Evgen Khruslov | Searcharxiv

arXiv2023

Inclination of subspaces and decomposition of electromagnetic fields into potential and vortex components

Using the notion of inclination of two subspaces $L$ and $M$ of Hilbert space $\mathcal{H}$, we prove the theorem on the extension of linear continuous functionals defined on the subspace $L$ to $\mathcal{H}$ so that the extended functionals vanish on the subspace $M$. We apply this theorem to study the question of decomposition of the electromagnetic field in resonator with ideally conducting walls into potential and vortex components and derive the Korn-type inequality for vortex fields.

math.AP↗

arXiv2023

Creating and controlling band gaps in periodic media with small resonators

We investigate spectral properties of the Neumann Laplacian $\mathscr{A}_\varepsilon$ on a periodic unbounded domain $Ω_\varepsilon$ depending on a small parameter $\varepsilon>0$. The domain $Ω_\varepsilon$ is obtained by removing from $\mathbb{R}^n$ $m\in\mathbb{N}$ families of $\varepsilon$-periodically distributed small resonators. We prove that the spectrum of $\mathscr{A}_\varepsilon$ has at least $m$ gaps. The first $m$ gaps converge as $\varepsilon\to 0$ to some intervals whose location and lengths can be controlled by a suitable choice the resonators; other gaps (if any) go to infinity. An application to the theory of photonic crystals is discussed.

math.SP↗