Nodal Sets of Steklov

We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in Rn – the eigenfunctions of the Dirichlet-to-Neumann map Λ. For a bounded Lipschitz domain Ω ⊂ Rn, this map associates to each function u defined on the boundary ∂Ω, the normal derivative of the harmonic function on Ω with boundary data u. Under the assumption that the domain Ω is C2, we prove a doubling property for the eigenfunction u. The main goal of this Thesis is to estimate the Hausdorff Hn−2-measure of the nodal set of u|∂Ω in terms of the eigenvalue λ as λ grows to infinity, provided Ω is fixed. In case that the domain Ω is analytic, we prove a polynomial bound (Cλ6). My methods, which build on the work of Lin, Garofalo and Han [Garofalo and Lin, CPAM 40 (1987), no. 3; Lin, CPAM 42 (1989), no. 6; Han and Lin, JPDE 7 (1994), no. 2], can be used also for solutions to more general elliptic equations and/or boundary conditions.