View source for Logarithmic potential

A potential with the logarithmic kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l0606301.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l0606302.png" /> is the distance between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l0606303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l0606304.png" /> of the Euclidean plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l0606305.png" />, that is, a potential of 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/l/l060/l060630/l0606306.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where, generally, speaking, the integration is carried out with respect to an arbitrary [[Borel measure|Borel measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l0606307.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l0606308.png" /> with compact support <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l0606309.png" />. Physically one can assume that the logarithmic potential arises from the [[Newton potential|Newton potential]] of the forces of gravitation when the distribution of the attracting masses in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063010.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063011.png" /> does not depend, for example, on the coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063012.png" />. Of course the total mass is infinite, but if one performs a regularization of the resulting attracting force <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063013.png" />, which can be regarded as acting in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063014.png" />, consisting in discarding the infinite term, then the potential of the finite part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063015.png" /> will invariably have the form (1) (see [[#References|[2]]]). In contrast to the Newton kernel, the logarithmic kernel has a singularity not only as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063016.png" />, but also as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063017.png" />, which causes some differences in the behaviour of the logarithmic potential as compared with the Newton potential. They occur mainly in the solution of exterior boundary value problems (see [[Exterior and interior boundary value problems|Exterior and interior boundary value problems]]). The main applications of the logarithmic potential occur in the solution of planar [[Boundary value problems in potential theory|boundary value problems in potential theory]] (see also [[Boundary value problem, elliptic equations|Boundary value problem, elliptic equations]]).

The main properties of the logarithmic potential are: 1) outside the support <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063018.png" /> of the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063019.png" /> the logarithmic potential is a regular solution of the [[Laplace equation|Laplace equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063020.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063021.png" /> is a [[Harmonic function|harmonic function]] on the open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063022.png" />, but is not regular at infinity, however; 2) if the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063023.png" /> is absolutely continuous, that is, the integral (1) 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/l/l060/l060630/l06063024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063025.png" /> is a finite domain, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063026.png" /> is the area element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063027.png" /> and the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063028.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063029.png" />, then the second derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063030.png" /> are continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063031.png" /> and satisfy the [[Poisson equation|Poisson equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063032.png" />.

If the integral in (2) extends along a closed Lyapunov curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063033.png" /> (see [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]), 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/l/l060/l060630/l06063034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

one talks of the logarithmic potential of a single (or simple) layer, distributed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063035.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063036.png" />, then the logarithmic potential of the single layer (3) is continuous everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063037.png" />. Its normal derivative has limits from the inside and the outside of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063038.png" />, respectively:

<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/l/l060/l060630/l06063039.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/l/l060/l060630/l06063040.png" /></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/l/l060/l060630/l06063041.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

is the so-called direct value of the normal derivative of the logarithmic potential of a single layer and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063042.png" /> is the angle between the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063043.png" /> and the outward normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063044.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063045.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063046.png" />. The integral (4) is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063047.png" />.

The logarithmic potential of a double layer 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/l/l060/l060630/l06063048.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063049.png" /> is the outward normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063050.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063051.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063052.png" />, then the logarithmic potential of the double layer (5) is a regular harmonic function inside and outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063053.png" /> and has normal (non-angular) limits from the inside and the outside of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063054.png" />, respectively:

<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/l/l060/l060630/l06063055.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/l/l060/l060630/l06063056.png" /></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/l/l060/l060630/l06063057.png" /></td> </tr></table>

is the direct value of the logarithmic potential of the double layer at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063058.png" />. The normal derivative of the logarithmic potential of a double layer is continuous under transition through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060630/l06063059.png" />.

The listed boundary properties of the logarithmic potential of a simple and a double layer are completely analogous to the corresponding properties of the Newton potential (see also [[Potential theory|Potential theory]]). From (5) it is obvious that the logarithmic potential of a double layer is a harmonic function that is regular at infinity.

The logarithmic potential is also directly connected with [[Boundary value problems of analytic function theory|boundary value problems of analytic function theory]], since an integral of Cauchy type can be expressed in terms of the logarithmic potential of a single and a double layer (see [[#References|[3]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.L. Sobolev,   "Partial differential equations of mathematical physics" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Webster,   "Partial differential equations of mathematical physics" , Hafner  (1955)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.I. Muskhelishvili,   "Singular integral equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Ch.J. de la Vallée-Poussin,   "Le potentiel logarithmique, balayage et répresentation conforme" , Libraire Univ. Louvain  (1949)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G.C. Evans,   "The logarithmic potential, discontinuous Dirichlet and Neumann problems" , New York  (1927)</TD></TR></table>



====Comments====
See also [[Lyapunov theorem|Lyapunov theorem]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.K. Hayman,   P.B. Kennedy,   "Subharmonic functions" , '''1''' , Acad. Press  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Král,   "Integral operators in potential theory" , ''Lect. notes in math.'' , '''823''' , Springer  (1980)</TD></TR></table>