Il 30 ottobre 2023, alle 4:30 pm (CET), nel quadro della serie di seminari Insalate di Matematica, in aula u1-05 e in diretta streaming, Camilla Polvara (Università di Milano) parlerà di
Title: "Symmetry breaking for semilinear elliptic equations"
Abstract: We consider semilinear elliptic equations in two different domains: in spherical sectors inside a cone with mixed boundary conditions and in cones with Neumann boundary conditions.
The aim of the talk is to show that a radial symmetry result of Gidas-Ni-Nirenberg type for positive solutions does not hold in general nonconvex cones.
This symmetry breaking result is achieved by studying the Morse index of radial positive solutions and analyzing how it depends on the domain D on the unit sphere which spans the cone.
In particular, it is proved that the Neumann eigenvalues of the Laplace Beltrami operator on D play a role in computing the Morse index.
A similar breaking of symmetry result is obtained for the positive solutions of the critical Neumann problem in the whole unbounded cone. In this case, it is proved that the standard bubbles, which are the only radial solutions, become unstable for a class of nonconvex cones.
Keywords: Semilinear elliptic problem, critical and subcritical nonlinearities, spherical sectors and cones, symmetry breaking