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- W2968686935 abstract "This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem $$ -Delta phi_{sigma_j}=0,quadtext{ on }Omega,qquadqquad partial_nu phi_{sigma_j}=sigma_j phi_{sigma_j}quad text{ on }partialOmega $$ in two-dimensional domains $Omega$. In particular, this paper presents a dense family $mathcal{A}$ of simply-connected two-dimensional domains with analytic boundaries such that, for each $Omegain mathcal{A}$, the nodal set of the eigenfunction $phi_{sigma_j}$ $not$ dense at scale $sigma_j^{-1}$. This result addresses a question put forth under Open Problem 10 in Girouard and Polterovich, J. Spectr. Theory, 321-359 (2017). In fact, the results in the present paper establish that, for domains $Omegain mathcal{A}$, the nodal sets of the eigenfunctions $phi_{sigma_j}$ associated with the eigenvalue $sigma_j$ have starkly different character than anticipated: they are not dense at any shrinking scale. More precisely, for each $Omegain mathcal{A}$ there is a value $r_1>0$ such that for each $j$ there is $x_jin Omega$ such that $phi_{sigma_j}$ does not vanish on the ball of radius $r_1$ around $x_j$." @default.
- W2968686935 created "2019-08-22" @default.
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- W2968686935 date "2019-08-09" @default.
- W2968686935 modified "2023-10-16" @default.
- W2968686935 title "Domains without dense Steklov nodal sets" @default.
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