Laplace-Beltrami equation on hypersurfaces and $\Gamma$-convergence

TL;DR

This paper analyzes the asymptotic behavior of a heat transfer boundary value problem in thin layers as their thickness approaches zero, deriving a limit problem on the hypersurface involving the Laplace-Beltrami operator.

Contribution

It explicitly describes the $ ext{Gamma}$-limit of the boundary value problem for heat transfer in thin layers, connecting it to the Laplace-Beltrami equation on the hypersurface.

Findings

Limit Dirichlet problem is the Laplace-Beltrami equation on the surface.

Neumann boundary conditions transform in the $ ext{Gamma}$-limit.

Variational methods and tangential differential operators are used for analysis.

Abstract

We investigate a mixed boundary value problem for the stationary heat transfer equation in a thin layer with a mid hypersurface $\mathcal{C}$ in $\mathbb{R}^3$ with the boundary. The main object is to trace what happens in $\Gamma$-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 $\Gamma$-limit. For this we apply the variational formulation and the calculus of G\"unter'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 $\mathbb{R}^n$.