Plasmon resonance with finite frequencies: a validation of the quasi-static approximation for diametrically small inclusions (Q2799042)

This paper extends [the authors, J. Math. Anal. Appl. 435, No. 1, 162--178 (2016; Zbl 1327.81193)], to the Helmholtz operator under two asymptotic regimes.NEWLINENEWLINEThe frame is a dielectric medium whose permittivity is \(\epsilon_m>0\) outside some bounded inclusion \(\Omega\subset\mathbb{R}^d,\;d=2,3\) whose permittivity is \(\epsilon_c+i\delta\) with \(\epsilon_c<0\).NEWLINENEWLINETo state the main result of this paper, first define the function \(\lambda(t)=(t+1)/(2(t-1))\). So if the frequency is small or if the diameter of \(\Omega\) is small both compared to the dissipation \(\delta\) then the outgoing solution \(u\) to the Helmholtz equation with dipole source satisfies \(\|\nabla u\|_{L^2(\Omega)}=O(1/\delta)\) when \(\delta\rightarrow 0\) if \(\lambda(\epsilon_c/\epsilon_m)\) is an eigenvalue of the Neumann-Poincaré operator. If not, the solution remains bounded as \(\delta\) goes to zero.