Compactness of the Neumann-Poincaré operator
Edward John Specht, H. T. Jones · Transactions of the American Mathematical Society · 1969
The Neumann-Poincaré integral equation arises in connection with the Dirichlet and Neumann problems of potential theory and in connection with conformai mapping.Warschawski [6] has proved the compactness of the integral operator involved here under what seem to be natural smoothness conditions on the boundary curve, for the case where the boundary is a single contour.Because this proof relies heavily upon complex function theory, it does not extend easily to higher dimensions.It is the purpose of the present paper to give a proof, for the case of several contours, which will extend readily to higher dimensions.2. Definitions.Let 38x,...,3Sm be bounded nonintersecting contours in the plane whose interiors are disjoint, and let l¡ be the standard representation^) of 3d¡.Let s0=0, let s¡ be the sum of the lengths of 38x,..., 39 h and let J = [0, sm].Let £ be the function defined on J so that t,s= t,xs for all * in [0, sx] and £i = £/s -i/_1) for all s in (sj-x, Sj],j=2,..., m, and let 38 be the range of £.Let A be the function defined for all ordered pairs (s, t) such that l,s and £/ both belong to J1, for some7= 1,..., m as follows:A function a defined on J will be said to satisfy a Holder condition on 3d if and only if \as -at\ -Sa\Ais, t)\" for some numbers a and b such that a>0 and 0 |£s -£/1 for all s and / in J', and let the function A be defined on J x J (except at points is, t) where £i=£/) by the equality A(s, /) = log (c/|£s-£/|).For any function a defined