A. Logunov

activity

20132026

most citedOn ratios of harmonic functions

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

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14 papers · 1 filter

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.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.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.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…