Functional Analysis and Applications Seminar 

31/03/2006, 15:00 — 16:00 — Room P3.10, Mathematics Building
Roland Duduchava, University of Saarland, Saarbruecken

Differential operators and boundary value problems on hypersurfaces

We explore the extent to which basic differential operators (such as Laplace-Beltrami, Lamé, Navier-Stokes, etc.) and boundary value problems on a hypersurface in ${ℝ}^n$ can be expressed globally, in terms of the standard spatial coordinates in ${ℝ}^n$. The approach we develop also provides, in some important cases, useful simplifications as well as new interpretations of classical operators and equations. In particular, we obtain explicit representations of the surface deformation tensor, the Laplace-Beltrami operator, the Lamé-Beltrami operator, the operator of isotropic elasticity and the Stokes system on the hypersurface. The obtained representations are helpful in proving existence of fundamental solutions on the manifold and in writing explicit Green formulae for the classical boundary value problems on an open subsurface. Combined with the potential method, the obtained results allow to prove the unique solvability of the aforementioned boundary value problems by a standard scheme. The lecture is based upon joint research with with Dorina Mitrea and Marius Mitrea.