Oscillatory functions vanish on a large set

Let $(M, g)$ be an $n$-dimensional, compact Riemannian manifold. We will show that functions that are orthogonal to the first few Laplacian eigenfunctions have to have a large zero set. Let us assume $f \in C^0 (M)$ is orthogonal $\langle f, \phi_k \rangle = 0$ to all eigenfunctions $\phi_k$ with eigenvalue $\lambda_k \leq \lambda$. If $\lambda$ is large, then the function $f$ has to vanish on a large set\[\mathcal{H}^{n-1} {\lbrace x : f(x) = 0 \rbrace} \gtrsim\frac{\sqrt{\lambda}}{(\operatorname{log} \lambda)^{n/2}}{\left (\frac{{\lVert f \rVert}_{L^1}}{{\lVert f \rVert}_{L^\infty}}\right )}^{2-\frac{1}{n}} \quad \textrm{.}\]Trigonometric functions on the flat torus $\mathbb{T}^d$ show that the result is sharp up to a logarithm if ${\lVert f \rVert}_{L^1} \sim {\lVert f \rVert}_{L^\infty}$. We also obtain a stronger result conditioned on the geometric regularity of $\lbrace x : f(x) = 0 \rbrace$. This may be understood as a very general higher-dimensional extension of the Sturm oscillation theorem.

Keywords

Sturm oscillation theorem, nodal set, Laplacian eigenfunction

2010 Mathematics Subject Classification

28A75, 34B24, 34C10, 35B05, 35J05, 35K08, 46E35

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This work is supported by the NSF (DMS-1763179) and by the Alfred P. Sloan Foundation.

Received 8 May 2018

Accepted 10 May 2019

Published 21 August 2020