A Dynamical Approach to the Berezin--Li--Yau Inequality

TL;DR

This paper introduces a dynamical method using volume-preserving mean curvature flow to prove the sharp Berezin--Li--Yau inequality for convex domains, establishing new monotonicity principles and geometric inequalities.

Contribution

The paper develops a novel dynamical approach based on mean curvature flow and monotonicity principles to prove the sharp Berezin--Li--Yau inequality for convex domains.

Findings

Monotonicity of Riesz mean along the flow for convex domains

Geometric correlation inequality between boundary spectral density and mean curvature

Convergence of the flow to the ball implies the sharp inequality

Abstract

We develop a dynamical method for proving the sharp Berezin--Li--Yau inequality. The approach is based on the volume-preserving mean curvature flow and a new monotonicity principle for the Riesz mean $R_\Lambda(\Omega_t)$. For convex domains we show that $R_\Lambda$ is monotone non-decreasing along the flow. The key input is a geometric correlation inequality between the boundary spectral density $Q_\Lambda$ and the mean curvature $H$, established in all dimensions: in $d=2$ via circular symmetrization, and in $d\ge 3$ via the boundary Weyl expansion together with the Laugesen--Morpurgo trace minimization principle. Since the flow converges smoothly to the ball, the monotonicity implies the sharp Berezin--Li--Yau bound for every smooth convex domain. As an application, we obtain a sharp dynamical Ces\`aro--P\'olya inequality for eigenvalue averages.