FOREWORD
Xlll
inverse source problems. Often an identification problem can be reduced to an
inverse source problem if we are interested only in uniqueness. For example,
consider two differential equations of parabolic type:
(d/dt)u. - AUj + ajUj = F, 7 = 1, 2.
Letting u = u2- u{, f ax- a2, and a = u{ we get
(d/dt)u - Au + a2u = af.
Also we have additional information, for example, boundary data. Then,
from a uniqueness theorem for the pair (u, f) (an inverse source problem),
one can derive a uniqueness result for the inverse problem concerning iden-
tification of the coefficient a . This is why we are focusing on inverse source
problems.
Now we just mention certain well-known and important inverse problems
of mathematical physics referring to the books and expository papers given
below: the inverse seismic problem (see the books of Lavrent'ev, Romanov,
and Shishatskii [90], and of Romanov [132, 133]); integral geometry and
tomography (the books of Gel'fand, Graev, and Vilenkin [33] and of Helgason
[38], the papers of Anikonov [6], Lavrent'ev and Buhgeim [89], Muhometov
[101], Natterer [104], and of Smith, Solmon, and Wagner [145]); the inverse
spectral problem (the books of Levitan [92] and of Poschel and Trubowitz
[117], the papers of Eskin, Ralston and Trubowitz [29], of Guillemin and
Melrose [36], of M. Kac [73], of Prosser [125], and of Sleeman and Zayed
[144]; and the inverse scattering problem (the books of Chadan and Sabatier
[21], of Colton and Kress [24], of L. Faddeev [30], of Lax and Phillips, the
papers of Angell, Kleinman, and Roach [4], of J. Keller [77], of Majda [98], of
Nachman [102], and of R. Newton [106]). We emphasize that in the theory of
non-well-posed problems for differential equations, uniqueness in the Cauchy
problem is of great importance; the contemporary state of this problem is
described by Hormander [42, 45], Nirenberg [109], and Zuily [170].
The field of inverse problems is growing very rapidly. In 1979 there were
two international conferences, in Newark, U.S.A., and in Halle, Germany
[46], and in 1983 the conferences in New York [47] and in Samarkand, the
U.S.S.R. [159], have been organized. New formulations and results were
obtained. As a bright example we mention the inverse problem posed by
Calderon in 1980 and investigated by R. Kohn and Vogelius [83]. Recently,
very strong results including a complete solution of this problem in the three-
dimensional case were obtained by Sylvester and Uhlmann [156].
Applications are growing very rapidly as well and now they include physics,
geophysics, chemistry, medicine, and engineering. We refer to the books and
expository articles of Baltes [10], Bolt [11], Bukhgeim [16], J. Keller [78], R.
Newton [105], Payne [114], Talenti [157], and Tarantola [158].
This book is devoted mainly to the inverse problem of potential theory
and closely related questions such as coefficient identification problems. In
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