Measure Upper Bounds of Nodal Sets of Robin Eigenfunctions

2018/01/07 by Liu, Fang, Tian, Long, Yang, Xiaoping · 1 citation
#35J30 #58E10 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1801.02114

Abstract

In this paper, we obtain the upper bounds for the Hausdorff measures of nodal sets of eigenfunctions with the Robin boundary conditions, i.e., \\triangle u+λu=0, in Ω,
u_ν+μu=0, on ∂Ω, . where the domain Ω⊆ℝⁿ, u_ν means the derivative of u along the outer normal direction of ∂Ω. We show that, if Ω is bounded and analytic, and the corresponding eigenvalue λ is large enough,then the measure upper bounds for the nodal sets of eigenfunctions are C√λ, where C is a positive constant depending only on n and Ω but not on μ We also show that, if ∂Ω is C^∞ smooth and ∂Ω∖Γ is piecewise analytic, where Γ⊆∂Ω is a union of some n-2 dimensional submanifolds of ∂Ω, μ>0, and λ is large enough, then the corresponding measure upper bounds for the nodal sets of u are C(√λ+μ^α+μ^-cα) for some positive number α, where c is a positive constant depending only on n, and C is a positive constant depending on n, Ω, Γ and α.

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