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Alexander I. Nazarov

Publications and source records attributed to Alexander I. Nazarov.

At least 19 recordsLinked to original sources
arXiv2026

Truncations for fractional Laplacians

Let $Ω\subset\mathbb R^n$ be a bounded Lipschitz domain. We prove and widely generalize a conjecture of A.\,I.~Nazarov \cite{Naz21}: for $s\in(1,\frac 32)$ the quadratic form $Q^{\rm SP}_s[u]$ of the spectral fractional Dirichlet Laplacian strictly increases under the map $u\mapsto|u|$ provided $u\in\tilde H^s(Ω)$ changes sign in $Ω$.

math.AP↗
arXiv2026

The Neumann problem for the generalized Hénon equation. Local analysis

For the boundary value problem $$\left\{ \begin{array}{rcll} -Δ_p u+u^{p-1}&=&|x|^αu^{q-1}&\mbox{in }Ω,\\ \frac{\displaystyle\partial u}{\displaystyle\partial{\bf n}}&=&0&\mbox{on }\partial Ω, \end{array}\right. $$ in the unit ball $Ω$, we investigate the properties of the positive radial solution. It is known, that for $1 2$ be sufficiently close to $2$. Then for all $p p$ is sufficiently close to $p$.

math.AP↗
arXiv2026

Hardy-Sobolev inequalities involving mixed radially and cylindrically symmetric weights

We deal with weighted Hardy-Sobolev type inequalities for functions on $\mathbb{R}^d$, $d\geq 2$. The weights involved are anisotropic, given by products of powers of the distance to the origin and to a nontrivial subspace. We establish necessary and sufficient conditions for validity of these inequalities, and investigate the existence/nonexistence of extremal functions.

math.AP↗
arXiv2025

Steklov-type 1D inequalities (a survey)

We give a survey of classical and recent results on sharp constants and symmetry/asymmetry of extremal functions in $1$-dimensional functional inequalities.

math.CA↗
arXiv2024

Hardy type inequalities with mixed cylindrical-spherical weights: the general case

We continue our investigation of Hardy-type inequalities involving combinations of cylindrical and spherical weights. Compared to [Cora-Musina-Nazarov, Ann. Sc. Norm. Sup., 2024], where the quasi-spherical case was considered, we handle the full range of allowed parameters. This has led to the observation of new phenomena related to lack of compactness.

math.AP↗
arXiv2023

Hardy type inequalities with mixed weights in cones

We study Hardy type inequalities involving mixed cylindrical and spherical weights, for functions supported in cones. These inequalities are related to some singular or degenerate differential operators.

math.AP↗
arXiv2022

Fractional operators as traces of operator-valued curves

We relate non integer powers ${\mathcal L}^{s}$, $s>0$ of a given (unbounded) positive self-adjoint operator $\mathcal L$ in a real separable Hilbert space $\mathcal H$ with a certain differential operator of order $2\lceil{s}\rceil$, acting on even curves $\mathbb R\to \mathcal H$. This extends the results by Caffarelli--Silvestre and Stinga--Torrea regarding the characterization of fractional powers of differential operators via an extension problem.

math.AP↗
arXiv2022

Around the normal derivative lemma

This survey provides a description of the history and the state of the art of one of the most important fields in the qualitative theory of elliptic partial differential equations including the strong maximum principle, the boundary point principle (the normal derivative lemma) and related topics.

math.AP↗
arXiv2021

Variety of fractional Laplacians

This paper is a survey of recent results on comparison of various fractional Laplacians prepared for Proceedings of ICM2022.

math.AP↗
arXiv2021

On comparison of fractional Laplacians

For $s>-1$, $s\notin\mathbb N_0$, we compare two natural types of fractional Laplacians $(-Δ)^s$, namely, the restricted Dirichlet and the spectral Neumann ones. We show that for the quadratic form of their difference taken on the space $\tilde{H}^s(Ω)$ is positive or negative depending on whether the integer part of $s$ is even or odd. For $s\in(0,1)$ and convex domains we prove also that the difference of these operators is positivity preserving on $\tilde{H}^s(Ω)$. This paper complements [10] and [11] where similar statements were proved for the spectral Dirichlet and the restricted Dirichlet fractional Laplacians.

math.AP↗
arXiv2020

Venttsel boundary value problems with discontinuous data

We study linear and quasilinear Venttsel boundary value problems involving elliptic operators with discontinuous coefficients. On the base of the a priori estimates obtained, maximal regularity and strong solvability in Sobolev spaces are proved.

math.AP↗
arXiv2019

A tool for symmetry breaking and multiplicity in some nonlocal problems

We prove some basic inequalities relating the Gagliardo-Nirenberg seminorms of a symmetric function $u$ on $\mathbb R^n$ and of its perturbation $uφ_μ$, where $φ_μ$ is a suitably chosen eigenfunction of the Laplace-Beltrami operator on the sphere $\mathbb S^{n-1}$, thus providing a technical but rather powerful tool to detect symmetry breaking and multiplicity phenomena in variational equations driven by the fractional Laplace operator. A concrete application to a problem related to the fractional Caffarelli-Kohn-Nirenberg inequality is given.

math.AP↗