Volume 333, 2003
Giovanni Alessandrini, Antonino Morassi and Edi Rosset
We deal with a class of inverse boundary problems of detection of
inclusions or cavities in electrical conductors or elastic bodies from boundary
measurements. We review the results and the methods of the so-called ap-
that is of upper and lower bounds on the volume of
the unknown inclusion and cavities in terms of work measurements taken from
The inverse conductivity problem.
Suppose that a given electrically
conducting body n having, for simplicity of discussion, uniform conductivity u
might contain an unknown inclusion
having different conductivity, for instance
We ask whether D can be determined by the knowledge of a prescribed current
on the boundary
and of the corresponding voltage
This is the prototype of various inverse boundary problems which arise in many
applied fields, from geophysical prospection to medical imaging.
Let us first formulate analytically the direct problem from which it originates.
We suppose that the region containing the body is represented by a bounded open
with Lipschitz boundary, and let
sent the prescribed current density on
If the inclusion D is present, then the
is determined (up to an additive constant) as the
(n) solution to the Neumann problem
where v denotes the exterior unit normal to
The inverse problem, also known as the inverse conductivity problem with one
measurement, consists of determining D when, for a prescribed nontrivial
Mathematics Subject Classification.
Primary 35R30; Secondary 35R25, 35B05, 74B05,
Key words and phrases.
Inverse boundary problems, inclusions, cavities, volume bounds.
Work supported in part by MIUR, grant n. 2002013279.
2003 American Mathematical Society