Doubling inequalities and critical sets of Dirichlet eigenfunctions

TL;DR

This paper establishes sharp doubling inequalities for gradients and bounds for critical sets of Dirichlet eigenfunctions on Riemannian manifolds, introducing new techniques to handle smooth manifolds without double manifolds.

Contribution

It develops novel methods to prove sharp doubling inequalities for gradients and bounds critical sets of eigenfunctions on smooth manifolds, overcoming previous technical limitations.

Findings

Sharp doubling inequalities for gradients are established.

Upper bounds for critical sets are derived in analytic manifolds.

New techniques address challenges in smooth manifold settings.

Abstract

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.