27 apr 2021 -- 14:30   [open in google calendar]

Zoom seminar

Please write to Andrea.pinamonti@unitn.it or to Andrea.marchese@unitn.it if you want to attend the seminar.

Abstract.

The torsion problem consists in the study of pairs (Ω, u), where Ω ⊂ R² is a bounded domain and u : Ω → R is a function with constant nonzero laplacian and such that u = 0 on the boundary ∂Ω. A celebrated result due to Serrin states that, if one assumes the additional hypothesis that the normal derivative of u is constant on ∂Ω, then Ω must be a ball and u is rotationally symmetric. We discuss the characterization of rotationally symmetric solutions to the torsion problem on a ring-shaped domain. In contrast with Serrin’s result, we show that having locally constant Neumann boundary data is not sufficient for this purpose. Nevertheless, we prove that rotational symmetry can be forced by means of an additional assumption on the number of maximum points.