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

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 $\Om$ is constant, then $\Om$ must be a ball. One of the main aims of the paper is 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 Alexandrov) and of the symmetry for overdetermined Serrin-type problems for Hessian equations (originally established by Brandolini, Nitsch, Salani, and Trombetti), but also bring several benefits, including new interesting symmetry results and quantitative stability estimates. In fact, leveraging the analysis performed in the classical case (i.e., with classical mean curvature and classical Laplacian) by Magnanini and Poggesi in a series of papers, we will extend their approach to the higher order setting (i.e., with $k$-order mean curvature and $k$-Hessian operator, for $k \ge 1$) achieving various quantitative estimates of closeness to the symmetric configuration. Finally, leveraging the quantitative analysis in presence of bubbling phenomena performed in arXiv:2405.06376, we also provide a quantitative stability result of closeness of almost constant $k$-mean curvature boundaries to a set given by the union of a finite number of disjoint balls of equal radii. In passing, we will also provide two alternative proofs of the result established by Brandolini, Nitsch, Salani, and Trombetti, one of which provides the extension to Hessian operators of the approach famously pioneered by Weinberger for the classical Laplacian.