Geometrical Versions of improved Berezin-Li-Yau Inequalities

We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in $\R^d$, $d \geq 2$. In particular, we derive upper bounds on Riesz means of order $\sigma \geq 3/2$, 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.