MIXED IMPEDANCE BOUNDARY VALUE PROBLEMS FOR THE LAPLACE-BELTRAMI EQUATION

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 H-1(l) 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 H-p(s)(l), for s > 1= p, 1 < p < infinity. 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 halfplane. 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 > 1= p and 1 < p < infinity 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 H-1(l) for arbitrary complex values of the nonzero constant in the impedance condition.