Mixed type boundary value problems for Laplace–Beltrami equation on a surface with the Lipschitz boundary

Abstract

Abstract The purpose of the present research is to investigate a general mixed type boundary value problem for the Laplace–Beltrami equation on a surface with the Lipschitz boundary 𝒞 {\mathcal{C}} in the non-classical setting when solutions are sought in the Bessel potential spaces ℍ p s ⁢ ( 𝒞 ) {\mathbb{H}^{s}_{p}(\mathcal{C})} , 1 p < s < 1 + 1 p {\frac{1}{p}<s<1+\frac{1}{p}} , 1 < p < ∞ {1<p<\infty} . Fredholm criteria and 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 Ω α ⊂ ℝ 2 {\Omega_{\alpha}\subset\mathbb{R}^{2}} of magnitude α. The model mixed BVP is investigated in the earlier paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain, Georgian Math. J. 27 2020, 2, 211–231], and the model Dirichlet and Neumann boundary value problems are studied in the non-classical setting. The problems are investigated by the potential method and reduction to locally equivalent 2 × 2 {2\times 2} systems of Mellin convolution equations with meromorphic kernels on the semi-infinite axes ℝ + {\mathbb{R}^{+}} in the Bessel potential spaces. Such equations were recently studied by R. Duduchava [Mellin convolution operators in Bessel potential spaces with admissible meromorphic kernels, Mem. Differ. Equ. Math. Phys. 60 2013, 135–177] and V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl. 443 2016, 2, 707–731].