On the Serrin problem for ring-shaped domains

Abstract

In this paper, we deal with the open problem of characterising rotationally symmetric solutions to $\Delta u=-2$, when Dirichlet boundary conditions are imposed on a ring-shaped planar domain. In contrast with Serrin’s classical result, we show that the simplest possible set of overdetermining conditions, namely the prescription of locally constant Neumann boundary data, is not sufficient to obtain a complete characterisation of the solutions. A further requirement on the number of maximum points arises in our analysis as a necessary and sufficient condition for the rotational symmetry. Some new arguments are also introduced in the spirit of comparison geometry, that we believe of independent interest. In particular, the notion of expected core radius is defined and employed to achieve a qualitative description of the solutions, eventually leading to new classification results.