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

斯梅尔在六十年代研究了线性椭圆型边值问题。流形上的微分算子，并研究了当基础流形被一族微分同胚收缩时解的存在性。这个项目的主管采纳了这一想法，并利用分支理论研究了当收缩区域时非线性方程的非平凡解的存在性。到目前为止，这只适用于欧几里得空间中星形区域上的Dirichlet边界条件。这个项目的目的是得到在更一般的方程、更一般的边界条件和更一般的区域上收缩区域时非线性偏微分方程解的存在性。

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.