Nodal sets of Robin and Neumann eigenfunctions

We investigate the measure of nodal sets for Robin and Neumann eigenfunctions in the domain and on the boundary of the domain. A polynomial upper bound for the interior nodal sets is obtained for Robin eigenfunctions in the smooth domain. For the analytic domain, the sharp upper bounds of the interior nodal sets was shown for Robin eigenfunctions. More importantly, we obtain the sharp upper bounds for the boundary nodal sets of Neumann eigenfunctions with new quantitative global Carleman estimates. Furthermore, the sharp doubling inequality and vanishing order of Robin eigenfunctions on the boundary of the domain are obtained.