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Domains Without Dense Steklov Nodal Sets
- Source :
- Journal of Fourier Analysis and Applications. 26
- Publication Year :
- 2020
- Publisher :
- Springer Science and Business Media LLC, 2020.
-
Abstract
- This 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.
- Subjects :
- Applied Mathematics
General Mathematics
Open problem
010102 general mathematics
Sigma
Mathematics::Spectral Theory
Eigenfunction
01 natural sciences
Omega
Combinatorics
0103 physical sciences
010307 mathematical physics
Ball (mathematics)
0101 mathematics
Analysis
Eigenvalues and eigenvectors
Mathematics
Subjects
Details
- ISSN :
- 15315851 and 10695869
- Volume :
- 26
- Database :
- OpenAIRE
- Journal :
- Journal of Fourier Analysis and Applications
- Accession number :
- edsair.doi.dedup.....5f7a318469c06e9bf1c6f8b9c7e774a0