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A. I. Nazarov

Publications and source records attributed to A. I. Nazarov.

At least 19 recordsLinked to original sources
arXiv2025

Non-radial minimizers for Hardy--Sobolev inequalities in non-convex cones

The symmetry breaking is obtained for Neumann problems driven by $p$-Laplacian in certain non-convex cones. These problems are generated by the Hardy--Sobolev inequalities. In the case of the Sobolev inequality for the ordinary Laplacian this problem was investigated in (Ciraolo, Pacella, Polvara, 2024). Such problems have obvious radial solutions -- Talenti--Bliss type functions of $|x|$. However, under a certain restriction on the first Neumann eigenvalue $λ_1(D)$ of the Beltrami--Laplace operator on the spherical cross-section $D$ of the cone we prove this radial solution cannot be an extremal function, therefore minimizer must be non-radial. This leads to multiple solutions for the corresponding Neumann problem.

math.AP↗
arXiv2024

Entire solutions to the Swift--Hohenberg equation via variational approach

We study the stationary Swift--Hohenberg equation $(Δ+ 1)^2 u - αu - βu^2 + u^3=0$ in the whole space $\mathbb R^n$, $2\le n \le 7$. We develop and modify the variational approach introduced by Lerman, Naryshkin and Nazarov (2020) and obtain a series of periodic solutions with certain additional symmetries.

math.AP↗
arXiv2023

Dirichlet Fractional Laplacian in multi-tubes

We describe the spectrum structure for the restricted Dirichlet fractional Laplacian in multi-tubes, i.e. domains with cylindrical outlets to infinity. Some new effects in comparison with the local case are discovered. In this version, Theorem 4 is essentially improved.

math.SP↗
arXiv2023

Nonstationary Venttsel problems with discontinuous data

We study linear and quasilinear Venttsel initial-boundary value problems for parabolic operators with discontinuous coefficients. On the basis of the a priori estimates obtained, strong solvability in composite Sobolev spaces is proved.

math.AP↗
arXiv2021

Solutions with various structures for semilinear equations in $\mathbb R^n$ driven by fractional Laplacian

We study bounded solutions to the fractional equation $(-Δ)^s u + u - |u|^{q-2}u = 0$ in $\mathbb R^n$ for $n\ge2$ and subcritical exponent $q>2$. Applying the variational approach based on concentration arguments and symmetry considerations which was introduced by Lerman, Naryshkin and Nazarov (2020) we construct several types of solutions with various structures (radial, rectangular, triangular, hexagonal, quasi-periodic, breather type, etc.).

math.AP↗
arXiv2021

Spectral equivalence of Gaussian random functions: operator approach

We introduce a new approach to the spectral equivalence of Gaussian processes and fields, based on the methods of operator theory in Hilbert space. Besides several new results including identities in law of quadratic norms for integrated and multiply integrated Gaussian random functions we give an application to goodness-of-fit testing.

math.PR↗
arXiv2020

Spectral analysis for some multifractional Gaussian processes

We study the small ball asymptotics problem in $L_2$ for two generalizations of the fractional Brownian motion with variable Hurst parameter. To this end, we perform careful analysis of the singular values asymptotics for associated integral operators.

math.PR↗
arXiv2019

On the trace formula for higher-order ODO

A first order trace formula is obtained for a higher-order differential operator on a segment in the case where the perturbation is an operator of multiplication by a finite complex-valued measure. For the operators of even order $n\ge4$ a new term in the final formula is discovered.

math.SP↗
arXiv2018

Abundance of entire solutions to nonlinear elliptic equations by the variational method

We study entire bounded solutions to the equation $Δu - u + u^3 = 0$ in $\mathbb R^2$. Our approach is purely variational and is based on concentration arguments and symmetry considerations. This method allows us to construct in a unified way several types of solutions with various symmetries (radial, breather type, rectangular, triangular, hexagonal, etc.), both positive and sign-changing. The method is also applicable for more general equations in any dimension.

math.AP↗
arXiv2015

A counterexample to the Hopf-Oleinik lemma (elliptic case)

We construct a new counterexample confirming the sharpness of the Dini-type condition for the boundary of $Ω$. In particular, we show that for convex domains the Dini-type assumption is the necessary and sufficient condition which guarantees the Hopf-Oleinik type estimates.

math.AP↗
arXiv2015

On the maximum principle for parabolic equations with unbounded coefficients

Translation of the paper "Interpolation of linear spaces and maximum estimates for solutions to parabolic equations" published in Russian in the collected volume "Partial differential equations", Akad. Nauk SSSR, Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1987, 50--72. Original proofs are essentially simplified. Some gaps are fixed and some comments are added.

math.AP↗
arXiv2014

On symmetry of extremals in several embedding theorems

We study the symmetry/asymmetry of functions providing sharp constants in the embedding theorems ${\stackrel{\circ}{W}}\vphantom{W}_2^r(-1,1)\hookrightarrow{\stackrel{\circ}{W}}\vphantom{W}_\infty^k(-1,1)$ for various $r$ and $k$. The sharp constants for all $r>k$ in the cases $k=4$ and $k=6$ are calculated explicitly as well.

math.CA↗