Geometry and interior nodal sets of Steklov eigenfunctions
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Calculus of Variations
Geometry and interior nodal sets of Steklov eigenfunctions Jiuyi Zhu1 Received: 28 April 2020 / Accepted: 30 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We investigate the geometric properties of Steklov eigenfunctions in smooth manifolds. We derive the refined doubling estimates and Bernstein’s inequalities. For the real analytic manifolds, we are able to obtain the sharp upper bound for the measure of interior nodal sets. Mathematics Subject Classification 35P20 · 35P15 · 58C40 · 28A78
1 Introduction In this paper, we address the geometric properties and interior nodal sets of Steklov eigenfunctions g eλ (x) = 0, x ∈ M, (1.1) ∂eλ ∂ν (x) = λeλ (x), x ∈ ∂ M, where ν is a unit outward normal on ∂ M. Assume that (M, g) is a n-dimensional smooth, connected and compact manifold with smooth boundary ∂ M, where n ≥ 2. The Steklov eigenfunctions were first studied by Steklov in 1902 for bounded domains in the plane. It is also regarded as eigenfunctions of the Dirichlet-to-Neumann map, which is a first order homogeneous, self-adjoint and elliptic pseudodifferential operator. The spectrum λ j of Steklov eigenvalue problem consists of an infinite increasing sequence with 0 = λ0 < λ1 ≤ λ2 ≤ λ3 , . . . , and lim λ j = ∞. j→∞
The eigenfunctions {eλ j } form an orthonormal basis such that eλ j eλk d Vg = δ kj . eλ j ∈ C ∞ (M), ∂M
Communicated by F.H. Lin. Research is partially supported by the NSF Grant DMS 1656845.
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Jiuyi Zhu [email protected] Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA 0123456789().: V,-vol
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J. Zhu
Recently, the study of nodal geometry of eigenfunctions has been attracting much attention. Estimating the Hausdorff measure of nodal sets has always been an important subject concerning the nodal geometry of eigenfunctions. The celebrated problem about nodal sets centers around the famous Yau’s conjecture for smooth manifolds. Let φλ be eigenfunctions of − g φλ = λ2 φλ
(1.2)
on compact manifolds (M, g) without boundary, Yau conjectured that the upper and lower bounds of nodal sets of eigenfunctions in (1.2) satisify cλ ≤ H n−1 ({x ∈ M|φλ (x) = 0}) ≤ Cλ
(1.3)
where C, c depend only on the manifold M. The conjecture is shown to be true for real analytic manifolds by Donnelly-Fefferman in [8,10]. Lin [19] also proved the upper bound for the analytic manifolds using a different approach. Note that we use λ2 for the eigenvalue of Laplacian eigenfunctions to reflect the order of the elliptic operator, since the Dirichletto-Neumann map is a first order elliptic pseudodifferential operator and Laplace operator is a second order elliptic operator. Let us briefly review the recent literature concerning the progress of Yau’s conjecture on nodal sets of Laplacian eigenfunctions (1.2). For the conjecture (1.3) on the measure of nodal sets on smooth manifolds, there are important breakthrough made by Logunov and Malinnikova [20,22] and [21] in recent years. For the upper bound of nodal se
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