Doubling Property and Vanishing Order of Steklov Eigenfunctions

Zhu, Jiuyi

doi:10.1080/03605302.2015.1025980

Cited by 24 publications

(8 citation statements)

References 20 publications

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“…Here C = C(C 0 ) > 0 and r ∈ (0, r 0 ) for some sufficiently small constant r 0 > 0. Results of this flavour have been proved for eigenfunctions or equations with differentiable potentials (with dependences on the C 1 norm of the potentials, that is, M in (6) is the size of q C 1 (R n ) ) in [Rül17a,Zhu15] (on compact manifolds or bounded domains, respectively). For the spectral fractional Laplacian and its eigenfunctions on compact manifolds these dependences are indeed immediate consequences from the corresponding ones of the Laplacian.…”

Section: Introductionmentioning

confidence: 99%

On the fractional Landis conjecture

Rüland

Wang

2019

Journal of Functional Analysis

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In this paper we study a Landis-type conjecture for fractional Schrödinger equations of fractional power s ∈ (0, 1) with potentials. We discuss both the cases of differentiable and non-differentiable potentials. On the one hand, it turns out for differentiable potentials with some a priori bounds, if a solution decays at a rate e −|x| 1+ , then this solution is trivial. On the other hand, for s ∈ (1/4, 1) and merely bounded non-differentiable potentials, if a solution decays at a rate e −|x| α with α > 4s/(4s − 1), then this solution must again be trivial. Remark that when s → 1, 4s/(4s − 1) → 4/3 which is the optimal exponent for the standard Laplacian. For the case of non-differential potentials and s ∈ (1/4, 1), we also derive a quantitative estimate mimicking the classical result by Bourgain and Kenig.Wang is supported in part by MOST

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Section: Introductionmentioning

confidence: 99%

On the fractional Landis conjecture

Rüland

Wang

2019

Journal of Functional Analysis

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“…Whereas Zelditch obtains the (conjectured) optimal vanishing order, Bellova and Lin lose polynomial factors in their estimates which thus results in only almost optimal bounds. Simultaneously and independently of the present work the doubling inequalities of Bellova and Lin have been improved to an optimal scaling in the eigenvalue by Zhu [Zhu14]. His results lead to optimal bounds on the vanishing order of eigenfunctions.…”

Section: Introductionmentioning

confidence: 75%

On quantitative unique continuation properties of fractional Schrödinger equations: Doubling, vanishing order and nodal domain estimates

Rüland

2016

Trans. Amer. Math. Soc.

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Abstract. In this article we determine bounds on the maximal order of vanishing for eigenfunctions of a generalized Dirichlet-to-Neumann map (which is associated with fractional Schrödinger equations) on a compact, smooth Riemannian manifold, (M, g), without boundary. Moreover, with only slight modifications these results generalize to equations with C 1 potentials. Here Carleman estimates are a key tool. These yield a quantitative three balls inequality which implies quantitative bulk and boundary doubling estimates and hence leads to the control of the maximal order of vanishing. Using the boundary doubling property, we prove upper bounds on the H n−1 -measure of nodal domains of eigenfunctions of the generalized Dirichlet-to-Neumann map on analytic manifolds.

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“…This is true for S λ ∩ M since e λ is harmonic in M (see e.g. [10, Chapter 4]), while one can, for instance see that the same is true for S λ ∩ ∂M using the doubling lemma in [24]. In addition, for each λ, there are only finitely many nodal domains (see e.g.…”

Section: Introductionmentioning

confidence: 92%

Lower bounds for interior nodal sets of Steklov eigenfunctions

Sogge

Wang

Zhu

2016

Proc. Amer. Math. Soc.

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Abstract. We study the interior nodal sets, Z λ of Steklov eigenfunctions in an ndimensional relatively compact manifolds M with boundary and show that one has the lower bounds |Z λ | ≥ cλ 2−n 2for the size of its (n − 1)-dimensional Hausdorff measure. The proof is based on a Dong-type identity and estimates for the gradient of Steklov eigenfunctions, similar to those in [18] and [19], respectively.

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Doubling Property and Vanishing Order of Steklov Eigenfunctions

Cited by 24 publications

References 20 publications

On the fractional Landis conjecture

On the fractional Landis conjecture

On quantitative unique continuation properties of fractional Schrödinger equations: Doubling, vanishing order and nodal domain estimates

Lower bounds for interior nodal sets of Steklov eigenfunctions

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