Alberto Farina, Enrico Valdinoci
On partially and globally overdetermined problems of elliptic type
(347K, pdf)
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.