Mathematical Analysis Seminars @ DISIM

Title: Hessian operators, overdetermined problems, and higher order mean curvatures: symmetry and stability results

Abstract: This is a joint work with Nunzia Gavitone, Gloria Paoli and Giorgio Poggesi. It is well known that there is a deep connection between Serrin’s symmetry result – dealing with overdetermined problems involving the Laplacian – and the celebrated Alexandrov’s Soap Bubble Theorem (SBT) – stating that, if the mean curvature H of the boundary of a smooth bounded connected open set Ω is constant, then Ω must be a ball. We want to extend the study of such a connection to the broader case of overdetermined problems for Hessian operators and constant higher order mean curvature boundaries. Our analysis will not only provide new proofs of the higher order SBT (originally established by Ros in [2]) and of the symmetry for overdetermined Serrin-type problems for Hessian equations (originally established by Brandolini, Nitsch, Salani, and Trombetti in [1]), but also bring several benefits, including new interesting symmetry results and quantitative stability estimates.

[1] B. Brandolini, C. Nitsch, P. Salani, and C. Trombetti. On the stability of the Serrin problem. J. Differential Equations, 245(6):1566–1583, 2008.

[2] A. Ros. Compact hypersurfaces with constant higher order mean curvatures. Rev. Mat. Iberoamericana, 3(3-4):447–453, 1987.