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Nikolai Filonov

Publications and source records attributed to Nikolai Filonov.

4 recordsLinked to original sources
arXiv2020

On the Pólya conjecture for the Neumann problem in tiling sets

In 1954, G. Pólya conjectured that the counting function of the eigenvalues of the Laplace operator of Dirichlet (resp. Neumann) boundary value problem in a bounded set $Ω\subset{\mathbb R}^d$ is lesser (resp. greater) than $C_W |Ω| λ^{d/2}$. Here $λ$ is the spectral parameter, and $C_W$ is the constant in the Weyl asymptotics. In 1961, Pólya proved this conjecture for tiling sets in the Dirichlet case, and for tiling sets under some additional restrictions for the Neumann case. We prove the Pólya conjecture in the Neumann case for all tiling sets.

math.SP↗
arXiv2020

Non-uniqueness of Leray-Hopf solutions for a dyadic mode

The dyadic model $\dot u_n + λ^{2n}u_n - λ^{βn}u_{n-1}^2 + λ^{β(n+1)}u_nu_{n+1} = f_n$, $u_n(0)=0$, is considered. It is shown that in the case of non-trivial right hand side the system can have two different Leray-Hopf solutions.

math.AP↗
arXiv2004

Absolutely continuous spectrum for the isotropic Maxwell operator with coefficients that are periodic in some directions and decay in others

The purpose of this paper is to prove that the spectrum of an isotropic Maxwell operator with electric permittivity and magnetic permeability that are periodic along certain directions and tending to a constant super-exponentially fast in the remaining directions is purely absolutely continuous. The basic technical tools is a new ``operatorial'' identity relating the Maxwell operator to a vector-valued Schrodinger operator. The analysis of the spectrum of that operator is then handled using ideas developed by the same authors in a previous paper.

math-ph↗