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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|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|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|Laplace equation]] in the three-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p0741601.png" /> are stated, though the above-mentioned problems are also studied in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p0741602.png" />-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p0741603.png" /> Euclidean spaces for the potential of general elliptic equations (see ).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p0741604.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p0741605.png" />, be simply-connected bounded domains with piecewise-smooth boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p0741606.png" />; let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p0741607.png" /></td> </tr></table>

be a [[Newton potential|Newton potential]]; and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p0741608.png" /></td> </tr></table>

be a single-layer potential (cf. [[Simple-layer potential|Simple-layer potential]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p0741609.png" /> is the distance between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416013.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416014.png" />) almost-everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416015.png" /> (on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416016.png" />). Further, let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416017.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416018.png" /> are real numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416019.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416020.png" />. To obtain uniqueness conditions for the solution to this problem it is formulated in the following way: Find conditions on the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416021.png" /> and on the densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416022.png" />, such that from the equality of exterior potentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416024.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

would follow the equalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416028.png" />. If the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416029.png" /> consists of one component, then condition (1) holds when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416030.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416032.png" /> is sufficiently large, or when the data obtained on the boundary of the ball, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416033.png" />, ensure equality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416035.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416037.png" /> consist of one component.

A solution to the general inverse problem in potential theory is unique if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416039.png" /> and if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416040.png" /> are domains of contact, i.e. are such that for each of the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416042.png" /> there exists a common segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416043.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416044.png" />) of the boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416045.png" />, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416046.png" />.

To obtain the inverse problem in potential theory for Newton potentials one has to assume in (1) that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416048.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416050.png" />, be star-like domains with respect to a common point and let the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416051.png" /> have the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416054.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416055.png" />. If the Newton potentials satisfy the conditions (1) and, moreover, if there exists a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416056.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416057.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416059.png" />.

If in the conditions (1) one assumes that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416062.png" />, then one obtains the problem of the determination of the shape of the attracting body from known values of the exterior Newton potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416063.png" /> with given density. In the case of given densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416064.png" /> which are monotone non-decreasing with increasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416065.png" />, the solution of this problem is unique in the class of domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416066.png" /> that are star-like with respect to a common point.

If one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416069.png" /> in (1), one obtains the problem of determining the shape of the attracting body from the known values of the exterior single-layer potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416070.png" /> with given density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416071.png" />. For convex bodies with a constant density, the solution to this problem is unique.

If in the condition (1) one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416074.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416075.png" /> have the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416076.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416078.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416079.png" />. To obtain existence theorems one uses the following formulation of this problem. Find conditions on the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416080.png" /> and on the densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416082.png" />, such that from the equality of the interior potentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416083.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416084.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416085.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

would follow the equalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416088.png" />.

If in conditions (2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416090.png" />, then the solution is unique in the class of convex bodies with variable positive density. If in conditions (2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416092.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416093.png" />, then the solution is also unique in the class of convex bodies.

Let a body be sought such that its exterior Newton potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416094.png" /> of a given density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416095.png" /> outside the body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416096.png" /> be equal to a given harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416098.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p07416099.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160100.png" /> close in the sense of some function metric to the exterior Newton potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160101.png" /> of a given body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160102.png" /> with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160103.png" />. For simply-connected domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160104.png" /> with a smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160105.png" />, under the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160106.png" /> the solution of this problem exists and is unique.

The interior problem is stated similarly to the exterior one, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160107.png" /> is a solution of the inhomogeneous equation in a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160108.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160109.png" /></td> </tr></table>

Find a body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160110.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160111.png" /></td> </tr></table>

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|Branching of solutions]].

The planar inverse problems in potential theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160112.png" /> are stated similarly to those in space, taking into account the corresponding behaviour at infinity. Accordingly, a series of statements mentioned above for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160113.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160114.png" /> be a given density, and consider instead of a logarithmic mass potential its derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160115.png" />; let an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160117.png" />, on the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160118.png" /> outside a disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160119.png" /> be given whose singular points under analytic continuation are situated inside a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160120.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160121.png" />. It is required to find a bounded simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160122.png" /> with Jordan boundary, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160123.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160124.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160125.png" />, where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160126.png" /></td> </tr></table>

The solution to this problem is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160127.png" /> which conformally maps the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160128.png" /> in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160129.png" />-plane onto the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160130.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160131.png" />-plane and which satisfies the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160132.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160133.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160134.png" /> be a given bounded simply-connected domain with Jordan boundary and let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160135.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160136.png" />. Then the function satisfies the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160137.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160138.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160139.png" /> is a solution of equation (3) in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160140.png" /> is replaced by the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160141.png" /> mentioned above, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160142.png" /> is univalent for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160143.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160144.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160145.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160146.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160147.png" />.

From equation (3) one can obtain a number of relations between the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160148.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160149.png" />. For instance, if the exterior potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160150.png" /> can be continued analytically inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160151.png" /> across the entire boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160152.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160153.png" /> is an analytic function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160154.png" />; and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160155.png" /></td> </tr></table>

implies

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160156.png" /></td> </tr></table>

This sometimes allows one to solve planar inverse problems in potential theory in closed form. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160157.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160158.png" />. Then the associated non-linear equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160159.png" /> is, generally speaking, equivalent to a non-linear system of algebraic equations with respect to the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160160.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160161.png" />, which is, in general, not univalent for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160162.png" />, is obtained as the solution of this algebraic system of equations. The class of univalent solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160163.png" /> in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160164.png" /> which meet the requirements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160165.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160166.png" /> 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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Novikov,   "Sur le problème inverse de potentiel"  ''Dokl. Akad. Nauk SSSR'' , '''18''' :  3  (1938)  pp. 165–168</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Tikhonov,   "On the stability of inverse problems"  ''Dokl. Akad. Nauk SSSR'' , '''39''' :  5  (1943)  pp. 195–198  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.N. Sretenskii,   "Theory of the Newton potential" , Moscow-Leningrad  (1946)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.K. Ivanov,   "The inverse problem of potential of a body close to a given body"  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''20''' :  6  (1956)  pp. 793–818  (In Russian)</TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top">  V.K. Ivanov,   "An integral equation in the inverse problem for the logarithmic potential in closed form"  ''Dokl. Akad. Nauk SSSR'' , '''105''' :  3  (1955)  pp. 409–411  (In Russian)</TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top">  V.K. Ivanov,   "On the solvability of the inverse problem for the logarithmic potential in closed form"  ''Dokl. Akad. Nauk SSSR'' , '''106''' :  4  (1956)  pp. 598–599  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M.M. [M.M. Lavrent'ev] Lavrentiev,   "Some improperly posed problems of mathematical physics" , Springer  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7a]</TD> <TD valign="top">  A.I. Prilepko,   "Uniqueness of the solution of the exterior inverse problem of the Newtonian potential"  ''Diff. Eq.'' , '''2'''  (1966)  pp. 56–64  ''Differentsial'nye Uravneniya'' , '''2'''  (1966)  pp. 107–124</TD></TR><TR><TD valign="top">[7b]</TD> <TD valign="top">  A.I. Prilepko,   "On inverse problems in potential theory"  ''Diff. Eq.'' , '''3'''  (1967)  pp. 14–20  ''Differentsial'nye Uravneniya'' , '''3'''  (1967)  pp. 30–44</TD></TR><TR><TD valign="top">[7c]</TD> <TD valign="top">  A.I. Prilepko,   "Inverse external contact problems for generalized magnetic potentials generated by variable magnetic densities"  ''Diff. Eq.'' , '''6'''  (1970)  pp. 31–39  ''Differentsial'nye Uravneniya'' , '''6'''  (1970)  pp. 27–49</TD></TR><TR><TD valign="top">[7d]</TD> <TD valign="top">  A.I. Prilepko,   "The interior inverse potential problem for a body differing slightly from a given body"  ''Diff. Eq.'' , '''8'''  (1972)  pp. 90–96  ''Differentsial'nye Uravaneniya'' , '''8'''  (1972)  pp. 118–125</TD></TR><TR><TD valign="top">[7e]</TD> <TD valign="top">  A.I. Prilepko,   "The inverse problem of a metaharmonic potential for a body close to a given body"  ''Sibirsk. Mat. Zh.'' , '''6''' :  6  (1965)  pp. 1332–1356  (In Russian)</TD></TR><TR><TD valign="top">[7f]</TD> <TD valign="top">  A.I. Prilepko,   "Interior inverse problems of generalized potentials"  ''Siberian Math. J.'' , '''12''' :  3  (1971)  pp. 447–460  ''Sibirsk. Mat. Zh.'' , '''12'''  (1971)  pp. 630–647</TD></TR><TR><TD valign="top">[7g]</TD> <TD valign="top">  A.I. Prilepko,   "On the stability and uniqueness of a solution of inverse problems of generalized potentials of a simple layer"  ''Siberian Math. J.'' , '''12''' :  4  (1971)  pp. 594–601  ''Sibirsk. Mat. Zh.'' , '''12'''  (1971)  pp. 828–836</TD></TR><TR><TD valign="top">[7h]</TD> <TD valign="top">  A.I. Prilepko,   "Mixed inverse problems of potential theory in the case of stellar bodies"  ''Siberian Math. J.'' , '''12''' :  6  (1971)  pp. 969–978  ''Sibirsk. Mat. Zh.'' , '''12'''  (1971)  pp. 1341–1353</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.N. Tikhonov,   V.I. [V.I. Arsenin] Arsenine,   "Solution of ill-posed problems" , Winston  (1977)  (Translated from Russian)</TD></TR></table>



====Comments====
Transformations from single-layer to double-layer potentials and vice versa are considered in [[#References|[a4]]].

For the inverse problem for Newton potentials the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160167.png" /> need not be star-like in order that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160168.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074160/p074160169.png" />, cf. [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Anger,   "Inverse and improperly posed problems in differential equations" , Akademie Verlag  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.M. Lavrent'ev,   "Some improperly posed problems of mathematical physics and analysis" , Amer. Math. Soc.  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.W. Schulze,   G. Wildenhain,   "Methoden der Potentialtheorie für elliptische Differentialgleichungen beliebiger Ordnung" , Birkhäuser  (1977)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.G. Ramm,   "Scattering by obstacles" , Reidel  (1986)  pp. 71</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D. Aharonov,   M. Schiffer,   L. Zalcman,   "Potato kugel"  ''Israel J. of Math.'' , '''40'''  (1981)  pp. 331–339</TD></TR></table>