A counterexample to symmetry of $L^p$ norms of eigenfunctions

TL;DR

This paper constructs a sequence of Laplacian eigenfunctions on a flat torus in dimensions three and higher, demonstrating that the $L^p$ norms of positive and negative parts can behave asymmetrically, countering previous assumptions.

Contribution

It provides a counterexample showing the asymmetry of $L^p$ norms of eigenfunction parts, answering a question by Jakobson and Nadirashvili.

Findings

Existence of eigenfunctions with asymmetric $L^p$ norm ratios

Counterexample on flat $d$-torus for $d\u2265 3$

Elementary and computer-assisted proof approach

Abstract

We answer a question of Jakobson and Nadirashvili on the asymptotic behavior of the $L^p$ norms of positive and negative parts of eigenfunctions of the Laplacian. More precisely, we show that there exists a sequence of eigenfunctions $\psi_n$ on the flat $d$-torus for $d\geq 3$, with eigenvalues $\lambda_n\to\infty$ as $n\to\infty$, such that the ratio $\|\psi_n\chi_{\{\psi_n>0\}}\|_p / \|\psi_n\chi_{\{\psi_n<0\}}\|_p $ does not tend to $1$ as $n\to\infty$ for $1<p\leq \infty$. Our argument is elementary and computer-assisted.