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

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 $\mathbb{R}^n$.