Potential theory, inverse problems in

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 $ \mathbf R ^ {3} $ are stated, though the above-mentioned problems are also studied in $ n $- dimensional $ ( n > 2 ) $ Euclidean spaces for the potential of general elliptic equations (see ).

Let $ T _ \alpha $, $ \alpha = 1 , 2 $, be simply-connected bounded domains with piecewise-smooth boundaries $ S _ \alpha $; let

$$ U _ \alpha = \int\limits _ {T _ \alpha } \frac{1}{| x - y | } \mu _ \alpha ( y) dy $$

be a Newton potential; and let

$$ V _ \alpha ( x) = \int\limits _ {S _ \alpha } \frac{1}{| x - y | } \zeta _ \alpha ( y) dS _ {y} $$

be a single-layer potential (cf. Simple-layer potential), where $ | x - y | $ is the distance between the points $ x = ( x _ {1} , x _ {2} , x _ {3} ) $ and $ y = ( y _ {1} , y _ {2} , y _ {3} ) $ in $ \mathbf R ^ {3} $, $ \mu _ \alpha \neq 0 $( $ \zeta _ \alpha ( y) \neq 0 $) almost-everywhere in $ T _ \alpha $( on $ S _ \alpha $). Further, let

$$ Z _ \alpha ( x) = \beta U _ \alpha ( x) + \gamma V _ \alpha ( x) , $$

where $ \beta , \gamma $ are real numbers, $ \beta ^ {2} + \gamma ^ {2} \neq 0 $.

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 $ Z ( x) $. To obtain uniqueness conditions for the solution to this problem it is formulated in the following way: Find conditions on the domains $ T _ \alpha $ and on the densities $ \mu _ \alpha , \zeta _ \alpha $, such that from the equality of exterior potentials $ Z _ {1} ( x) $ and $ Z _ {2} ( x) $:

$$ \tag{1 } Z _ {1} ( x) = Z _ {2} ( x) \ \ \textrm{ for } \ x \in \mathbf R ^ {3} \setminus ( \overline{T}\; _ {1} \cup \overline{T}\; _ {2} ) $$

would follow the equalities $ T _ {1} = T _ {2} $, $ \mu _ {1} = \mu _ {2} $, $ \zeta _ {1} = \zeta _ {2} $. If the set $ \mathbf R ^ {3} \setminus ( \overline{T}\; _ {1} \cup \overline{T}\; _ {2} ) $ consists of one component, then condition (1) holds when $ Z _ {1} ( x) = Z _ {2} ( x) $ for $ | x | > R $, where $ R $ is sufficiently large, or when the data obtained on the boundary of the ball, $ | x | = R $, ensure equality of $ Z _ {1} ( x) $ and $ Z _ {2} ( x) $ 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 $ T ^ { \prime } = T _ {1} \cap T _ {2} $ and $ T ^ { \prime\prime } = \mathbf R ^ {3} \setminus ( \overline{T}\; _ {1} \cup \overline{T}\; _ {2} ) $ consist of one component.

A solution to the general inverse problem in potential theory is unique if $ \mu _ {1} = \mu _ {2} = \mu > 0 $, $ \zeta _ {1} = \zeta _ {2} = \zeta > 0 $ and if the $ T _ \alpha $ are domains of contact, i.e. are such that for each of the domains $ T ^ { \prime } $ and $ T ^ { \prime\prime } $ there exists a common segment $ S _ {*} $( $ \mathop{\rm mes} S _ {*} \neq 0 $) of the boundaries $ S _ \alpha $, moreover, $ \mathop{\rm mes} [ ( S _ {1} \cup S _ {2} ) \setminus S _ {*} ] = 0 $.

To obtain the inverse problem in potential theory for Newton potentials one has to assume in (1) that $ \beta = 1 $ and $ \gamma = 0 $. Let $ T _ \alpha $, $ \alpha = 1 , 2 $, be star-like domains with respect to a common point and let the functions $ \mu _ \alpha ( y) $ have the form $ \mu _ \alpha ( y) = \delta _ \alpha \nu ( y) $, where $ \delta _ \alpha = \textrm{ const } $ and $ \nu > 0 $ is independent of $ \rho = | y | $. If the Newton potentials satisfy the conditions (1) and, moreover, if there exists a point $ x _ {0} \in T _ {1} \cap T _ {2} $ such that $ U _ {1} ( x _ {0} ) = U _ {2} ( x _ {0} ) $, then $ T _ {1} = T _ {2} $, $ \mu _ {1} = \mu _ {2} $.

If in the conditions (1) one assumes that $ \mu _ {1} = \mu _ {2} = \mu $, $ \beta = 1 $, $ \gamma = 0 $, then one obtains the problem of the determination of the shape of the attracting body from known values of the exterior Newton potential $ U ( x) $ with given density. In the case of given densities $ \mu ( y) $ which are monotone non-decreasing with increasing $ | y | $, the solution of this problem is unique in the class of domains $ T _ \alpha $ that are star-like with respect to a common point.

If one puts $ \beta = 0 $, $ \gamma = 1 $, $ S _ {1} = S _ {2} $ in (1), one obtains the problem of determining the shape of the attracting body from the known values of the exterior single-layer potential $ V ( x) $ with given density $ \zeta $. For convex bodies with a constant density, the solution to this problem is unique.

If in the condition (1) one puts $ T _ {1} = T _ {2} = T $, $ \beta = 1 $, $ \gamma = 0 $, 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 $ \mu _ \alpha ( y) $ have the form $ \mu _ \alpha ( y) = \eta ( y) \nu _ \alpha ( y) $, where $ \partial \nu _ \alpha / \partial \rho = 0 $, $ \partial \eta / \partial \rho \geq 0 $.

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 $ Z ( x) $. To obtain existence theorems one uses the following formulation of this problem. Find conditions on the domains $ T _ \alpha $ and on the densities $ \mu _ \alpha $, $ \zeta _ \alpha $, such that from the equality of the interior potentials $ Z _ {1} ( x) $ and $ Z _ {2} ( x) $:

$$ \tag{2 } Z _ {1} ( x) = Z _ {2} ( x) \ \ \textrm{ for } \ x \in T _ {1} \cap T _ {2} $$

would follow the equalities $ T _ {1} = T _ {2} $, $ \mu _ {1} = \mu _ {2} $, $ \zeta _ {1} = \zeta _ {2} $.

If in conditions (2) $ \beta = 1 $, $ \gamma = 0 $, then the solution is unique in the class of convex bodies with variable positive density. If in conditions (2) $ \beta = 0 $, $ \gamma = 1 $, $ \zeta _ {1} = \zeta _ {2} = \zeta = \textrm{ const } $, then the solution is also unique in the class of convex bodies.

Let a body be sought such that its exterior Newton potential $ U ( x ; T _ {1} , \mu ) $ of a given density $ \mu ( x) $ outside the body $ T _ {1} $ be equal to a given harmonic function $ H ( x) $, $ H ( x) \rightarrow 0 $ as $ | x | \rightarrow \infty $, and $ H ( x) $ close in the sense of some function metric to the exterior Newton potential $ U ( x ; T , \mu ) $ of a given body $ T $ with density $ \mu $. For simply-connected domains $ T $ with a smooth boundary $ S $, under the condition $ \mu ( x) \mid _ {S} \neq 0 $ the solution of this problem exists and is unique.

The interior problem is stated similarly to the exterior one, moreover, $ H ( x) $ is a solution of the inhomogeneous equation in a bounded domain $ G _ {0} \supset \overline{T}\; $:

$$ \Delta H = - \mu ( x) \ \ \textrm{ for } \ x \in G _ {0} . $$

Find a body $ T _ {1} $ such that

$$ \mathop{\rm grad} H ( x) = \mathop{\rm grad} U ( x ; T _ {1} , \mu ). $$

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 $ ( n = 2 ) $ are stated similarly to those in space, taking into account the corresponding behaviour at infinity. Accordingly, a series of statements mentioned above for $ n = 3 $ 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 $ \mu = 1 $ be a given density, and consider instead of a logarithmic mass potential its derivative $ \partial / \partial z = [ ( \partial / \partial x) - i ( \partial / \partial y ) ] / 2 $; let an analytic function $ H ( z) $, $ H ( \infty ) = 0 $, on the complex plane $ z = x + iy $ outside a disc $ K ( 0 , R ) = \{ {z } : {| z | < R } \} $ be given whose singular points under analytic continuation are situated inside a domain $ D _ {*} $, $ 0 \in D _ {*} $. It is required to find a bounded simply-connected domain $ D $ with Jordan boundary, $ \overline{D}\; _ {*} \subset D \subset \overline{D}\; \subset K ( 0 , R ) $, such that $ H ( z) = U ( z , D ) $ for $ | z | > R $, where

$$ U ( z , D ) = - \frac{1} \pi {\int\limits \int\limits } _ { D } \frac{1}{z - \zeta } d \xi d \eta ,\ \ \zeta = \xi + i \eta . $$

The solution to this problem is a function $ z ( t) $ which conformally maps the unit disc $ | t | < 1 $ in the complex $ t $- plane onto the domain $ D $ in the $ ( z = x + iy) $- plane and which satisfies the conditions $ z ( 0) = 0 $, $ z ^ \prime ( 0) > 0 $.

Let $ D $ be a given bounded simply-connected domain with Jordan boundary and let the function $ U _ \alpha ( z) = U ( z , D ) $ for $ z \in \mathbf R ^ {2} \setminus D $. Then the function satisfies the equation

$$ \tag{3 } z ^ {*} ( s) = - \frac{1}{2 \pi i } \int\limits _ {| t| = 1 } \frac{U _ \alpha [ z ( t) ] dt }{t-s} ,\ | s | > 1 , $$

where

$$ z ^ {*} ( s) = {z \left ( \frac{1}\overline{ {s}}\; \right ) } bar \ \ \textrm{ for } \ | s | \geq 1 . $$

If $ z ( t) $ is a solution of equation (3) in which $ U _ \alpha ( z) $ is replaced by the function $ H ( z) $ mentioned above, and if $ z ( t) $ is univalent for $ | t | < 1 $, $ z ( 0) = 0 $, $ z ^ \prime ( 0) > 0 $, then $ H ( z) = U ( z , D ) $ for $ | z | > R $.

From equation (3) one can obtain a number of relations between the functions $ U _ \alpha ( z) $ and $ z ( t) $. For instance, if the exterior potential $ U _ \alpha ( z) $ can be continued analytically inside $ D $ across the entire boundary $ \partial D $, then $ z ( t) $ is an analytic function for $ | t | = 1 $; and

$$ U _ \alpha ( z) = \sum_{k=1}^ { m } \frac{c _ k}{z ^ {k}} \ \ \textrm{ for } \ | z | > R ,\ c _ {m} \neq 0 , $$

implies

$$ z ( t) = \alpha _ {1} t + \dots + \alpha _ {m} t ^ {m} ,\ \ \alpha _ {m} \neq 0. $$

This sometimes allows one to solve planar inverse problems in potential theory in closed form. Let $ H ( z) = \sum_{k=1}^ {m} c _ {k} / z ^ {k} $, $ c _ {m} \neq 0 $. Then the associated non-linear equation for $ z ( t) $ is, generally speaking, equivalent to a non-linear system of algebraic equations with respect to the coefficients $ \alpha _ {1} \dots \alpha _ {m} $. The function $ z ( t) $, which is, in general, not univalent for $ | t | < 1 $, is obtained as the solution of this algebraic system of equations. The class of univalent solutions $ z ( t) $ in the disc $ | t | < 1 $ which meet the requirements $ z ( 0) = 0 $, $ z ^ \prime ( 0) > 0 $ 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.

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Comments[edit]

Transformations from single-layer to double-layer potentials and vice versa are considered in [a4].

For the inverse problem for Newton potentials the domains $ T _ \alpha $ need not be star-like in order that $ T _ {1} = T _ {2} $ and $ \mu _ {1} = \mu _ {2} $, cf. [a5].

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