Mathematical analysis of transverse EM field concentration for adjacent obstacles with nonlocal boundary conditions in the quasistatic regime

TL;DR

This paper rigorously analyzes electromagnetic field concentration between obstacles with nonlocal boundary conditions, revealing how nonlocal effects and frequency influence gradient blowup and field enhancement in nanophotonics.

Contribution

It introduces new mathematical models with nonlocal boundary conditions and derives sharp conditions and rates for electromagnetic field gradient blowup.

Findings

Nonlocal boundary conditions alter classical gradient estimates.

Wave frequency reduces the severity of field concentration.

Sharp asymptotic formulas for field enhancement are established.

Abstract

This paper presents a rigorous mathematical analysis of transverse electromagnetic (EM) field concentration between two adjacent obstacles within the framework of the quasi-static approximation. We investigate three degenerate conductivity models recently introduced in [22], two of these incorporating nonlocal boundary conditions to capture fundamental physical phenomena, such as surface nonlocality and thin-layer interactions. Our primary results establish sharp conditions for gradient blowup and derive the corresponding optimal blowup rates. These findings elucidate how nonlocal boundary conditions modify classical gradient estimates. Furthermore, we analyze the influence of wave frequency, demonstrating that it mitigates the severity of field concentration even in the limit of a vanishing gap distance. Consequently, this work extends the classical theory of field enhancement in plasmonic and metamaterial systems to incorporate nonlocal surface effects, yielding precise asymptotic formulas that are essential for the quantitative design of nanophotonic devices.