A. Logunov

activity

20132026

most citedOn ratios of harmonic functions

15 citations · 47 across the 16 of their papers we have counts for

collaborators

23 papers

math.CA2026

Estimating the number of real zeros of linear combinations of radicals of polynomials

Gal Binyamini, Avner Kiro, Alexander Logunov +2

We obtain upper bounds for the number of real zeros of functions of the form $f (x) = k = 1 \sum n c_{k} (P_{k} (x))^{α_{k}},$ where $c_{k}, α_{k} \in R$ and each $P_k…

math.AP2026

Nested nodal loops of biharmonic functions

Javier Gómez-Serrano, Robert Koirala, Alexander Logunov

Given any $n\in\mathbb{N}$, we construct a real-valued biharmonic polynomial on $\mathbb{R}^2$ whose zero set contains a nest of $n$ smooth, disjoint topological loops, meani…

math.AP2023

Almost sharp lower bound for the nodal volume of harmonic functions

Alexander Logunov, Lakshmi Priya, Andrea Sartori

This paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let $u$ be a real-valued harmonic function in $R^{n}$ …

math.AP2021

The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions

A. Logunov, E. Malinnikova, N. Nadirashvili +1

Let $Ω$ be a bounded domain in $R^{n}$ with $C^{1}$ boundary and let $u_{λ}$ be a Dirichlet Laplace eigenfunction in $Ω$ with eigenvalue $λ$ . We show that the $(n - 1)$ -dimensio…

math.AP2021★ 3 cited

An elliptic adaptation of ideas of Carleman and Domar from complex analysis related to Levinson's log log theorem

Alexander Logunov, Hristo Papazov

Using the three balls inequality, we adapt the elegant ideas of Carleman and Domar from complex analysis to linear elliptic PDE and generalize the classical Levinson's loglog theor…

math.AP2020★ 3 cited

Local version of Courant's nodal domain theorem

S. Chanillo, A. Logunov, E. Malinnikova +1

Let $(M, g)$ be a closed Riemannian manifold, where g is $C^{1}$ -smooth metric. Consider the sequence of eigenfunctions $u_{k}$ of the Laplace operator on M. Let $B$ be a ball on $M$ .…

math.AP2019

Review of Yau's conjecture on zero sets of Laplace eigenfunctions

Alexander Logunov, Eugenia Malinnikova

This is a review of old and new results and methods related to the Yau conjecture on the zero set of Laplace eigenfunctions. The review accompanies two lectures given at the confer…

math.AP2019★ 2 cited

Lecture notes on quantitative unique continuation for solutions of second order elliptic equations

Alexander Logunov, Eugenia Malinnikova

In these lectures we present some useful techniques to study quantitative properties of solutions of elliptic PDEs. Our aim is to outline a proof of a recent result on propagation…

math.SP2018

Eigenfunctions with infinitely many isolated critical points

Lev Buhovsky, Alexander Logunov, Mikhail Sodin

We construct a Riemannian metric on the $2$ -dimensional torus, such that for infinitely many eigenvalues of the Laplace-Beltrami operator, a corresponding eigenfunction has infin…

cs.DS2018

Collapsing Superstring Conjecture

Alexander Golovnev, Alexander S. Kulikov, Alexander Logunov +2

In the Shortest Common Superstring (SCS) problem, one is given a collection of strings, and needs to find a shortest string containing each of them as a substring. SCS admits $2\fr…

math.CA2017★ 7 cited

A discrete harmonic function bounded on a large portion of $Z^{2}$ is constant

Lev Buhovsky, Alexander Logunov, Eugenia Malinnikova +1

An improvement of the Liouville theorem for discrete harmonic functions on $Z^{2}$ is obtained. More precisely, we prove that there exists a positive constant $ε$ …

math.AP2017★ 3 cited

Quantitative propagation of smallness for solutions of elliptic equations

Alexander Logunov, Eugenia Malinnikova

Let $u$ be a solution to an elliptic equation $div (A \nabla u) = 0$ with Lipschitz coefficients in $R^{n}$ . Assume $∣ u ∣$ is bounded by $1$ in the ball $B = {∣ x ∣ \leq 1}$ .…

math.SG2017

Poisson Brackets of Partitions of Unity on Surfaces

Lev Buhovsky, Alexander Logunov, Shira Tanny

Given an open cover of a closed symplectic manifold, consider all smooth partitions of unity consisting of functions supported in the covering sets. The Poisson bracket invariant o…

math.CV2017

Translation-invariant probability measures on entire functions

Lev Buhovsky, Adi Glucksam, Alexander Logunov +1

We study non-trivial translation-invariant probability measures on the space of entire functions of one complex variable. The existence (and even an abundance) of such measures was…

math.AP2016★ 4 cited

Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three

Alexander Logunov, Eugenia Malinnikova

Let $Δ_{M}$ be the Laplace operator on a compact $n$ -dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions $u : Δ u + λ u = 0$ . In dimension $n = 2$ …

math.AP2016

Nodal sets of Laplace eigenfunctions: proof of Nadirashvili's conjecture and of the lower bound in Yau's conjecture

Alexander Logunov

Let $u$ be a harmonic function in the unit ball $B (0, 1) \subset R^{n}$ , $n \geq 3$ , such that $u (0) = 0$ . Nadirashvili conjectured that there exists a positive constant $c$ , d…

math.AP2016

Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure

Alexander Logunov

Let $M$ be a compact $C^{\infty}$ -smooth Riemannian manifold of dimension $n$ , $n \geq 3$ , and let $φ_{λ} : Δ_{M} φ_{λ} + λ φ_{λ} = 0$ denote the Laplace eigenfunction on $M$ co…

math.AP2015★ 10 cited

Ratios of harmonic functions with the same zero set

Alexander Logunov, Eugenia Malinnikova

We study the ratio of harmonic functions $u, v$ , which have the same zero set $Z$ in the unit ball $B \subset R^{n}$ . The ratio $f = u / v$ can be extended to a real analytic nowh…

math.AP2014

On the higher-dimensional harmonic analog of the Levinson log log theorem

Alexander Logunov

Let $M : (0, 1) \to [e, + \infty)$ be a decreasing function such that $0 \int 1 lo g lo g M (y) d y < + \infty$ . Consider the set $H_{M}$ of all functions $u$ harmonic in $P:=\{(…

math.CA2014

On a theorem of M. Cartwright in higher dimensions

A. Logunov, E. Malinnikova, P. Mozolyako

We consider harmonic functions in the unit ball of $R^{n + 1}$ that are unbounded near the boundary but can be estimated from above by some (rapidly increasing) radial weigh…

math.AP2014★ 15 cited

On ratios of harmonic functions

Alexander Logunov, Eugenia Malinnikova

Let $u$ and $v$ be harmonic in $Ω \subset R^{n}$ functions with the same zero set $Z$ . We show that the ratio $f$ of such functions is always well-defined and is real analyt…

math.CA2013

Weak integral conditions for BMO

Alexander A. Logunov, Leonid Slavin, Dmitriy M. Stolyarov +2

We study the question of how much one can weaken the defining condition of BMO. Specifically, we show that if $Q$ is a cube in $R^{n}$ and $h : [0, \infty) \to [0, \infty)$ is suc…

math.CA2013

On the Boundary Behavior of Positive Solutions of Elliptic Differential Equations

A. A. Logunov

Let $u$ be a positive harmonic function in the unit ball $B_{1} \subset R^{n}$ and let $μ$ be the boundary measure of $u$ . Consider a point $x \in \partial B_{1}$ and let $n (x)$ …

Estimating the number of real zeros of linear combinations of radicals of polynomials

Nested nodal loops of biharmonic functions

Almost sharp lower bound for the nodal volume of harmonic functions

The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions

An elliptic adaptation of ideas of Carleman and Domar from complex analysis related to Levinson's log log theorem

Local version of Courant's nodal domain theorem

Review of Yau's conjecture on zero sets of Laplace eigenfunctions

Lecture notes on quantitative unique continuation for solutions of second order elliptic equations

Eigenfunctions with infinitely many isolated critical points

Collapsing Superstring Conjecture

A discrete harmonic function bounded on a large portion of Z2 is constant

Quantitative propagation of smallness for solutions of elliptic equations

Poisson Brackets of Partitions of Unity on Surfaces

Translation-invariant probability measures on entire functions

Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three

Nodal sets of Laplace eigenfunctions: proof of Nadirashvili's conjecture and of the lower bound in Yau's conjecture

Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure

Ratios of harmonic functions with the same zero set

On the higher-dimensional harmonic analog of the Levinson log log theorem

On a theorem of M. Cartwright in higher dimensions

On ratios of harmonic functions

Weak integral conditions for BMO

On the Boundary Behavior of Positive Solutions of Elliptic Differential Equations

A discrete harmonic function bounded on a large portion of $Z^{2}$ is constant