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2015 Nodal sets and growth exponents of Laplace eigenfunctions on surfaces
Guillaume Roy-Fortin
Anal. PDE 8(1): 223-255 (2015). DOI: 10.2140/apde.2015.8.223


We prove a result, announced by F. Nazarov, L. Polterovich and M. Sodin, that exhibits a relation between the average local growth of a Laplace eigenfunction on a closed surface and the global size of its nodal set. More precisely, we provide a lower and an upper bound to the Hausdorff measure of the nodal set in terms of the expected value of the growth exponent of an eigenfunction on disks of wavelength-like radius. Combined with Yau’s conjecture, the result implies that the average local growth of an eigenfunction on such disks is bounded by constants in the semiclassical limit. We also obtain results that link the size of the nodal set to the growth of solutions of planar Schrödinger equations with small potential.


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Guillaume Roy-Fortin. "Nodal sets and growth exponents of Laplace eigenfunctions on surfaces." Anal. PDE 8 (1) 223 - 255, 2015.


Received: 6 September 2014; Accepted: 26 November 2014; Published: 2015
First available in Project Euclid: 28 November 2017

zbMATH: 1316.58025
MathSciNet: MR3336925
Digital Object Identifier: 10.2140/apde.2015.8.223

Primary: 58J50

Rights: Copyright © 2015 Mathematical Sciences Publishers


Vol.8 • No. 1 • 2015
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