Logarithmic potential

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 $\mu$ on $\mathbf R^2$ with compact support $S=S(\mu)$. Physically one can assume that the logarithmic potential arises from the 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 [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). The main applications of the logarithmic potential occur in the solution of planar boundary value problems in potential theory (see also 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 $\Delta u=0$, that is, $u$ is a 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 $\Delta u=-2\pi f$.

If the integral in \eqref{2} extends along a closed Lyapunov curve $L$ (see 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). 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, since an integral of Cauchy type can be expressed in terms of the logarithmic potential of a single and a double layer (see [3]).

References

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

Comments

See also Lyapunov theorem.

References

[a1]	W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976)
[a2]	J. Král, "Integral operators in potential theory" , Lect. notes in math. , 823 , Springer (1980)