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.