On the remainder term of the Berezin inequality on a convex domain

Artikel i vetenskaplig tidskrift, 2017

We study the Dirichlet eigenvalues of the Laplacian on a convex
domain in Rn, with n ≥ 2. In particular, we generalize and improve upper
bounds for the Riesz means of order σ ≥ 3/2 established in an article by
Geisinger, Laptev and Weidl. This is achieved by refining estimates for a
negative second term in the Berezin inequality. The obtained remainder term
reflects the correct order of growth in the semi-classical limit and depends only
on the measure of the boundary of the domain. We emphasize that such an
improvement is for general Ω ⊂ Rn not possible and was previously known to
hold only for planar convex domains satisfying certain geometric conditions.
As a corollary we obtain lower bounds for the individual eigenvalues λk ,
which for a certain range of k improves the Li–Yau inequality for convex do-
mains. However, for convex domains one can by using different methods obtain
even stronger lower bounds for λk .