Laplace–Beltrami equation on hypersurfaces and Γ‐convergence |
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| Authors: | Tengiz Buchukuri Roland Duduchava George Tephnadze |
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| Affiliation: | 1. A. Razmadze Mathematical Institute, Tbilisi State University, Tbilisi, Georgia;2. University of Georgia, Tbilisi, Georgia;3. Department of Engineering Sciences and Mathematics, Lule? 4. University of Technology, Lule?, Sweden;5. Department of Mathematics, Faculty of Exact and Natural Sciences, Tbilisi State University, Tbilisi, Georgia |
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| Abstract: | A mixed boundary value problem for the stationary heat transfer equation in a thin layer around a surface  with the boundary is investigated. The main objective is to trace what happens in Γ‐limit when the thickness of the layer converges to zero. The limit Dirichlet BVP for the Laplace–Beltrami equation on the surface is described explicitly, and we show how the Neumann boundary conditions in the initial BVP transform in the Γ‐limit. For this, we apply the variational formulation and the calculus of Günter's tangential differential operators on a hypersurface and layers, which allow global representation of basic differential operators and of corresponding boundary value problems in terms of the standard Euclidean coordinates of the ambient space  . Copyright © 2017 John Wiley & Sons, Ltd. |
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| Keywords: | hypersurface Gü nter's derivatives Laplace– Beltrami equation Γ ‐convergence heat transfer equation |
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with the boundary is investigated. The main objective is to trace what happens in Γ‐limit when the thickness of the layer converges to zero. The limit Dirichlet BVP for the Laplace–Beltrami equation on the surface is described explicitly, and we show how the Neumann boundary conditions in the initial BVP transform in the Γ‐limit. For this, we apply the variational formulation and the calculus of Günter's tangential differential operators on a hypersurface and layers, which allow global representation of basic differential operators and of corresponding boundary value problems in terms of the standard Euclidean coordinates of the ambient space 
. Copyright © 2017 John Wiley & Sons, Ltd.