Overdetermined elliptic problems in onduloid-type domains with general nonlinearities

Abstract

In this paper, we prove the existence of solutions to a general semilinear elliptic problem with overdetermined boundary conditions. The proof uses a local bifurcation argument from the straight cylinder, in analogy with the onduloids and the theory of Constant Mean Curvature surfaces. Such examples have been found already for linear problems or with nonlinearity 𝑓 (𝑢) = 1. In this work we are able to extend this phenomenon for a large class of functions 𝑓 (𝑢). Remark: This manuscript, especially the whole proof, is a work in progress in collaboration with David Ruiz and Pieralberto Sicbaldi