Functional Analysis and Applications Seminar 

22/04/2008, 15:00 — 16:00 — Room P3.31, Mathematics Building
Roland Duduchava, Andrea Razmadze Mathematical Institute and IB Euro-Caucasian University, Tbilisi, Georgia

Boundary value problems on hypersurfaces

Partial differential equations on Riemannian manifolds are usually written in intrinsic coordinates, involving metric tensor and Christoffel symbols. But if we deal with a hypersurface, the cartesian coordinates of the ambient space can be applied. 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. This enables, for example, more transparent proofs of Korn's inequalities, tightly connected with solvability and uniqueness of some boundary value problems. Moreover, based on the principle that the displacement minimizes the total free elastic energy at equilibrium, was derived the Lamé operator on the surface (R. Duduchava, D. Mitrea and M. Mitrea). The equation is represented in terms of Günter's derivatives. The Killing's vector fields, solutions of the homogeneous Lamé equation are investigated. Relatively simple form of operators in terms of Günter's and Stoke's derivatives enable simplified treatment of corresponding boundary value problems with the Lax-Milgram lemma and Korn's inequalities with and without boundary conditions. Another approach, the potential method, using fundamental solutions, potential operators, Green formulae and boundary integral equations, available in explicit form, is developed as well. A special accent is made on a thin flexural shell problems in elasticity. We suggest for their study the approach applied to the derivation of Lamé equation. The approach differs from the ones proposed before for modeling linearly elastic flexural shells, suggested by Cosserats (1909), Goldenveiser (1961), Naghdi (1963), Vekua (1965), Novozhilov (1970), Koiter (1970) and many others.