Localization and geometrization in plasmon resonances and geometric structures of Neumann-Poincaré eigenfunctions

This paper reports some interesting discoveries about the localization and geometrization phenomenon in plasmon resonances and the intrinsic geometric structures of Neumann-Poincaré eigenfunctions. It is known that plasmon resonance generically occurs in the quasi-static regime where the size of the plasmonic inclusion is sufficiently small compared to the wavelength. In this paper, we show that the global smallness condition on the plasmonic inclusion can be replaced by a local high-curvature condition, and the plasmon resonance occurs locally near the high-curvature point of the plasmonic inclusion. We link this phenomenon with the geometric structures of the Neumann-Poincaré (NP) eigenfunctions. The spectrum of the Neumann-Poincaré operator has received significant attentions in the literature. We show that the Neumann-Poincaré eigenfunctions possess some intrinsic geometric structures near the high-curvature points. We mainly rely on numerics to present our findings. For a particular case when the domain is an ellipse, we can provide the analytic results based on the explicit solutions.

On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures

We study positive solutions to the fractional Lane-Emden system $$\begin{array}{} \displaystyle \left\{ \begin{aligned} (-{\it\Delta})^s u &= v^p+\mu \quad &\text{in } {\it\Omega} \\(-{\it\Delta})^s v &= u^q+\nu \quad &\text{in } {\it\Omega}\\u = v &= 0 \quad &&\!\!\!\!\!\!\!\!\!\!\!\!\text{in } {\it\Omega}^c={\mathbb R}^N \setminus {\it\Omega}, \end{aligned} \right. \end{array}$$(S) where Ω is a C2 bounded domains in ℝN, s ∈ (0, 1), N > 2s, p > 0, q > 0 and μ, ν are positive measures in Ω. We prove the existence of the minimal positive solution of (S) under a smallness condition on the total mass of μ and ν. Furthermore, if p, q ∈ $\begin{array}{} (1,\frac{N+s}{N-s}) \end{array}$ and 0 ≤ μ, ν ∈ Lr(Ω), for some r > $\begin{array}{} \frac{N}{2s}, \end{array}$ we show the existence of at least two positive solutions of (S). The novelty lies at the construction of the second solution, which is based on a highly nontrivial adaptation of Linking theorem. We also discuss the regularity of the solutions.

Classifying Toposes for First Order Theories

By a classifying topos for a first-order theory T, we mean a topos<br />E such that, for any topos F, models of T in F correspond exactly to<br />open geometric morphisms F ! E. We show that not every (infinitary)<br />first-order theory has a classifying topos in this sense, but we<br />characterize those which do by an appropriate `smallness condition',<br />and we show that every Grothendieck topos arises as the classifying<br />topos of such a theory. We also show that every first-order theory<br /> has a conservative extension to one which possesses<br /> a classifying topos, and we obtain a Heyting-valued completeness<br /> theorem for infinitary first-order logic.