Measure Upper Bounds of Nodal Sets of Robin Eigenfunctions

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In this paper, we obtain the upper bounds for the Hausdorff measures of nodal sets of eigenfunctions with the Robin boundary conditions, i.e., equation* {arrayl u+λu=0, in Ω, uν+μu=0, on∂Ω, array. equation* where the domain Ωⁿ, uν 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 ∂ΩΓ 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+μα+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.