We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in RdRd, d≥2d≥2. In particular, we derive upper bounds on Riesz means of order σ≥3/2σ≥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.