On the polarizability and capacitance of the cube

Abstract

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.

Spectral bounds for the Neumann-Poincaré operator on planar domains with corners

“…In fact, P s (P e − P i )P s is intimately connected to Poincaré's variational problem of inner and outer energies. To explain our point of view, let us briefly recall and combine some of the ideas of [9] and [11]. The analogies with the previous work of this section should be clear.…”

“…The arguments essentially follow those of [11], with some additional technicalities, addressed in [9], arising from the fact that K is no longer a compact operator on the scale H s of Besov spaces, 0 ≤ s ≤ 1.…”

“…Specifically we will see its validity on closed Lipschitz hypersurfaces of R n , n ≥ 2, in Section 3 below. Nonetheless, precise bounds for the spectrum of this operator are very scarce, see for instance [9]. Two real dimensions are however special, due to the large group of conformal mappings.…”

“…It is therefore of great interest to obtain information about the essential spectrum of the Neumann-Poincaré operator. See for example [4], [9].…”

Numerical Analysis of the Potential Formulation of the Volume Integral Equation for Electromagnetic Scattering

“…Fuchs [1975] showed that there are six major static resonances between r = −0.42, and r = −3.68. However, as pointed out by Helsing and Perfekt [2013], some of them may be artifacts due to insufficient resolution or unintended rounding of the corners. Be that as it may, there are no such things as infinitely sharp corners or wedges described by the bulk permittivity in the physical world.…”