Overdetermined elliptic problems: bifurcation of solutions and modica type estimates

Resumen

This thesis is concerned with the study of semilinear overdetermined elliptic problems under the boundary conditions: the solution vanishes at the boundary and the normal derivative is constant. In this thesis, we have proved some results of local bifurcation of solutions to overdetermined elliptic problems, which give nontrivial solutions. Moreover, a gradient estimate in the spirit of Modica's classical result is presented. In Eucliean setting, we first obtained the existence of nontrivial unbounded domains, bifurcating from the straight cylinder such that the overdetermined elliptic problem has a positive bounded solution. We proved such a result for a very general class of functions f by making use of the Crandall-Rabinowitz Bifurcation Theorem. We also have treated the case of the complement of a cylinder for f=0, where we proved the existence of nontrivial unbounded exceptional domains in the Euclidean space for d>3. Moreover, we have established a kind of Modica type estimate for bounded solutions to the overdetermined elliptic problem. As we have seen, the presence of the boundary changes the usual form of the Modica estimate for entire solutions. The case of equality has also been discussed. From such estimates we will derive information about the curvature of the boundary under a certain condition on the normal derivative and the nonlinearity. The proof uses the maximum principle together with scaling arguments and a careful passage to the limit in the arguments by contradiction. Concerning the problem on the sphere, we have constructed nontrivial contractible domains where the overdetermined elliptic problem could admit a positive solution. These domains are perturbations of a subdomain of the sphere. This shows in particular that Serrin's theorem for overdetermined problems in the Euclidean space cannot be generalized to the sphere even for contractible domains.