A lower bound for sums of eigenvalues of the Laplacian

Let $\lambda _{k}(\Omega )$ be the $k$th Dirichlet eigenvalue of a bounded domain $\Omega$ in $\mathbb{R} ^{n}$. According to Weyl's asymptotic formula we have

The optimal in view of this asymptotic relation lower estimate for the sums $\sum_{j=1}^{k}\lambda _{j}(\Omega )$ has been proven by P.Li and S.T.Yau (Comm. Math. Phys. 88 (1983), 309-318). Here we will improve this estimate by adding to its right-hand side a term of the order of $k$ that depends on the ratio of the volume to the moment of inertia of $\Omega$.