Functional Analysis and Applications Seminar 

26/06/2014, 14:30 — 15:15 — Room P3.10, Mathematics Building
Roland Duduchava, A. Razmadze Mathematical Institute

Calculus of tangential differential operators on hypersurfaces

Partial differential equations on surfaces in the Euclidean space and corresponding boundary value problems (BVPs), encounter rather often in applications. For example: heat conduction by a thin conductive surface or deformation a of thin elastic surface are governed by some differential equations on these surfaces. To rigorously derive equations which govern the above mentioned processes we need a calculus of tangential partial differential operators on a hypersurface (i.e., a surfaces in the Euclidean space $\mathbb{R}^n$ of co-dimension $1$). There are known many approaches to this problem, but the main task is to find the one which gives simplest results. We suggest a calculus of partial differential operators on a hypersurface based on Günter's and Stoke's tangential derivatives. We will expose basics of this calculus and show how classical differential operators, such as Laplace-Beltrami operator (governing the heat conduction), Lamé-Beltrami operator (governing the deformation of an elastic surface), Schrödinger equation, are written with the help of Günter's derivatives. We will end up with the demonstration of $\Gamma$-convergence of a BVP for the Laplace equation in a curved layer to a BVP for Laplace-Beltrami equation on a mid-surface when the thickness of the layer