Contemporary Mathematics
Volume 333, 2003
Size Estimates
Giovanni Alessandrini, Antonino Morassi and Edi Rosset
ABSTRACT.
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-
proach of
size estimates,
that is of upper and lower bounds on the volume of
the unknown inclusion and cavities in terms of work measurements taken from
the exterior.
1.
Introduction
1.1.
The inverse conductivity problem.
Suppose that a given electrically
conducting body n having, for simplicity of discussion, uniform conductivity u
=
1,
might contain an unknown inclusion
D,
having different conductivity, for instance
(}' =
2.
We ask whether D can be determined by the knowledge of a prescribed current
density
p
on the boundary
an
and of the corresponding voltage
u
measured on
an.
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
set n
c
!Rn,
with Lipschitz boundary, and let
p
E
H-!
(an) '
Ian
p
=
0,
repre-
sent the prescribed current density on
an.
If the inclusion D is present, then the
electrostatic potential
u
=
u( x)
is determined (up to an additive constant) as the
H
1
(n) solution to the Neumann problem
{
div((1+XD)V'u)=O inn,
(1.1)
V'u. v
=
p
on
an,
where v denotes the exterior unit normal to
an.
The inverse problem, also known as the inverse conductivity problem with one
measurement, consists of determining D when, for a prescribed nontrivial
p,
the
trace
ulan
is measured.
2000
Mathematics Subject Classification.
Primary 35R30; Secondary 35R25, 35B05, 74B05,
74G75.
Key words and phrases.
Inverse boundary problems, inclusions, cavities, volume bounds.
Work supported in part by MIUR, grant n. 2002013279.
©
2003 American Mathematical Society
http://dx.doi.org/10.1090/conm/333/05951
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