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.
