Jack Thompson: On an overdetermined problem involving the fractional Laplacian

Overdetermined problems are a type of boundary value problem where `too many' conditions are imposed on the solution. In general, such a problem is ill-posed so the main objective is to classify sets in which the problem is well-posed. A classical result due to J. Serrin says that a bounded domain in $\mathbb R^n$ that admits a function with constant Laplacian, zero Dirichlet data, and constant Neumann data must be a ball.

We consider a semi-linear generalisation of Serrin's problem driven by the fractional Laplacian where the value of the solution is prescribed on hyperplane parallel to the boundary. We prove that the existence of a non-negative solution forces the region to be a ball. We also discuss some further related results. This is joint work with S. Dipierro, G. Poggesi, and E. Valdinoci.