Lower Bounds for Nodal Sets of Eigenfunctions
By T. Colding, W. Minicozzi. 2010
100 citations recorded by the source corpus.
We prove lower bounds for the Hausdorff measure of nodal sets of eigenfunctions.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/ccc23206c71482018175478306784824369c246c
How researchers see this paper
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In Applied Mathematics
- Christopher D. Sogge
- Steve Zelditch
- Hamid Hezari
- Xuehua Chen
- Eugenia Malinnikova
- Alexander Logunov
- Boris Hanin
- Y. Canzani
In Geometry and Topology
- Christopher D. Sogge
- Hamid Hezari
- Eugenia Malinnikova
- Alexander Logunov
- Steve Zelditch
- Zhiqin Lu
- Bernard Shiffman
- Kento Fujita
Citations in the corpus
- A Few Endpoint Geodesic Restriction Estimates for Eigenfunctions (2012, background): “We also would like to point out that the problem of finding improvements for estimates that are saturated by the highest weight spherical harmonics is related to the problem of obtaining lower bounds for the size of nodal sets of eigenfunctions (see [11], [33] and [16]).”
- EIGENFUNCTIONS AND NODAL SETS (2012, methodology): “sequently, (22) 3 n 2 .Hn 1(Z ’ ); >1: Inequality (22) follows from (21) and the lower bounds in [SoZ] (23) 1 n 4 . Z M j’ jdV g: The lower bound (22) was rst proved by Colding and Minicozzi [CM]. A slightly weaker result was proved in [SoZ] by a dierent method inspired by the article of R. T. D”
- A natural lower bound for the size of nodal sets (2011, background): “The first “polynomial type” lower bounds appear to be those given in [Colding and Minicozzi 2011] and [Sogge and Zelditch 2011a] (see also [Mangoubi 2011]).”
- Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three (2016, methodology): “The two known approaches are: (1) follow the ideas of Donnelly and Fefferman and find many balls on the wave-scale λ−1/2 with bounded doubling index, as it is done in [2] or (2) use the Green formula 2 ∫ Eλ |∇Muλ| = λ ∫ M |uλ| and the estimate ‖uλ‖∞ ‖uλ‖1 ≤ Cλ (n−1)/4, see [16].”
- Lower bounds on the Hausdorff measure of nodal sets II (2010, background): “Independently Colding and Minicozzi [2] obtained the more favorable lower bound (1) λ1− n−1 2 .”