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      Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three

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          Abstract

          Let \(\Delta_M\) be the Laplace operator on a compact \(n\)-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions \(u:\Delta u + \lambda u =0\). In dimension \(n=2\) we refine the Donnelly-Fefferman estimate by showing that \(H^1(\{u=0 \})\le C\lambda^{3/4-\beta}\), \(\beta \in (0,1/4)\). The proof employs the Donnelli-Fefferman estimate and a combinatorial argument, which also gives a lower (non-sharp) bound in dimension \(n=3\): \(H^2(\{u=0\})\ge c\lambda^\alpha\), \(\alpha \in (0,1/2)\). The positive constants \(c,C\) depend on the manifold, \(\alpha\) and \(\beta\) are universal.

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          Nodal sets of solutions of elliptic and parabolic equations

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            Nodal sets for eigenfunctions of the Laplacian on surfaces

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              Sign and area in nodal geometry of Laplace eigenfunctions

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                Author and article information

                Journal
                2016-05-09
                Article
                1605.02595
                77db8295-4d82-4c76-ace7-d1937b89fb2a

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                math.AP math.CA math.DG math.SP

                Analysis,Functional analysis,Geometry & Topology
                Analysis, Functional analysis, Geometry & Topology

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