Problems in which one has to find the form and densities
of an attracting body from given values of
the exterior (interior) potential of this body (see
Potential theory).
Stated otherwise, one of these problems consists in finding a body such that
its exterior volume potential with a given density coincides outside this
body with a given harmonic function. Originally, inverse problems in potential
theory were considered in the framework of the theory of the Earth's
shape and in celestial mechanics. Inverse problems in potential theory are
related to problems of the equilibrium shape of
a rotating fluid and to problems in geophysics.
The central place in studies of inverse problems in potential theory
is occupied by the problems of the existence, uniqueness and
stability, and also by creating efficient numerical methods for their solution. Existence
theorems have been obtained for local solutions for the case of a
body close to a given body, but significant difficulties are
encountered in the studies of the non-linear equations to which these
problems are generally reduced. There are no existence criteria for
global solutions
(1983).
In many cases the existence of global
solutions is assumed beforehand (this is natural in many
applications) and one considers the problems of uniqueness and stability.
One of the principal stages in studies of uniqueness
is to discover additional conditions which ensure the uniqueness of a
solution. Closely related to the problem of uniqueness is the problem
of stability. For problems written in the form of an equation
of the first kind, generally speaking, finite variations of solutions may
correspond to an arbitrary small variation of the
right-hand side, i.e. these problems are ill-posed (cf.
Ill-posed problems).
To make a problem well-posed one can impose a series of
additional restrictions on the solutions; under these restrictions one obtains different characteristics
of the deviation of a solution as a function of the deviation of the right-hand side.
Below
inverse problems for a
Newton (volume) potential and a single-layer potential for the
Laplace equation
in the three-dimensional Euclidean space
are stated, though the above-mentioned problems are also studied in
-dimensional
Euclidean spaces for the potential of general elliptic equations (see
).
Let
,
,
be simply-connected bounded domains with piecewise-smooth boundaries
;
let
be a
Newton potential;
and let
be a single-layer potential (cf.
Simple-layer potential),
where
is the distance between the points
and
in
,
(
)
almost-everywhere in
(on
).
Further, let
where
are real numbers,
.
The general
exterior inverse problem in potential theory
consists in finding the shapes and densities of an
arbitrary body by given values of an exterior potential
.
To obtain uniqueness conditions for the solution to this problem it
is formulated in the following way: Find conditions on the domains
and on the densities
,
such that from the equality of exterior potentials
and
:
would follow the equalities
,
,
.
If the set
consists of one component, then condition
(1)
holds when
for
,
where
is sufficiently large, or when the data obtained on the boundary of the ball,
,
ensure equality of
and
outside this sphere. As such data one can choose Dirichlet
data on the entire boundary of the closed ball, Cauchy data on
a piece of the boundary of the closed ball, etc. In
the sequel, it is assumed for simplicity that the sets
and
consist of one component.
A solution to the general inverse problem in potential theory is unique if
,
and if the
are domains of contact, i.e. are such that for each of the domains
and
there exists a common segment
(
)
of the boundaries
,
moreover,
.
To obtain the
inverse problem in potential theory for Newton potentials
one has to assume in
(1)
that
and
.
Let
,
,
be star-like domains with respect to a common point and let the functions
have the form
,
where
and
is independent of
.
If the Newton potentials satisfy the conditions
(1)
and, moreover, if there exists a point
such that
,
then
,
.
If in the conditions
(1)
one assumes that
,
,
,
then one obtains the problem of the determination of
the shape of the attracting body from known values of the exterior Newton potential
with given density. In the case of given densities
which are monotone non-decreasing with increasing
,
the solution of this problem is unique in the class of domains
that are star-like with respect to a common point.
If one puts
,
,
in
(1),
one obtains the problem of determining the shape of the
attracting body from the known values of the exterior single-layer potential
with given density
.
For convex bodies with a constant density, the solution to this problem is unique.
If in the condition
(1)
one puts
,
,
,
then one obtains the problem of determining the density of
an arbitrary body from known values of the exterior Newton potential.
The solution of this problem is unique if the functions
have the form
,
where
,
.
The general
interior inverse problem in potential theory
consists in finding the shape and density of an
attracting body from given values of an interior potential
.
To obtain existence theorems one uses the following formulation
of this problem. Find conditions on the domains
and on the densities
,
,
such that from the equality of the interior potentials
and
:
would follow the equalities
,
,
.
If in conditions
(2)
,
,
then the solution is unique in the class of
convex bodies with variable positive density. If in conditions
(2)
,
,
,
then the solution is also unique in the class of convex bodies.
Let a body be sought such that its exterior Newton potential
of a given density
outside the body
be equal to a given harmonic function
,
as
,
and
close in the sense of some function metric to the exterior Newton potential
of a given body
with density
.
For simply-connected domains
with a smooth boundary
,
under the condition
the solution of this problem exists and is unique.
The interior problem is stated similarly to the exterior one, moreover,
is a solution of the inhomogeneous equation in a bounded domain
:
Find a body
such that
Unlike exterior problems, an interior problem, in general, does not have
a unique solution; the number of solutions is
determined by the corresponding bifurcation equation; cf.
Branching of solutions.
The
planar inverse problems in potential theory
are stated similarly to those in space, taking into account
the corresponding behaviour at infinity. Accordingly, a
series of statements mentioned above for
are modified. Planar inverse problems in potential theory can sometimes be studied
conveniently by methods of the theory of functions of
a complex variable and by methods of conformal mapping.
The
planar exterior inverse problem in potential theory.
Let
be a given density, and consider instead of a logarithmic mass potential its derivative
;
let an analytic function
,
,
on the complex plane
outside a disc
be given whose singular points under analytic continuation are situated inside a domain
,
.
It is required to find a bounded simply-connected domain
with Jordan boundary,
,
such that
for
,
where
The solution to this problem is a function
which conformally maps the unit disc
in the complex
-plane
onto the domain
in the
-plane
and which satisfies the conditions
,
.
Let
be a given bounded simply-connected domain with Jordan boundary and let the function
for
.
Then the function satisfies the equation
where
If
is a solution of equation
(3)
in which
is replaced by the function
mentioned above, and if
is univalent for
,
,
,
then
for
.
From equation
(3)
one can obtain a number of relations between the functions
and
.
For instance, if the exterior potential
can be continued analytically inside
across the entire boundary
,
then
is an analytic function for
;
and
implies
This sometimes allows one to solve planar inverse
problems in potential theory in closed form. Let
,
.
Then the associated non-linear equation for
is, generally speaking, equivalent to a non-linear system
of algebraic equations with respect to the coefficients
.
The function
,
which is, in general, not univalent for
,
is obtained as the solution of this algebraic
system of equations. The class of univalent solutions
in the disc
which meet the requirements
,
is the solution of the stated inverse problem in potential theory.
Similar studies can be carried out in the case of the
exterior inverse problem for a logarithmic single-layer potential and also
in the case of interior inverse problems for logarithmic potentials;
moreover, for both exterior and interior inverse
problems one can consider variable densities.