Functional Analysis and Applications Seminar 

02/03/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Roland Duduchava, A. Razmadze Mathematical Institute, Academy of Sciences, Tbilisi, Georgia

Boundary value problems for shell equations

We propose writing partial differential equations on a hypersurface in cartesian coordinates of the ambient space instead of more customary local coordinates and the Riemannian metric tensor of the underlying surface. This seemingly trivial idea simplifies the form of many classical differential equations on the surface (Laplace-Beltrami, Lamé, Maxwell etc.), which turn out to have constant coefficients, and enables more transparent proofs of Korn's inequalities, tightly connected with solvability and uniqueness of some boundary value problems. The obtained results are applied to the Dirichlet and Neumann boundary value problems for the Laplace-Beltrami operator, for its square, and for the elasticity Lamé operators, describing thin shells in the form of an open smooth hypersurface with smooth boundary. An explicit Green formula is derived and it is proved that the Dirichlet boundary value problems has a unique solution in the Sobolev space of weak solutions while the Neumann boundary value problems are solvable under the usual orthogonality constraints on the data.