- April 22, 2008 , Tuesday - 15h00 (Room P
3.31):
Boundary value
problems on hypersurfaces
Roland Duduchava
(Andrea Razmadze Mathematical Institute and IB
Euro-Caucasian
University, Tbilisi, Georgia)
Abstract. 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.
Seminars take place in Lisbon, I.S.T. -
Post Graduation Building
Webpage: http://www.math.ist.utl.pt/funcional
