The objective of the present chapter is to investigate the general Mixed type boundary value problems for the Laplace–Beltrami equation on a surface with the Lipschitz boundary C in a non-classical setting, when solutions are sought in the Bessel potential spaces Hsp(C ), 1p < s < 1 + 1p , 1 < p < ∞. Fredholm criteria and the unique solvability criteria are found. By the localization the problem is reduced to the investigation of Model Dirichlet, Neumann and mixed boundary value problems for the Laplace equation in a planar angular domain ?α ? R2 of magnitude α. The model mixed BVP is investigated in earlier paper [69] and here we study Model Dirichlet and Neumann boundary value problems in a non-classical setting. The problems are investigated by the potential method and by reducing to locally equivalent 2×2 systems of Mellin convolution equations with meromorphic kernels on the semi-infinite axes R+ in the Bessel potential spaces. Such equations were studied recently by R. Duduchava in [59] and V. Didenko and R. Duduchava in [37].
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