On partially and globally overdetermined problems of elliptic type

Abstract. We consider some elliptic pde’s with Dirichlet and Neumann data prescribed on some por-tion 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 overdeter-mined problem is, in fact, globally overdetermined: this enables us to use several classical results in order to classify all the domains that admit a solution of suitable, general, partially overdetermined problems. These results may be seen as solutions of suitable inverse problems—that is to say, given that an overdetermined system possesses a solution, we find the shape of the admissible domains. Models of this type arise in several areas of mathematical physics and shape optimization. 1. Introduction. Th