Doubling inequalities and critical sets of Dirichlet eigenfunctions

We study the sharp doubling inequalities for the gradients and upper bounds for the critical sets of Dirichlet eigenfunctions on the boundary and in the interior of compact Riemannian manifolds. Most efforts are devoted to obtaining the sharp doubling inequalities for the gradients. New technique is developed to overcome the difficulties on the unavailability of the double manifold in obtaining doubling inequalities in smooth manifolds. The sharp upper bounds of critical sets in analytic Riemannian manifolds are consequences of the doubling inequalities.