Radial symmetry and partially overdetermined problems in a convex cone

Counterexamples to Symmetry for Partially Overdetermined Elliptic Problems

We exhibit several counterexamples showing that the famous Serrin’s symmetry result for semilinear elliptic overdetermined problems may not hold for partially overdetermined problems, that is when both Dirichlet and Neumann boundary conditions are prescribed only on part of the boundary. Our counterexamples enlighten subsequent positive symmetry results obtained by the first two authors for suc...

Approximate radial symmetry for overdetermined boundary value problems

Stability of Radial Symmetry for a Monge-ampère Overdetermined Problem

Recently, following a new approach, the symmetry of the solutions to overdetermined problems has been established for a class of Hessian type operators. In this paper we prove that the radial solution of the overdetermined Dirichlet problem for the Monge-Ampère equation is stable under suitable perturbations of the data.

On partially and globally overdetermined problems of elliptic type

We consider some elliptic PDEs with Dirichlet and Neumann data prescribed on some portion of the boundary of the domain and we obtain rigidity results that give a classification of the solution and of the domain. In particular, we find mild conditions under which a partially overdetermined problem is, in fact, globally overdetermined: this enables to use several classical results in order to cl...

Solving Overdetermined Eigenvalue Problems

We propose a new interpretation of the generalized overdetermined eigenvalue problem (A− λB)v ≈ 0 for two m × n (m > n) matrices A and B, its stability analysis, and an efficient algorithm for solving it. Usually, the matrix pencil {A− λB} does not have any rank deficient member. Therefore we aim to compute λ for which A − λB is as close as possible to rank deficient; i.e., we search for λ that...