EIGENFUNCTIONS AND NODAL SETS

By Steve Zelditch. 2012

213 citations recorded by the source corpus.

This is a survey of recent results on eigenfunctions of the Laplacian on compact Riemannian manifolds and their nodal sets. It is the write-up of my talk at JDG 2011.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/06ef2766aa76ab9ebbf815426b847051fe6c8049

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Optimal Shape and Location of Sensors for Parabolic Equations with Random Initial Data (2014, methodology): “Here, we are concerned only with weak limits in Ω, and following [65] we use the wording ”on the base”.”
Variational aspects of Laplace eigenvalues on Riemannian surfaces (2011, methodology): “µn,c)are bounded by a quantity that depends on the topology of M only. When the measure µis the genuine volume measure of a C∞-smooth Riemannian metric, the statement is classical and is due to Cheng [5]. Claim 5.3 is a partial case of a more general result proved in [28, Sect. 5]. 28 5.3. Proof of ”
Sharp eigenvalue bounds and minimal surfaces in the ball (2012, background): “ϕ−1(0)∪{p 1,...,pk}, j= 1,...,d. Ifd>2γ+i, then thereexist n1,...,n2γ+i+1 ∈ Z not all zero such that the homology class of Srepresented by P2γ+i+1 j=1 njΦj is zero. It follows from the argument of [Ch] (Lemma 3.1) that S\ Φ1(S1) ∪ ...∪ Φd(S1) has at least i+ 2 connected components. Therefore, M\ ”
Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces (2013, background): “Note that the number of eigenfunctions k( ) depends on but since the multiplicity of eigenvalues is bounded by a constant which only depends on the topology of the surface (see [6]), up to the extraction of a subsequence, we assume in the following that k is fixed.”
Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces (2012, unknown): “A classical result by Cheng in [7] says that the multiplicities mk(g) of these eigenvalues are bounded by quantities depending on the genus γ of M only.”
Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution (2013, background): “(cosθ,sinθ) : r>0,pN|S1(v) = 0}. It follows that the local structure of the nodal set in a small disc around a singular point p is C1 equivalent to Nequi-angular rays emanating from p. We refer to [Ch] for further details. NUMBER OF NODAL DOMAINS AND SINGULAR POINTS 13 6.1. Surfaceswithisometrici”
SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM (2014, unknown): “In two dimensions, the estimate on the number of nodal domains allows to control the eigenvalue multiplicities (see [12, 19]).”