  We study certain "geometric-invariant resonant cavitie"' introduced by
Liberal et. al in a 2016 Nature Comm. paper, modeled using the transverse
magnetic reduction of Maxwell's equations. The cross-section consists of a
dielectric inclusion surrounded by an "epsilon-near-zero" (ENZ) shell. When the
shell has the right area, its interaction with the inclusion produces a
resonance. Mathematically, the resonance is a nontrivial solution of a 2D
divergence-form Helmoltz equation $\nabla \cdot
\left(\varepsilon^{-1}(x,\omega) \nabla u \right) + \omega^2 \mu u = 0$, where
$\varepsilon(x,\omega)$ is the (complex-valued) dielectric permittivity,
$\omega$ is the frequency, $\mu$ is the magnetic permeability, and a
homogeneous Neumann condition is imposed at the outer boundary of the shell.
This is a nonlinear eigenvalue problem, since $\varepsilon$ depends on
$\omega$. Use of an ENZ material in the shell means that
$\varepsilon(x,\omega)$ is nearly zero there, so the PDE is rather singular.
Working with a Lorentz model for the dispersion of the ENZ material, we put the
discussion of Liberal et.~al.~on a sound foundation by proving the existence of
the anticipated resonance when the loss is sufficiently small. Our analysis is
perturbative in character despite the apparently singular form of the PDE.
While the existence of the resonance depends only on the area of the ENZ shell,
the rate at which it decays depends on the shape of the shell. We consider an
associated optimal design problem: what shape shell gives the slowest-decaying
resonance? We prove that if the dielectric inclusion is a ball then the optimal
shell is a concentric annulus. For an inclusion of any shape, we study a convex
relaxation of the design problem using tools from convex duality, and discuss
the conjecture that our relaxed problem amounts to considering
homogenization-like limits of nearly optimal designs.
