Recently, two papers [2, 9] appeared which give lower bounds on the size of the nodal sets of eigenfunctions. The purpose of this short note is to point out a third method to obtain a power law lower bound on the volume of the nodal sets. Our method is based on the Donnelly-Fefferman growth bound for eigenfunctions and a growth vs. volume relation we proved in .
Bibliographical noteFunding Information:
I would like to thank Iosif Polterovich for drawing my attention to this problem and for useful discussions. I thank Joseph Bernstein for an illuminating discussion. I am grateful to Leonid Polterovich for valuable remarks on a preliminary version of this note. This work was partially supported by the Israel Science Foundation (grant no. 225/10).
- Hausdorff measure
- Lower bound
- Nodal set