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

Mixed type boundary value problems for Laplace–Beltrami equation on a surface with the Lipschitz... AbstractThe 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 ℍps⁢(𝒞){\mathbb{H}^{s}_{p}(\mathcal{C})}, 1p<s<1+1p{\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 Ωα⊂ℝ2{\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]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Georgian Mathematical Journal de Gruyter

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

Duduchava, Roland
Georgian Mathematical Journal , Volume 28 (2): 14 – Apr 1, 2021
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References (3)

Publisher
de Gruyter
Copyright
© 2021 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1572-9176
eISSN
1572-9176
DOI
10.1515/gmj-2020-2074
Publisher site
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Abstract

AbstractThe 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 ℍps⁢(𝒞){\mathbb{H}^{s}_{p}(\mathcal{C})}, 1p<s<1+1p{\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 Ωα⊂ℝ2{\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].

Journal

Georgian Mathematical Journal – de Gruyter

Published: Apr 1, 2021

Keywords: Model BVP; surface; Lipschitz boundary; angular domain; mixed problem; Dirichlet problem; Neumann problem; potential method; boundary integral equation; Mellin convolution equation; Bessel potential space; Fredholm criteria; unique solvability; 35J57; 45E10; 47B35

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