Local asymmetry and the inner radius of nodal domains

Dan Mangoubi

doi:10.1080/03605300802038577

Local asymmetry and the inner radius of nodal domains

Dan Mangoubi^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

28 Scopus citations

Abstract

Let M be a closed Riemannian manifold of dimension n. Let φ_λ be an eigenfunction of the Laplace-Beltrami operator corresponding to an eigenvalue λ. We show that the volume of {φ_λ > 0} ∪ B is ≥ C|B|/λⁿ, where B is any ball centered at a point of the nodal set. We apply this result to prove that each nodal domain contains a ball of radius ≥ C/λⁿ. The results in this paper extend previous results of Nazarov, Polterovich, Sodin and of the author.

Original language	English
Pages (from-to)	1611-1621
Number of pages	11
Journal	Communications in Partial Differential Equations
Volume	33
Issue number	9
DOIs	https://doi.org/10.1080/03605300802038577
State	Published - Sep 2008
Externally published	Yes

Keywords

Asymmetry
Growth of eigenfunctions
Inner radius
Nodal domains

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10.1080/03605300802038577

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AB - Let M be a closed Riemannian manifold of dimension n. Let φλ be an eigenfunction of the Laplace-Beltrami operator corresponding to an eigenvalue λ. We show that the volume of {φλ > 0} ∪ B is ≥ C|B|/λn, where B is any ball centered at a point of the nodal set. We apply this result to prove that each nodal domain contains a ball of radius ≥ C/λn. The results in this paper extend previous results of Nazarov, Polterovich, Sodin and of the author.

KW - Asymmetry

KW - Growth of eigenfunctions

KW - Inner radius

KW - Nodal domains

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JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

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Local asymmetry and the inner radius of nodal domains

Abstract

Keywords

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