Robin problem

equilibrium problem, electrostatic problem

A problem on the distribution of a positive Borel measure $\lambda$ on the boundary $S$ of a compact set $K$ in the $n$-dimensional Euclidean space $\mathbf R^n$, $n\geq2$, which generates a constant Newton potential for $n\geq3$, or constant logarithmic potential for $n=2$, on any connected component of the interior of $K$, i.e. the problem on the equilibrium distribution of an electric charge $\lambda(K)$ on the surface $S$ of a conductor $K$.

In the simplest classical case when $K$ is a closed domain in $\mathbf R^n$ homeomorphic to the sphere, bounded by a smooth simple surface or (when $n=2$) by a curve $S$ of class $C^{1,\alpha}$, $0<\alpha<1$, $0\in K$, the solution of Robin's problem is reduced to finding a non-trivial solution $\nu(x)$, $x\in S$, of the homogeneous Fredholm-type integral equation of the second kind

$$\frac12\nu(x)+\frac{1}{k_n}\int\limits_S\nu(y)\frac{\partial}{\partial n_x}E_n(x,y)dS(y)=0,\quad x\in S,\label{1}\tag{1}$$

under the normalization condition

$$\lambda(S)=\int\limits_S\nu(y)dS(y)=1.\label{2}\tag{2}$$

Here

$$E_2(x,y)=\ln\frac{1}{|x-y|},\quad E_n=\frac{1}{|x-y|^{n-2}}$$

for $n\geq3$, $|x-y|$ is the distance between two points $x,y\in\mathbf R^n$, $n_x$ is the direction of the exterior normal to $S$ at the point $x\in S$, $\nu(x)$ is the derivative, or density, of the absolutely-continuous measure $\lambda$ with respect to the Lebesgue measure on $S$,

$$k_2=2\pi,\quad k_n=\frac{2(n-2)\pi^{n/2}}{\Gamma(n/2)}$$

for $n\geq3$, and $dS(y)$ is the area element of the surface $S$. Equation \eqref{1} is obtained when one considers the interior Neumann problem for the domain bounded by $S$ under vanishing boundary conditions, since the simple-layer potential

$$u(x)=u(x,K)=\int\limits_S\nu(y)E_n(x,y)dS(y),$$

called the Robin potential, equilibrium potential or capacity potential, should, according to the condition of Robin's problem, have a constant value on $K$ (see Potential theory, and also [2]). The solution $\nu(x)$ for the problem \eqref{1}, \eqref{2} under the indicated conditions always exists in the class of continuous functions $C(S)$. The measure

$$\lambda(E)=\int\limits_E\nu(y)dS(y),\quad E\subset S,$$

which provides a solution of the Robin problem, is called the equilibrium measure. In a more complicated case, when the boundary of the compact set $K$ consists of a finite number of non-intersecting simple closed surfaces or (when $n=2$) curves of class $C^{1,\alpha}$, $0<\alpha<1$ (see [2]), the Robin problem is solved in a similar way. Moreover, on bounded connected components of the open set $G=CK=\mathbf R^n\setminus K$ the Robin potential $u(x)$ also preserves its constant value, i.e. on the boundaries of these components the density $\nu(x)=0$.

Let the compact set $K$ be connected. The constant value of the Robin potential $u(x)$ on $K$,

$$\gamma=\int\limits_S\nu(y)E_n(x,y)dS(y),\quad x\in K,$$

is called the Robin constant of the compact set $K$. For $n\geq3$ it is related to the harmonic, or Newton, capacity $C(K)$ of $K$ by the simple relation $C(K)=1/\gamma$; moreover, $0<\gamma<+\infty$, $0<C(K)<+\infty$. For $n=2$, the Robin constant can assume all values $-\infty<\gamma<+\infty$; the harmonic capacity is then expressed by the formula $C(K)=e^{-\gamma}$.

In another way, the equilibrium measure $\lambda$ is defined as the measure which yields the minimum of the energy integral

$$\iint\limits_{K\times K}E_n(x,y)d\mu(x)d\mu(y)$$

in the class of all measures $\mu$ concentrated on $K$ and such that $\mu\geq0$, $\mu(K)=1$. Such a measure $\lambda$ in the case of a compact set $K$ with a smooth boundary coincides with the one found above, but it exists also in the general case of an arbitrary compact set $K\subset\mathbf R^n$, $n\geq2$, if only $C(K)>0$. The corresponding equilibrium potential

$$u(x)=u(x;K)=\int E_n(x,y)d\lambda(y),$$

which is a generalization of the Robin potential, preserves the constant value $\gamma=1/C(K)$ for $n\geq3$, or $\gamma=-\ln C(K)$ for $n=2$, everywhere on $K$ except perhaps at the points of some set of capacity zero.

The name "Robin problem" is connected with studies of G. Robin (see [1]).

References

[1]	G. Robin, "Sur la distribution de l'électricité à la surface des conducteurs fermés et des conducteurs ouverts" Ann. Sci. Ecole Norm. Sup. , 3 (1886) pp. 31–358
[2]	N.M. Günter, "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar (1967) (Translated from Russian)
[3]	N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)
[4]	W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976)

Comments

In [1] Robin reconsiders and generalizes a problem formulated by S. Poisson (1811).

References

[a1]	M. Tsuji, "Potential theory in modern function theory" , Chelsea, reprint (1975)