A Dirichlet-to-Neumann operator maps prescribed boundary values of a solution of a partial differential
equation to its outward normal flux. For the conductivity equation 
on a bounded domain
with Lipschitz boundary 
, it is 
, where 
and 
is the outward unit normal
vector (Sylvester and Uhlmann 1987).
For a bounded measurable coefficient with 
, the operator
is defined weakly by
 |
where 
is the weak solution with boundary value 
and 
is any extension of 
. The weak equation makes the value independent of that extension.
The pairing is between the boundary Sobolev
spaces 
and 
.
For 
,
this maps data for the Dirichlet problem for
the Laplacian to Neumann
boundary conditions. Recovering 
from 
is Calderón's
inverse problem.
A corresponding boundary measurement operator can be defined for a connection Laplacian 
on a Hermitian vector
bundle over a Riemannian manifold.
