TY - CHAP

T1 - Nodal sets of laplace eigenfunctions

T2 - Estimates of the hausdorff measure in dimensions two and three

AU - Logunov, Alexander

AU - Malinnikova, Eugenia

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

KW - Harmonic functions

KW - Laplace eigenfunctions

KW - Nodal set

UR - http://www.scopus.com/inward/record.url?scp=85044743326&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85044743326&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-59078-3_17

DO - 10.1007/978-3-319-59078-3_17

M3 - Chapter

AN - SCOPUS:85044743326

T3 - Operator Theory: Advances and Applications

SP - 333

EP - 344

BT - Operator Theory

PB - Springer International Publishing

ER -