Domains Without Dense Steklov Nodal Sets

Abstract

This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem -Δϕσj=0,onΩ,∂νϕσj=σjϕσjon∂Ω\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} -\\Delta \\phi _{\\sigma _j}=0,\\quad \\hbox { on }\\,\\,\\Omega ,\\quad \\partial _\\nu \\phi _{\\sigma _j}=\\sigma _j \\phi _{\\sigma _j}\\quad \\hbox { on }\\,\\,\\partial \\Omega \\end{aligned}$$\\end{document}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 Omega in mathcal {A}, the nodal set of the eigenfunction phi _{sigma _j} “is 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 7(2):321–359, 2017). In fact, the results in the present paper establish that, for domains Omega in 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 Omega in 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.