Geometrical Versions of improved Berezin–Li–Yau Inequalities

Abstract

Abstract. We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set inIn particular, we derive upper bounds on Riesz means of order 3=2, that improve the sharp Berezin inequality by a negative second term. This remainder term depends on geometric properties of the boundary of the set and reflects the correct order of growth in the semi-classical limit.Under certain geometric conditions these results imply new lower bounds on individual eigenvalues, which improve the Li-Yau inequality.Mathematics Subject Classification (2010). Primary 35P15; Secondary 47A75.

A bound for the perimeter of inner parallel bodies

“…Deducing Corollary 1.3 is simply a matter of applying the following theorem due to Steinhagen [15]. We note that this theorem appeared in the case of planar convex bodies in earlier work by Blaschke [1], and this simpler case is sufficient for proving the inequality conjectured in [6]. Theorem 2.2 (Steinhagen's inequality [15]).…”

“…Our study of this problem is motivated by work of Geisinger, Laptev and Weidl in [6] where they use Theorem 1.1 to obtain bounds on the Riesz eigenvalue means for the Dirichlet Laplacian on a convex domain Ω ⊂ R n . For convex domains in the plane satisfying the inequality…”

“…In [9] the techniques of [6] are combined with this result to obtain further geometrical improvements of Berezin-type bounds for the Dirichlet eigenvalues of the Laplacian on convex domains.…”

Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain

“…The resulting upper bound can be seen as an improvement of an inequality going back to work of Berezin [4] and Li-Yau [25]. Such improved versions of the Berezin-Li-Yau inequality have been the topic of several recent papers [13,14,16,21,23,26,35]. Lower bounds in the same spirit are contained in [15].…”

Improved Berezin—Li—Yau inequalities with magnetic field

“…Several results in this direction were obtained recently both for the Berezin inequality [9,23] (for γ ≥ 3 2 ) and for the Li-Yau estimate [8,20,13,25,26]. In particular, Melas proved in [20] that there exists a positive constant M d such that…”