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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 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 these type arise in several areas of mathematical physics and shape optimization.