When does a double-layer potential equal to a single-layer one?

Abstract

<abstract><p>Let $ D $ be a bounded domain in $ {{\mathbb R}}^3 $ with a closed, smooth, connected boundary $ S $, $ N $ be the outer unit normal to $ S $, $ k &gt; 0 $ be a constant, $ u_{N^{\pm}} $ are the limiting values of the normal derivative of $ u $ on $ S $ from $ D $, respectively $ D': = {{\mathbb R}}^3\setminus \bar{D} $; $ g(x, y) = \frac{e^{ik|x-y|}}{4\pi |x-y|} $, $ w: = w(x, \mu): = \int_S g_{N}(x, s)\mu(s)ds $ be the double-layer potential, $ u: = u(x, \sigma): = \int_S g(x, s)\sigma(s)ds $ be the single-layer potential.</p> <p>In this paper it is proved that for every $ w $ there is a unique $ u $, such that $ w = u $ in $ D $ and vice versa. This result is new, although the potential theory has more than 150 years of history.</p> <p>Necessary and sufficient conditions are given for the existence of $ u $ and the relation $ w = u $ in $ D' $, given $ w $ in $ D' $, and for the existence of $ w $ and the relation $ w = u $ in $ D' $, given $ u $ in $ D' $.</p></abstract>