Geometrical Versions of improved Berezin–Li–Yau Inequalities

We study the eigenvalues of the Dirichlet Laplace operator on an arbitrarybounded, open set in $\R^d$, $d \geq 2$. In particular, we derive upper boundson Riesz means of order $\sigma \geq 3/2$, that improve the sharp Berezininequality by a negative second term. This remainder term depends on geometricproperties of the boundary of the set and reflects the correct order of growthin the semi-classical limit. Under certain geometric conditions these resultsimply new lower bounds on individual eigenvalues, which improve the Li-Yauinequality.