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Hausdorff measure bounds for nodal sets of Steklov eigenfunctions

Stefano Decio

Vol. 17 (2024), No. 4, 1237–1259
Abstract

We study nodal sets of Steklov eigenfunctions in a bounded domain with 𝒞2 boundary. Our first result is a lower bound for the Hausdorff measure of the nodal set: we show that, for uλ a Steklov eigenfunction with eigenvalue λ0, we have d1({uλ = 0}) cΩ, where cΩ is independent of λ. We also prove an almost sharp upper bound, namely, d1({uλ = 0}) CΩλlog (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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