We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in R n -the eigenfunctions of the Dirichlet-toNeumann map. Under the assumption that the domain Ω is C 2 , we prove a doubling property for the eigenfunction u. We estimate the Hausdorff H n−2 -measure of the nodal set of u| ∂Ω in terms of the eigenvalue λ as λ grows to infinity. In case that the domain Ω is analytic, we prove a polynomial bound O(λ 6 ). Our arguments, which make heavy use of Almgren's frequency functions, are built on the previous works [Garofalo and Lin, CPAM 40 (1987), no. 3; Lin, CPAM 42 (1989), no. 6].
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