An efficient integral equation based solver is constructed for the electrostatic problem on domains with cuboidal inclusions. It can be used to compute the polarizability of a dielectric cube in a dielectric background medium at virtually every permittivity ratio for which it exists. For example, polarizabilities accurate to between five and ten digits are obtained (as complex limits) for negative permittivity ratios in minutes on a standard workstation. In passing, the capacitance of the unit cube is determined with unprecedented accuracy. With full rigor, we develop a natural mathematical framework suited for the study of the polarizability of Lipschitz domains. Several aspects of polarizabilities and their representing measures are clarified, including limiting behavior both when approaching the support of the measure and when deforming smooth domains into a non-smooth domain. The success of the mathematical theory is achieved through symmetrization arguments for layer potentials.

Abstract. The boundary double layer potential, or the Neumann-Poincaré operator, is studied on the Sobolev space of order 1/2 along the boundary, coinciding with the space of charges giving rise to double layer potentials with finite energy in the whole space. Poincaré's program of studying the spectrum of the boundary double layer potential is developed in complete generality, on closed Lipschitz hypersurfaces in Euclidean space. Furthermore, the Neumann-Poincaré operator is realized as a singular integral transform bearing similarities to the Beurling-Ahlfors transform in 2D. As an application, bounds for the spectrum of the Neumann-Poincaré operator are derived from recent results in quasi-conformal mapping theory, in the case of planar curves with corners.

For the classical space of functions with bounded mean oscillation, it is well known that V M O * * = BM O and there are many characterizations of the distance from a function f in BM O to V M O. When considering the Bloch space, results in the same vein are available with respect to the little Bloch space. In this paper such duality results and distance formulas are obtained by pure functional analysis. Applications include general Möbius invariant spaces such as Q K -spaces, weighted spaces, Lipschitz-Hölder spaces and rectangular BM O of several variables.

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Exploiting the homogeneous structure of a wedge in the complex plane, we compute the spectrum of the anti-linear Ahlfors-Beurling transform acting on the associated Bergman space. Consequently, the similarity equivalence between the Ahlfors-Beurling transform and the Neumann-Poincaré operator provides the spectrum of the latter integral operator on a wedge. A localization technique and conformal mapping lead to the first complete description of the essential spectrum of the Neumann-Poincaré operator on a planar domain with corners, with respect to the energy norm of the associated harmonic field.