Solutions to overdetermined elliptic problems on small domains and some applications

Author: Alberto Enciso

Abstract: In this talk we consider the overdetermined boundary problem for a general second order semilinear elliptic equation on bounded domains (in a compact manifold or on \(R^n\)), where one prescribes both the Dirichlet and Neumann data of the solution modulo a multiplicative constant. We are interested in the case where the data are not necessarily constant and where the coefficients of the equation can depend on the position, so that the overdetermined problem does not generally admit a radial solution. The main idea is that, under minor technical hypotheses, nontrivial solutions to the overdetermined boundary problem can be constructed on domains that are small perturbations of small ellipsoids. As we will see, this has interesting applications in different fields. The talk is based on joint work with Miguel Domínguez-Vázquez and Daniel Peralta-Salas.