TY - JOUR

T1 - Sign and area in nodal geometry of Laplace eigenfunctions

AU - Nazarov, Fëdor

AU - Polterovich, Leonid

AU - Sodin, Mikhail

PY - 2005/8

Y1 - 2005/8

N2 - The paper deals with asymptotic nodal geometry for the Laplace-Beltrami operator on closed surfaces. Given an eigenfunction / corresponding to a large eigenvalue, we study local asymmetry of the distribution of sign(f) with respect to the surface area. It is measured as follows: take any disc centered at the nodal line {f = 0}, and pick at random a point in this disc. What is the probability that the function assumes a positive value at the chosen point? We show that this quantity may decay logarithmically as the eigenvalue goes to infinity, but never faster than that. In other words, only a mild local asymmetry may appear. The proof combines methods due to Donnelly-Fefferman and Nadirashvili with a new result on harmonic functions in the unit disc.

AB - The paper deals with asymptotic nodal geometry for the Laplace-Beltrami operator on closed surfaces. Given an eigenfunction / corresponding to a large eigenvalue, we study local asymmetry of the distribution of sign(f) with respect to the surface area. It is measured as follows: take any disc centered at the nodal line {f = 0}, and pick at random a point in this disc. What is the probability that the function assumes a positive value at the chosen point? We show that this quantity may decay logarithmically as the eigenvalue goes to infinity, but never faster than that. In other words, only a mild local asymmetry may appear. The proof combines methods due to Donnelly-Fefferman and Nadirashvili with a new result on harmonic functions in the unit disc.

UR - http://www.scopus.com/inward/record.url?scp=24144478993&partnerID=8YFLogxK

U2 - 10.1353/ajm.2005.0030

DO - 10.1353/ajm.2005.0030

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AN - SCOPUS:24144478993

SN - 0002-9327

VL - 127

SP - 879

EP - 910

JO - American Journal of Mathematics

JF - American Journal of Mathematics

IS - 4

ER -