Bifurcation of nonlinear elliptic equations on Smale's shrinking domains.

Smale considered in the sixties boundary value problems for linear elliptic. Differential operators on manifolds and studied the existence of solutions when the underlying manifold is shrunk by a family of diffeomorphism. The supervisor of this project has taken up this idea and focussed on the existence of non-trivial solutions for nonlinear equations when shrinking a domain using bifurcation theory. By now this has only be done for Dirichlet boundary conditions on star-shaped domains in Euclidean space. The aim of this project is to obtain the existence of solutions of nonlinear PDEs when shrinking the domain for more general equations, more general boundary conditions and more general domains as in our previous work.