Special Minicourse Lecture 2 - Dynamical Approach to an Overdetermined Problem

Abstract:

I will present a dynamical approach to an elliptic overdetermined problem, in which one seeks a bounded domain allowing a solution to an elliptic equaiton with both Dirichlet and Neumann boundary conditions. One of the key observations is that the deformation of a continuously varying domain for a parametrized overdetermined problem forms a parabolic semiflow. In particular, by this approach we can establish a rigidity and stability result for an overdetermined problem even in an asymmetric situation. Another application is the construction of a foliated family of solutions to Bernoulli's free boundary problem.

The talks will be accessible to graduate students with a good basis in Analysis and PDE.