Hausdorff measure bounds for nodal sets of Steklov eigenfunctions

Decio, Stefano

doi:10.2140/apde.2024.17.1237

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Hausdorff measure bounds for nodal sets of Steklov eigenfunctions

Stefano Decio

Vol. 17 (2024), No. 4, 1237–1259

DOI: 10.2140/apde.2024.17.1237

Abstract

We study nodal sets of Steklov eigenfunctions in a bounded domain with $𝒞^{2}$ boundary. Our first result is a lower bound for the Hausdorff measure of the nodal set: we show that, for $u_{λ}$ a Steklov eigenfunction with eigenvalue $λ \neq 0$ , we have $ℋ^{d - 1} ({u_{λ} = 0}) \geq c_{Ω}$ , where $c_{Ω}$ is independent of $λ$ . We also prove an almost sharp upper bound, namely, $ℋ^{d - 1} ({u_{λ} = 0}) \leq C_{Ω} λ \log (λ + e)$ .

Keywords

Steklov eigenfunctions, nodal set, frequency function

Mathematical Subject Classification

Primary: 35J15, 58J50

Milestones

Received: 3 May 2021

Revised: 13 July 2022

Accepted: 19 August 2022

Published: 17 May 2024

Authors

Stefano Decio
Department of Mathematical Sciences Norwegian University of Science and Technology Trondheim Norway	Institute for Advanced Study Princeton, NJ United States

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Vol. 17, No. 4, 2024