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On partially and globally overdetermined problems of elliptic type
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 135, Number 6, December 2013
- pp. 1699-1726
- 10.1353/ajm.2013.0052
- Article
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We consider some elliptic pde’s 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 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.