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Laplace–Beltrami equation on hypersurfaces and Γ‐convergence
Author(s) -
Buchukuri Tengiz,
Duduchava Roland,
Tephnadze George
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4331
Subject(s) - mathematics , mathematical analysis , hypersurface , boundary value problem , limit (mathematics) , laplace's equation , parabolic partial differential equation , dirichlet problem , partial differential equation , euclidean space
A mixed boundary value problem for the stationary heat transfer equation in a thin layer around a surface C 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 spaceR n . Copyright © 2017 John Wiley & Sons, Ltd.