- 11-52 Alberto Farina, Enrico Valdinoci
- On partially and globally overdetermined problems of elliptic type
Mar 29, 11
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Abstract. 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.