The Work of Logunov and Malinnikova in Relation to Yau's Conjecture (Fall 2019)

COURSE DESCRIPTION

We will discuss the recent work of Logunov and Malinnikova, for which they were awarded a Clay research award. This work led to the proof of Nadirashvili's conjecture and of the lower bound in Yau's conjecture. Nadirashvili's conjecture roughly states that if $u$ is a harmonic function in the unit ball in ${\mathbb{R}}^n$ which is zero at the center of the ball, then the $(n-1)$-dimensional Hausdorff measure of the zero set of $u$ in the ball is bounded below by a purely dimensional constant. Yau's conjecture concerns the $(n-1)$-dimensional Hausdorff measure of the zero set of Laplace eigenfunctions. In particular, Yau conjectured that the zero set of the eigenfunction associated to the eigenvalue $\lambda$ should have $(n-1)$-dimensional Hausdorff measure which is approximately $\sqrt{\lambda }$.

Primary Sources

Nodal Sets of Laplace Eigenfunctions: Estimates of the Hausdorff Measure in Dimensions Two and Three

Nodal Sets of Laplace Eigenfunctions: Polynomial Upper Estimates of the Hausdorff Measure

Nodal Sets of Laplace Eigenfunctions: proof of Nadirashvili's conjecture and the lower bound in Yau's conjectures

Secondary Sources

"A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order", Aronszajn; 1957

"Some Problems of the Qualitative Theory of Second Order Elliptic Equations (Case of Several Independent Variables)", Landis; 1963

"Uber Knoten yon Eigenfunktionen des Laplace-Beltrami-Operators", Bruning; 1978

"Monotonicity Properties of Variational Integrals, $A_p$ Weights and Unique Continuation", Garofalo, Lin; 1986

"Nodal sets of eigenfunctions on Riemannian manifolds", Donnelly, Fefferman; 1988

"Nodal Sets of Solutions of Elliptic Equations", Hardt, Simon; 1989

"Nodal Sets of Solutions of Elliptic and Parabolic Equations", Lin; 1991

"Logarithmic Convexity for Supremum Norms of Harmonic Functions", Korevaar, Meyers; 1994

"The effect of curvature on convexity properties of harmonic functions and eigenfunctions", Mangoubi; 2013

"Quantitative Propogation of Smallness for Solutions of Elliptic Equations.", Logunov, Malinnikova; 2017

Notes

Introduction

Preliminaries

Meta Argument

The Frequency Bound of Donnelly-Fefferman

Combinatorial Argument

Density of Zeroes