View source for Potential theory

Originally, studies related to the properties of forces which follow the law of gravitation. In the statement of this law given by I. Newton (1687) (cf. [[Newton laws of mechanics|Newton laws of mechanics]]) the only forces considered are the forces of mutual attraction acting upon two material particles of small size or two material points. These forces are directly proportional to the product of the masses of these particles and inversely proportional to the square of the distance between them. Thus, the first and the most important problem from the point of view of celestial mechanics and geodesy was to study the forces of attraction of a material point by a finite smooth material body — a spheroid and, in particular, an ellipsoid (since many celestial bodies have this shape). After first partial achievements by Newton and others, studies carried out by J.L. Lagrange (1773), A. Legendre (1784–1794) and P.S. Laplace (1782–1799) became of major importance. Lagrange has established that a field of gravitational forces, as it is called now, is a [[Potential field|potential field]] and has introduced a function which was later called by G. Green (1828) a potential function and by C.F. Gauss (1840) — just a [[Potential|potential]]. At present, the achievements of this initial period are included in courses on classical celestial mechanics (see also [[#References|[2]]]).

Already Gauss and his contemporaries discovered that the method of potentials (cf. [[Potentials, method of|Potentials, method of]]) can be applied not only to solve problems in the theory of gravitation but, in general, to solve a wide range of problems in mathematical physics, in particular in electrostatics and magnetism. In this connection, potentials became to be considered not only for the physically realistic problems concerning the mutual attraction of positive masses, but also for problems with  "masses"  of arbitrary sign, or charges. The principal boundary value problems were defined, such as the [[Dirichlet problem|Dirichlet problem]] and the [[Neumann problem]], the electrostatic problem of the static distribution of charges on conductors or the [[Robin problem|Robin problem]], and the problem of sweeping-out mass (see [[Balayage method|Balayage method]]). To solve the above-mentioned problems in the case of domains with sufficiently smooth boundaries certain types of potentials turned out to be efficient, i.e. special classes of parameter-dependent integrals such as volume potentials of distributed mass, single- and double-layer potentials, logarithmic potentials, Green potentials, etc. Results obtained by A.M. Lyapunov and V.A. Steklov at the end of the 19th century were fundamental for the creation of strong methods of solution of the principal boundary value problems. Studies in potential theory concerning properties of different potentials have acquired an independent significance.

In the first half of the 20th century, a great stimulus for the generalization of the principal problems and the completion of the existing formulations in potential theory was made on the basis of the general notions of a [[Radon measure|Radon measure]], a [[Capacity|capacity]] and generalized functions. Modern potential theory is closely related in its development to the theory of analytic, harmonic and subharmonic functions and to probability theory.

Together with further profound studies of classical boundary value problems and inverse problems (see [[Potential theory, inverse problems in|Potential theory, inverse problems in]]) the modern period in the development of potential theory is characterized by the application of methods and notions of topology and functional analysis, and the use of abstract axiomatic methods (see [[Potential theory, abstract|Potential theory, abstract]]).

==Principal classes of potentials and their properties.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p0741401.png" /> be a smooth closed surface, i.e. an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p0741402.png" />-dimensional smooth manifold without boundary, in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p0741403.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p0741404.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p0741405.png" />, which bounds a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p0741406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p0741407.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p0741408.png" /> be the exterior unbounded domain. 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/p074140/p0741409.png" /></td> </tr></table>

be a principal [[Fundamental solution|fundamental solution]] of the [[Laplace equation|Laplace equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414010.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414011.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/p074140/p07414012.png" /></td> </tr></table>

is the distance between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414016.png" /> is the area of the unit sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414018.png" /> is the gamma-function. The following three integrals, which depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414019.png" /> as a parameter,

<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/p074140/p07414020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414021.png" /> is the direction of the exterior (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414022.png" />) normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414023.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414024.png" />, are called the volume potential, the single-layer potential and the double-layer potential, respectively. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414027.png" /> are called the densities of the corresponding potentials; hereafter they are assumed to be absolutely integrable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414028.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414029.png" />, respectively. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414030.png" /> (and sometimes for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414031.png" />) the integrals (1) are called the Newton volume potential and the Newton single- and double-layer potentials; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414032.png" /> they are called logarithmic mass, single-layer or double-layer potentials, respectively. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414033.png" /> be of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414034.png" />. Then the volume potential (cf. [[Newton potential|Newton potential]]) and its first derivatives are continuous everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414035.png" />; moreover, they can be calculated by differentiation under the integral sign, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414036.png" />. Further,

<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/p074140/p07414037.png" /></td> </tr></table>

The second derivatives are continuous everywhere outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414038.png" />, but they have a discontinuity when passing across the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414039.png" />; moreover, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414040.png" /> they satisfy the [[Poisson equation|Poisson equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414042.png" />, and in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414043.png" /> — the Laplace equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414045.png" />. The above-mentioned properties characterize a volume potential.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414046.png" /> is a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414047.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414048.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414049.png" />, then Gauss' formula for a volume potential is valid:

<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/p074140/p07414050.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414051.png" />. The single-layer potential (cf. [[Simple-layer potential|Simple-layer potential]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414052.png" /> is a harmonic function when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414053.png" />; moreover,

<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/p074140/p07414054.png" /></td> </tr></table>

in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414055.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414056.png" />, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414057.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414058.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414059.png" />. A single-layer potential is continuous everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414061.png" />, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414062.png" /> and its tangential derivatives are continuous when passing across the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414063.png" />. The normal derivative of a single-layer potential has a discontinuity when passing across the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414064.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/p074140/p07414065.png" /></td> </tr></table>

<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/p074140/p07414066.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414068.png" /> are the limit values of the normal derivative from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414070.png" />, respectively, i.e.

<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/p074140/p07414071.png" /></td> </tr></table>

<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/p074140/p07414072.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414073.png" /> denotes the so-called direct value of the normal derivative of a single-layer potential calculated over the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414074.png" />, i.e.

<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/p074140/p07414075.png" /></td> </tr></table>

It is a continuous function of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414076.png" />, and the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414077.png" /> has a weak singularity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414078.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/p074140/p07414079.png" /></td> </tr></table>

These properties characterize a single-layer potential.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414080.png" />. The [[Double-layer potential|double-layer potential]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414081.png" /> is a harmonic function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414082.png" />; moreover,

<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/p074140/p07414083.png" /></td> </tr></table>

When passing across the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414084.png" /> the double-layer potential has a discontinuity (whence its name):

<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/p074140/p07414085.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414087.png" /> are the limit values of the double-layer potential from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414089.png" />, respectively, that is,

<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/p074140/p07414090.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414091.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414092.png" /> denotes the so-called direct value of the double-layer potential calculated over the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414093.png" />, that is,

<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/p074140/p07414094.png" /></td> </tr></table>

It is a continuous function of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414095.png" />, and the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414096.png" /> has a weak singularity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414097.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/p074140/p07414098.png" /></td> </tr></table>

The tangential derivatives of a double-layer potential also have a discontinuity when passing across the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p07414099.png" />, but the normal derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140100.png" /> retains its value when passing across <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140101.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/p074140/p074140102.png" /></td> </tr></table>

These properties characterize a double-layer potential.

In the case of a constant density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140103.png" /> Gauss' formula for a double-layer potential holds:

<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/p074140/p074140104.png" /></td> </tr></table>

The integral at the left-hand side of this equality is interpreted (when divided by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140105.png" />) as the solid angle at which the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140106.png" /> is seen from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140107.png" />.

Below, certain properties of potentials under weaker restrictions on the densities and the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140108.png" /> are given.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140109.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140110.png" /> is a harmonic function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140111.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140112.png" /> is summable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140113.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140114.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140115.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140118.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140119.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140120.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140121.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140122.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140123.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140124.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140125.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140126.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140127.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140128.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140129.png" />, then the generalized second derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140130.png" /> exist, they are also of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140131.png" /> and are expressed by singular integrals:

<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/p074140/p074140132.png" /></td> </tr></table>

<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/p074140/p074140133.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140134.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140135.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140136.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140137.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140138.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140139.png" />, then all generalized derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140140.png" /> also exist and belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140141.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140142.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140143.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140144.png" /> is a generalized solution of the Poisson equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140145.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140146.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140147.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140148.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140149.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140150.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140151.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140152.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140153.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140154.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140155.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140156.png" /> integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140157.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140158.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140159.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140160.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140161.png" /> be a closed bounded domain such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140162.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140163.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140164.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140165.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140166.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140167.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140168.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140169.png" />. If the density is bounded and summable, then

<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/p074140/p074140170.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140171.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140172.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140173.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140174.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140175.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140176.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140177.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140178.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140179.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140180.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140181.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140182.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140183.png" /> integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140184.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140185.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140186.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140187.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140188.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140189.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140190.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140191.png" /> integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140192.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140193.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140194.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140195.png" />.

For potentials and their derivatives extended by continuity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140196.png" /> the above-described properties of smoothness are also valid under the corresponding smoothness conditions on the density and the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140197.png" />.

==Representation of functions and solution of the principal boundary value problems in potential theory using potentials.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140198.png" /> be a function of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140199.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140200.png" /> be a smooth surface of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140201.png" />. Then the following integral identity (Green formula) holds:

<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/p074140/p074140202.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

<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/p074140/p074140203.png" /></td> </tr></table>

In particular, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140204.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140205.png" /> can be represented as the sum of a volume potential and single- and double-layer potentials, with respective densities

<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/p074140/p074140206.png" /></td> </tr></table>

For a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140207.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140208.png" /> that is harmonic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140209.png" /> the following identity holds:

<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/p074140/p074140210.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

Hence, such a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140211.png" /> can be represented in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140212.png" /> by the sum of single- and double-layer potentials with densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140213.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140214.png" />, respectively. However, the densities in (3) cannot be arbitrarily given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140215.png" />; they are connected by the integral relation obtained from (3) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140216.png" />.

A central place in potential theory is occupied by the Dirichlet and the Neumann boundary value problem (also called the first and the second boundary value problem (cf. also [[Dirichlet problem|Dirichlet problem]]; [[Neumann problem]])) for the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140217.png" /> (interior problems) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140218.png" /> (exterior problems) which, under the assumption of sufficient smoothness, can be completely studied by reducing them to the integral equations of potential theory.

The interior Dirichlet problem: Find a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140219.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140220.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140221.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140222.png" />, harmonic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140223.png" />, which satisfies the boundary condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140224.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140225.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140226.png" /> is a given continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140227.png" />. The solution to this problem always exists, is unique and can be obtained in the form of a double-layer potential

<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/p074140/p074140228.png" /></td> </tr></table>

with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140229.png" /> which is obtained as the unique solution of the Fredholm integral equation of the second kind

<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/p074140/p074140230.png" /></td> </tr></table>

The interior Neumann problem: Find a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140231.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140232.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140233.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140234.png" />, harmonic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140235.png" />, which satisfies the boundary condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140236.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140237.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140238.png" /> is a given continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140239.png" />. A solution to this problem exists if and only if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140240.png" /> satisfies the orthogonality condition

<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/p074140/p074140241.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

This solution is obtained up to an arbitrary additive constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140242.png" /> in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140243.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/p074140/p074140244.png" /></td> </tr></table>

is a single-layer potential whose density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140245.png" /> is obtained from the following Fredholm integral equation of the second kind:

<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/p074140/p074140246.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

The continuous homogeneous equation has a non-trivial solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140247.png" /> and the inhomogeneous equation (5) is solvable under the condition (4); moreover, its general solution has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140248.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140249.png" /> is an arbitrary constant.

The exterior Dirichlet problem: Find a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140250.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140251.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140252.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140253.png" />, harmonic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140254.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140255.png" />, which satisfies the boundary condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140256.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140257.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140258.png" /> is a given continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140259.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140260.png" /> is assumed to be regular at infinity, i.e.

<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/p074140/p074140261.png" /></td> </tr></table>

The solution of this problem always exists, is unique and can be obtained in the form

<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/p074140/p074140262.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140263.png" /> is a constant 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/p074140/p074140264.png" /></td> </tr></table>

is a double-layer potential whose density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140265.png" /> is a solution of the following Fredholm integral equation of the second kind:

<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/p074140/p074140266.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

<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/p074140/p074140267.png" /></td> </tr></table>

The corresponding homogeneous equation has the non-trivial solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140268.png" />. Under an adequate choice of the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140269.png" />, the solution of the inhomogeneous equation (6) takes the form

<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/p074140/p074140270.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140271.png" /> is an arbitrary constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140272.png" /> is a particular solution of (6). The constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140273.png" /> is chosen in the form

<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/p074140/p074140274.png" /></td> </tr></table>

where the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140275.png" /> must satisfy the condition

<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/p074140/p074140276.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

This density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140277.png" /> is a non-trivial solution of the equation (5) of the interior Neumann problem with data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140278.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140279.png" />, satisfying the normalization condition

<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/p074140/p074140280.png" /></td> </tr></table>

which is equivalent to (7) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140281.png" />. The single-layer potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140282.png" /> with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140283.png" /> is called an equilibrium potential or Robin potential. The density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140284.png" /> provides a solution to the [[Robin problem|Robin problem]] or the electrostatic problem on the distribution of charges on the conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140285.png" /> generating a constant equilibrium potential in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140286.png" />. A certain complexity in solving the exterior Dirichlet problem is due to the fact that, in general, the harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140287.png" /> that is regular at infinity decreases slower than the double-layer potential as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140288.png" /> and, thus, in the general case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140289.png" /> cannot be represented only by one double-layer potential.

The exterior Neumann problem: Find a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140290.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140291.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140292.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140293.png" />, harmonic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140294.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140295.png" />, which satisfies the boundary condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140296.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140297.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140298.png" /> is a given continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140299.png" />; in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140300.png" /> is assumed to be regular at infinity. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140301.png" /> the solution of this problem always exists and is unique; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140302.png" /> a solution exists if and only if the following condition holds:

<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/p074140/p074140303.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

Moreover, this solution is defined up to an arbitrary additive constant. This solution of the exterior Neumann problem can be represented in the form of a single-layer potential

<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/p074140/p074140304.png" /></td> </tr></table>

whose density is a solution of the following Fredholm integral equation of the second kind:

<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/p074140/p074140305.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140306.png" /> the solution of this equation always exists and is unique. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140307.png" /> the corresponding homogeneous equation has a non-trivial solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140308.png" />. Thus, the inhomogeneous equation (9) with the solvability condition (8) has a unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140309.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/p074140/p074140310.png" /></td> </tr></table>

and its general solution is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140311.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140312.png" /> is an arbitrary constant.

Boundary value problems in potential theory can also be solved using a [[Green function|Green function]]. For instance, for the (interior) Dirichlet problem the Green function has the form

<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/p074140/p074140313.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140314.png" /> is a harmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140315.png" /> that is continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140316.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140317.png" /> and that satisfies, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140318.png" />, the boundary condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140319.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140320.png" />. The solution of the (interior) Dirichlet problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140321.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140322.png" /> for the Poisson equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140323.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140324.png" />, with the boundary condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140325.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140326.png" />, can be represented in the form

<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/p074140/p074140327.png" /></td> </tr></table>

The integrals

<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/p074140/p074140328.png" /></td> </tr></table>

which depend on the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140329.png" />, are called the Green volume potential (of the Dirichlet problem) and the Green double-layer potential, respectively. Their properties are similar to the properties of the potentials (1).

Green functions allow one to reduce eigen value problems to integral equations. For instance, the Dirichlet problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140330.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140331.png" />, with boundary condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140332.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140333.png" />, is reduced to the following Fredholm integral equation of the second kind with a self-adjoint kernel:

<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/p074140/p074140334.png" /></td> </tr></table>

==Further generalization of some fundamental concepts in potential theory.==
Simultaneously with profound studies on the properties of the potentials (1), defined by densities of a more or less general form, and of their applications, the concept of potential itself has undergone a deep generalization related to the concepts of a [[Radon measure|Radon measure]] and a Radon integral. This process started in the 1920s.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140335.png" /> be a positive [[Borel measure|Borel measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140336.png" /> with compact support <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140337.png" />. The potential of the measure,

<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/p074140/p074140338.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>

exists everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140339.png" /> in the sense of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140340.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140341.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140342.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140343.png" /> (i.e. here the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140344.png" /> is also allowed), is a [[Superharmonic function|superharmonic function]] everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140345.png" /> and is harmonic outside the support <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140346.png" />. For a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140347.png" /> of arbitrary sign with compact support the potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140348.png" /> is defined on the basis of the canonical decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140349.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140350.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140351.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140352.png" />. At the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140353.png" /> where both potentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140354.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140355.png" /> assume the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140356.png" />, this potential is not defined. If the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140357.png" /> is concentrated on a smooth surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140358.png" />, then the double-layer potential of the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140359.png" /> is determined similarly to (10):

<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/p074140/p074140360.png" /></td> </tr></table>

The potential (10) is finite, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140361.png" /> everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140362.png" /> except at the points of a [[Polar set|polar set]], which is characterized as a set of outer [[Capacity|capacity]] zero. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140363.png" /> everywhere except on a set of outer capacity zero, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140364.png" />. If the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140365.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140366.png" />, is concentrated on a set of capacity zero, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140367.png" />. The following maximum principle is valid:

<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/p074140/p074140368.png" /></td> </tr></table>

i.e. the least upper bound of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140369.png" /> equals the least upper bound of the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140370.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140371.png" />. If this restriction is continuous (in the general case, including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140372.png" />) at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140373.png" />, then the potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140374.png" /> is continuous at that point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140375.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140376.png" />. The potentials of a measure, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140377.png" />, can be reduced to potentials of densities (1) if and only if the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140378.png" /> is absolutely continuous with respect to the Lebesgue measure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140379.png" /> or on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140380.png" />, respectively (see [[#References|[3]]]–[[#References|[6]]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140381.png" /> is a [[Generalized function|generalized function]], or distribution, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140382.png" />, then its potential is defined as the convolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140383.png" />, which is also a generalized function. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140384.png" /> is a generalized function with compact support, then the Poisson equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140385.png" /> is valid on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140386.png" /> in the sense of the theory of generalized functions. Potentials of measures can be considered as a particular case of potentials of distributions. For potentials of distributions see [[#References|[3]]], [[#References|[4]]], [[#References|[9]]].

For domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140387.png" /> with sufficiently smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140388.png" /> the method of potentials provides an efficient solution to the Dirichlet problem. One of the principal directions of development in potential theory consists in finding methods to prove the existence and uniqueness of a solution to the Dirichlet problem for wider classes of domains (see [[Balayage method|Balayage method]]; [[Dirichlet principle|Dirichlet principle]]; [[Perron method|Perron method]]; [[Schwarz alternating method|Schwarz alternating method]]). However, in 1910 S. Zaremba noted that for a plane domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140389.png" /> whose boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140390.png" /> has isolated points the Dirichlet problem is not always solvable in the above classical formulation; in addition, in 1912 H. Lebesgue has shown that it is not always solvable also for spatial domains homeomorphic to a closed sphere but with a sufficiently sharp edge at the boundary entering inside the domain (a so-called Lebesgue spine, see [[Irregular boundary point|Irregular boundary point]]), i.e. there exist continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140391.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140392.png" />, for which the Dirichlet problem cannot be solved in any way.

Thus, the generalized Perron–Wiener solution to the Dirichlet problem for the Laplace equation obtained in the course of development of the Perron method is of great importance. As has been shown by N. Wiener (1924), in this case any finite continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140393.png" /> prescribed on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140394.png" /> of an arbitrary bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140395.png" /> is resolutive, i.e. the generalized Perron–Wiener solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140396.png" /> for this function exists and, moreover, is unique. In general, in 1939 M. Brelot has shown that a finite measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140397.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140398.png" /> is resolutive if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140399.png" /> is integrable with respect to [[Harmonic measure|harmonic measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140400.png" />.

The generalized solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140401.png" /> does not take the prescribed values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140402.png" /> at all boundary points. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140403.png" /> is called a regular point if for any finite continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140404.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140405.png" /> the generalized solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140406.png" /> takes the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140407.png" />, that is,

<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/p074140/p074140408.png" /></td> </tr></table>

All other points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140409.png" /> are called irregular points; they include the isolated points of the boundary when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140410.png" /> and the Lebesgue spine for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140411.png" />. It turned out (the [[Kellogg–Evans theorem|Kellogg–Evans theorem]], 1933) that the set of irregular points has outer capacity zero, i.e. this set is in some sense thin. The set of regular points is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140412.png" />.

For the Dirichlet problem one can construct a generalized Green function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140413.png" />, which can be defined, e.g., for an arbitrary fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140414.png" /> in the following way:

<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/p074140/p074140415.png" /></td> </tr></table>

The generalized Green function preserves some properties of the classical Green function, for example, the symmetry property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140416.png" />, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140417.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140418.png" />, if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140419.png" /> is a regular point of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140420.png" /> (see [[#References|[4]]], [[#References|[6]]]).

The studies of potentials with other kernels, different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074140/p074140421.png" />, and the study of the Dirichlet problem for compacta and the stability of the Dirichlet problem are of great importance (see [[#References|[6]]], [[#References|[4]]]). Their application to the solution of boundary value problems is intensively developed (see [[Bessel potential|Bessel potential]]; [[Non-linear potential|Non-linear potential]]; [[Riesz potential|Riesz potential]]; and also [[#References|[3]]], [[#References|[11]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.M. [N.M. Gyunter] Günter,   "Potential theory and its application to basic problems of mathematical physics" , F. Ungar  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.N. Sretenskii,   "Theory of the Newton potential" , Moscow-Leningrad  (1946)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.S. Landkof,   "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Brélot,   "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  O.D. Kellogg,   "Foundations of potential theory" , F. Ungar  (1929)  (Re-issue: Springer, 1967)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M.V. Keldysh,   "On the resolutivity and the stability of the Dirichlet problem"  ''Uspekhi Mat. Nauk'' , '''8'''  (1941)  pp. 171–231  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.V. Bitsadze,   "Equations of mathematical physics" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.V. Bitsadze,   "Boundary value problems for second-order elliptic equations" , North-Holland  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  V.S. Vladimirov,   "Equations of mathematical physics" , M. Dekker  (1971)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  C. Miranda,   "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  S.G. Mikhlin,   "Linear partial differential equations" , Moscow  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  A.N. Tikhonov,   A.A. Samarskii,   "Equations of mathematical physics" , Pergamon  (1963)  (Translated from Russian)</TD></TR></table>



====Comments====
See also [[Potential theory, mixed boundary value problems of|Potential theory, mixed boundary value problems of]]; [[Potential theory, inverse problems in|Potential theory, inverse problems in]]; [[Potential theory, abstract|Potential theory, abstract]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.L. Helms,   "Introduction to potential theory" , Wiley (Interscience)  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Tsuji,   "Potential theory in modern function theory" , Chelsea, reprint  (1975)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Wermer,   "Potential theory" , ''Lect. notes in math.'' , '''408''' , Springer  (1974)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.L. Doob,   "Classical potential theory and its probabilistic counterpart" , Springer  (1984)  pp. 390</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  C. Dellacherie,   P.A. Meyer,   "Probabilities and potential" , '''A-B''' , North-Holland  (1978–1982)  (Translated from French)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C. Constantinescu,   A. Cornea,   "Potential theory on harmonic spaces" , Springer  (1972)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  V.I. Fabrikant,   "Applications of potential theory in mechanics. Selection of new results" , Kluwer  (1989)</TD></TR></table>