Michiaki Onodera - Dynamical approach to an overdetermined problem II

I will present a dynamical approach to an elliptic overdetermined problem, in which one seeks a bounded domain allowing a solution to an elliptic equation 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.