Author:
Bruno Oscar P.,Galkowski Jeffrey
Abstract
AbstractThis article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem $$\begin{aligned} -\Delta \phi _{\sigma _j}=0,\quad \hbox { on }\,\,\Omega ,\quad \partial _\nu \phi _{\sigma _j}=\sigma _j \phi _{\sigma _j}\quad \hbox { on }\,\,\partial \Omega \end{aligned}$$-Δϕσj=0,onΩ,∂νϕσj=σjϕσjon∂Ωin two-dimensional domains $$\Omega $$Ω. In particular, this paper presents a dense family $$\mathcal {A}$$A of simply-connected two-dimensional domains with analytic boundaries such that, for each $$\Omega \in \mathcal {A}$$Ω∈A, the nodal set of the eigenfunction $$\phi _{\sigma _j}$$ϕσj “is not dense at scale $$\sigma _j^{-1}$$σj-1”. This result addresses a question put forth under “Open Problem 10” in Girouard and Polterovich (J Spectr Theory 7(2):321–359, 2017). In fact, the results in the present paper establish that, for domains $$\Omega \in \mathcal {A}$$Ω∈A, the nodal sets of the eigenfunctions $$\phi _{\sigma _j}$$ϕσj associated with the eigenvalue $$\sigma _j$$σj have starkly different character than anticipated: they are not dense at any shrinking scale. More precisely, for each $$\Omega \in \mathcal {A}$$Ω∈A there is a value $$r_1>0$$r1>0 such that for each j there is $$x_j\in \Omega $$xj∈Ω such that $$\phi _{\sigma _j}$$ϕσj does not vanish on the ball of radius $$r_1$$r1 around $$x_j$$xj.
Funder
University College London
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,General Mathematics,Analysis
Reference21 articles.
1. Akhmetgaliyev, E.: Fast numerical methods for mixed, singular Helmholtz boundary value problems and Laplace eigenvalue problems-with applications to antenna design, sloshing, electromagnetic scattering and spectral geometry. PhD thesis, California Institute of Technology (2016)
2. Bell, S.R., Krantz, S.G.: Smoothness to the boundary of conformal maps. Rocky Mt. J. Math. 17(1), 23–40 (1987)
3. Bellová, K., Lin, F.-H.: Lin, F-H: Nodal sets of Steklov eigenfunctions. Calc. Var. Partial Differ. Equ. 54(2), 2239–2268 (2015)
4. Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers I. Springer, New York (1999). Asymptotic methods and perturbation theory, Reprint of the 1978 original
5. Galkowski, J., Toth, J.A.: Pointwise bounds for Steklov eigenfunctions. J. Geom. Anal. 29(1), 142–193 (2019)
Cited by
3 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献