A general integral identity with applications to a reverse Serrin problem

The talk aims to present a new general differential identity and an associated integral identity, which entails a pair of solutions of the Poisson equation with constant source term. This generalizes a formula that R. Magnanini and G. Poggesi previously proved and used to obtain quantitative estimates of spherical symmetry for the Serrin overdetermined boundary value problem.
As a first application of this new general differential identity, we prove a quantitative symmetry result for the ``reverse Serrin problem'', which we will introduce. In passing, we obtain a rigidity result for solutions of the aforementioned Poisson equation subject to a constant Neumann condition. This is a joint work with R. Magnanini and G. Poggesi.