@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 = "Publisher Copyright: {\textcopyright} Springer International Publishing AG, part of Springer Nature 2018.",
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",
}