View source for Logarithmic potential

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A potential with the logarithmic kernel $\ln 1/|x-y|$, where $|x-y|$ is the distance between the points $x$ and $y$ of the Euclidean plane $\mathbf R^2$, that is, a potential of the form

\begin{equation}u(x)=\int\ln\frac1{|x-y|}\,d\mu(y),\label{1}\end{equation}

where, generally, speaking, the integration is carried out with respect to an arbitrary [[Borel measure|Borel measure]] $\mu$ on $\mathbf R^2$ with compact support $S=S(\mu)$. 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 $\mathbf R^3$ of points $y=(y_1,y_2,y_3)$ does not depend, for example, on the coordinate $y_3$. Of course the total mass is infinite, but if one performs a regularization of the resulting attracting force $F$, which can be regarded as acting in the plane $(x_1,x_2,0)$, consisting in discarding the infinite term, then the potential of the finite part of $F$ will invariably have the form \eqref{1} (see [[#References|[2]]]). In contrast to the Newton kernel, the logarithmic kernel has a singularity not only as $|x-y|\to0$, but also as $|x-y|\to\infty$, 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 $S$ of the measure $\mu$ the logarithmic potential is a regular solution of the [[Laplace equation|Laplace equation]] $\Delta u=0$, that is, $u$ is a [[Harmonic function|harmonic function]] on the open set $\mathbf R^2\setminus S$, but is not regular at infinity, however; 2) if the measure $\mu$ is absolutely continuous, that is, the integral \eqref{1} takes the form

\begin{equation}u(x)=\int\limits_Df(y)\ln\frac1{|x-y|}\,d\sigma(y),\label{2}\end{equation}

where $D$ is a finite domain, $d\sigma$ is the area element of $D$ and the density $f$ belongs to the class $C^1(D\cup\partial D)$, then the second derivatives of $u$ are continuous in $D$ and satisfy the [[Poisson equation|Poisson equation]] $\Delta u=-2\pi f$.

If the integral in \eqref{2} extends along a closed Lyapunov curve $L$ (see [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]), that is,

\begin{equation}u(x)=\int\limits_Lf(y)\ln\frac1{|x-y|}\,ds(y),\label{3}\end{equation}

one talks of the logarithmic potential of a single (or simple) layer, distributed on $L$. If $f\in C^1(L)$, then the logarithmic potential of the single layer \eqref{3} is continuous everywhere in $\mathbf R^2$. Its normal derivative has limits from the inside and the outside of $L$, respectively:

$$\lim_{x\to y_0}\left.\frac{du}{dn_0}\right|_i=\frac{du(y_0)}{dn_0}+\pi f(y_0),$$

$$\lim_{x\to y_0}\left.\frac{du}{dn_0}\right|_o=\frac{du(y_0)}{dn_0}-\pi f(y_0),$$

where

\begin{equation}\frac{du(y_0)}{dn_0}=\int\limits_Lf(y)\frac{\cos(y-y_0,n_0)}{|y-y_0|}\,ds(y),\quad y_0\in L,\label{4}\end{equation}

is the so-called direct value of the normal derivative of the logarithmic potential of a single layer and $(y-y_0,n_0)$ is the angle between the vector $y-y_0$ and the outward normal $n_0$ to $L$ at the point $y_0\in L$. The integral \eqref{4} is continuous on $L$.

The logarithmic potential of a double layer has the form

\begin{equation}v(x)=\int\limits_Lg(y)\frac{\cos(y-x,n)}{|y-x|}\,ds(y),\label{5}\end{equation}

where $n$ is the outward normal to $L$ at $y\in L$. If $g\in C^1(L)$, then the logarithmic potential of the double layer \eqref{5} is a regular harmonic function inside and outside $L$ and has normal (non-angular) limits from the inside and the outside of $L$, respectively:

$$\lim_{x\to y_0}\left.v(x)\right|_i=v(y_0)+\pi f(y_0),$$

$$\lim_{x\to y_0}\left.v(x)\right|_o=v(y_0)-\pi f(y_0),$$

where

$$v(y_0)=\int\limits_Lg(y)\frac{\cos(y-y_0,n)}{|y-y_0|}\,ds(y),\quad y_0\in L,$$

is the direct value of the logarithmic potential of the double layer at the point $y_0\in L$. The normal derivative of the logarithmic potential of a double layer is continuous under transition through $L$.

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 \eqref{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>