For a dense family of analytic planar domains, Steklov eigenfunctions have fixed-size zero-free balls in the interior, so their nodal sets are not dense at any shrinking scale.
Geometry and interior nodal sets of Steklov eigenfunctions
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abstract
We investigate the geometric properties of Steklov eigenfunctions in smooth manifolds. We derive the refined doubling estimates and Bernstein's inequalities. For the real analytic manifolds, we are able to obtain the sharp upper bound for the measure of interior nodal sets.
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math.AP 1years
2019 1verdicts
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Domains without dense Steklov nodal sets
For a dense family of analytic planar domains, Steklov eigenfunctions have fixed-size zero-free balls in the interior, so their nodal sets are not dense at any shrinking scale.