# Robin problem

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

equilibrium problem, electrostatic problem

A problem on the distribution of a positive Borel measure on the boundary of a compact set in the -dimensional Euclidean space , , which generates a constant Newton potential for , or constant logarithmic potential for , on any connected component of the interior of , i.e. the problem on the equilibrium distribution of an electric charge on the surface of a conductor .

In the simplest classical case when is a closed domain in homeomorphic to the sphere, bounded by a smooth simple surface or (when ) by a curve of class , , , the solution of Robin's problem is reduced to finding a non-trivial solution , , of the homogeneous Fredholm-type integral equation of the second kind (1)

under the normalization condition (2)

Here for , is the distance between two points , is the direction of the exterior normal to at the point , is the derivative, or density, of the absolutely-continuous measure with respect to the Lebesgue measure on , for , and is the area element of the surface . Equation (1) is obtained when one considers the interior Neumann problem for the domain bounded by under vanishing boundary conditions, since the simple-layer potential called the Robin potential, equilibrium potential or capacity potential, should, according to the condition of Robin's problem, have a constant value on (see Potential theory, and also ). The solution for the problem (1), (2) under the indicated conditions always exists in the class of continuous functions . The measure 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 consists of a finite number of non-intersecting simple closed surfaces or (when ) curves of class , (see ), the Robin problem is solved in a similar way. Moreover, on bounded connected components of the open set the Robin potential also preserves its constant value, i.e. on the boundaries of these components the density .

Let the compact set be connected. The constant value of the Robin potential on , is called the Robin constant of the compact set . For it is related to the harmonic, or Newton, capacity of by the simple relation ; moreover, , . For , the Robin constant can assume all values ; the harmonic capacity is then expressed by the formula .

In another way, the equilibrium measure is defined as the measure which yields the minimum of the energy integral in the class of all measures concentrated on and such that , . Such a measure in the case of a compact set with a smooth boundary coincides with the one found above, but it exists also in the general case of an arbitrary compact set , , if only . The corresponding equilibrium potential which is a generalization of the Robin potential, preserves the constant value for , or for , everywhere on except perhaps at the points of some set of capacity zero.

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

How to Cite This Entry:
Robin problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Robin_problem&oldid=14200
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article