@inbook{0f256948cb804412a973db4062f2d1e1,

title = "Nodal sets of laplace eigenfunctions: Estimates of the hausdorff measure in dimensions two and three",

abstract = "Let ΔM be the Laplace operator on a compact n-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions u: ΔMu+λu = 0. In dimension n = 2 we refine the Donnelly–Fefferman estimate by showing that H1({u = 0}) ≤Cλ3/4−β for some β∈ (0, 1/4). The proof employs the Donnelly–Fefferman estimate and a combinatorial argument, which also gives a lower (non-sharp) bound in dimension n = 3: H2({u = 0}) ≥ cλα for some α∈ (0,1/2). The positive constants c, C depend on the manifold, α and β are universal.",

keywords = "Harmonic functions, Laplace eigenfunctions, Nodal set",

author = "Alexander Logunov and Eugenia Malinnikova",

note = "Funding Information: A. L. was supported in part by ERC Advanced Grant 692616 and ISF Grants 1380/13, 382/15. Eu. M. was supported by Project 213638 of the Research Council of Norway.",

year = "2018",

doi = "10.1007/978-3-319-59078-3_17",

language = "English (US)",

series = "Operator Theory: Advances and Applications",

publisher = "Springer International Publishing",

pages = "333--344",

booktitle = "Operator Theory",

}