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.