Hausdorff measure bounds for nodal sets of Steklov eigenfunctions

We study nodal sets of Steklov eigenfunctions in a bounded domain with C 2 boundary.Our first result is a lower bound for the Hausdorff measure of the nodal set: we show that for u λ a Steklov eigenfunction, with eigenvalue λ ≠ 0, H d-1 ({u λ = 0}) ≥ c Ω , where c Ω is independent of λ.We also prove an almost sharp upper bound, namely H d-1 ({u λ = 0}) ≤ C Ω λ log(λ + e).