vol.32 • No. 3 • Fall 2020
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