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 ℍ1(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 ℍps(𝒞), for s>1∕p, 1 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 ℍ1(𝒞) for arbitrary complex values of the nonzero constant in the impedance condition.
Highlights
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
By C we denote a subsurface of S, which has two faces C− and C+ and inherits the orientation from S: C+
The common end point of ΓN and ΓI will be denoted by PIN
Summary
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 will study the mixed impedance-Neumann-Dirichlet boundary value problem (MIND BVP) for the Laplace-Beltrami equation in different space settings In the present section we will prove the unique solvability of the MIND BVP (1.2) in the classical weak setting (3.1)–(3.2), based on the Lax-Milgram Lemma [31, 32].
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