Roland Duduchava

Partial Differential Equations on Hypersurfaces

abstract:
We propose an approach which allows global representation of basic differential operators (such as Laplace-Beltrami, Hodge-Laplacian, Lamé, Navier-Stokes, etc.) and of corresponding boundary value problems on a
hypersurface $\mathcal{S}$ in $\mathbb{R}^n$, in terms of the standard spatial coordinates in $\mathbb{R}^n$. The tools we develop also provide, in some important cases, useful simplifications as well as new interpretations of classical operators and equations.
The obtained results are applied to the Dirichlet and Neumann boundary value problems for the Laplace-Beltrami operator ${\mbox{\boldmath $\Delta$}}_{\mathcal C}$ and to the system of anisotropic elasticity on an open smooth hypersurface ${mathcal C}\subset{mathcal S}$ with the smooth boundary $\Gamma:=\partial{\mathcal C}$. We prove the solvability theorems for the Dirichlet and Neumann BVPs on open hypersurfaces in the Bessel potential spaces.

Mathematics Subject Classification: 58J05, 58J32, 35Q99, 58G15, 73B40, 73C15, 73C35

Key words and phrases: Guenter's derivative, Lame operator, anisotropic elasticity, open hypersurface, boundary value problem, Bessel potential space, Laplace-Beltrami operator