Abstract
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.
Original language | American English |
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Pages (from-to) | 1611-1621 |
Number of pages | 11 |
Journal | Communications in Partial Differential Equations |
Volume | 33 |
Issue number | 9 |
DOIs | |
State | Published - Sep 2008 |
Externally published | Yes |
Keywords
- Asymmetry
- Growth of eigenfunctions
- Inner radius
- Nodal domains