Multidimensional Linearized Inverse Problems
for Elastic Bodies


This project deals with initial boundary value problems for the Lame
differential equations system which describes the processes of the
elastic waves propagations in the half space or cylinder or other
regions of the three-dimentional Euclidean space.
Lame parameters and density in this system are assumed
to be the functions depending on space variables and representing
in the form
m=m0+m1,
l=l0+l1
r=r0+r1

where l0, m0, r0 are smooth (or piecewise smooth)
functions depending on one variable only.

The function l1, r1, m1, are supposed to be the
functions depending on the three variables and having smaller values than
the values of the functions m0, l0, r0.
The initial boundary value problems are rewritten in the linear approximation.
The linearized inverse problems consist in finding the functions
m1, r1, l1, if it is given the special boundary
conditions in the linearized initial boundary value problems for Lame
equations system and additional information about the solutions of these
linearized initial boundary value problems. In this case the functions
m0, r0, l0 are assumed to be the given ones.
From physical viewpoint these boundary conditions are impulse point
directional sourses concentrated on the boundary of the
domain in which wave processes are propagated. The additional information
for the inverse problems solving is given in the form of integral
characteristics of the wave fields observed on the boundary of given domain.
And the wave fields arise from acting of considered special sources.
Such types of the linearized inverse problems are called the inverse problems
in Born approximation also.

The questions of the uniqueness, stability estimates and existence of the
solution of these multidimensional linearized inverse problems are studied
and the numerical methods of its solvings are constructed.

This research continues the investigation of the monographies by Romanov
(1984); Lavrent'ev, Romanov, Shishatskii (1986); Yakhno (1990, 1998).

References

Lavrent'ev M.M., Romanov V.G., Shishatskii S.P., (1986)
Ill-Posed Problems of Mathematical Physics and Analysis,
American Mathematical Society Translations of Mathematical Monography,
Vol.64.

Romanov V.G., (1984) Inverse Problems of Mathematical Physics, Moscow,
Nauka, (in Russian).

Yakhno V.G., (1990) Inverse Problems for Differential Equations Systems of
Elastisity, Novosibirsk, Nauka, (in Russian).

Yakhno V.G. (1998) Inverse Problem for Differential Equations System of
Electromagnetoelasticity in Linear Approximation, in Book "Inverse Problems,
Tomography and Image Processing", Plenum Press, New York, 1998.

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