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arXiv · 1510.02376

Average $L^q$ growth and nodal sets of eigenfunctions of the Laplacian on surfaces

Guillaume Roy-Fortin
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Abstract

In a recent paper, we exhibit a link between the average local growth of Laplace eigenfunctions on surfaces and the size of their nodal set. In that paper, the average local growth is computed using the uniform - or $L^\infty$ - growth exponents on disks of wavelength radius. The purpose of this note is to prove similar results for a broader class of $L^q$ growth exponents with $q \in (1, \infty)$. More precisely, we show that the size of the nodal set is bounded above and below by the product of the average local $L^q$ growth with the frequency. We briefly discuss the relation between this new result and Yau's conjecture on the size of nodal sets.

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Guillaume Roy-Fortin. 2015-10-08. Average $L^q$ growth and nodal sets of eigenfunctions of the Laplacian on surfaces. https://arxiv.org/abs/1510.02376

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