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 with smooth boundary. We prove, using the Lax–Milgram Lemma, that this MIND BVP has a unique solution in the classical weak setting 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 , for , . 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 and 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 for arbitrary complex values of the nonzero constant in the impedance condition.
Citation
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
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