In the interesting paper under review the authors prove a two-term Weyl-type asymptotic formula for sums of eigenvalues \(0<\lambda_1\leq \lambda_2\leq\ldots\) repeated according to multiplicities, of the Dirichlet Laplacian \(-\Delta_\Omega\) in a bounded open set \(\Omega\subset\mathbb{R}^d,\) \(d\geq2,\) with Lipschitz boundary. Precisely, setting \(L_d=\frac{2}{2+d}\frac{\omega_d}{(2\pi)^d},\) it is proved that \[ \sum_{\lambda_k<\lambda}(\lambda-\lambda_k)=L_d|\Omega|\lambda^{1+\frac{d}{2}}- \frac{L_{d-1}}{4}\mathcal{H}^{d-1}(\partial\Omega)\lambda^{1+\frac{d-1}{2}}+o\left(\lambda^{1+\frac{d-1}{2}}\right) \] as \(\lambda\to\infty,\) where \(\omega_d\) is the measure of the unit ball in \(\mathbb{R}^d\) and \(\mathcal{H}^{d-1}(\partial\Omega)\) stands for the \((d-1)\)-dimensional Hausdorff measure of \(\partial\Omega.\)
In the case when \(\Omega\) is a convex domain, the universal bound \[ \left| \sum_{\lambda_k<\lambda}(\lambda-\lambda_k)-L_d|\Omega|\lambda^{1+\frac{d}{2}}+ \frac{L_{d-1}}{4}\mathcal{H}^{d-1}(\partial\Omega)\lambda^{1+\frac{d-1}{2}}\right| \leq C \mathcal{H}^{d-1}(\partial\Omega)\lambda^{1+\frac{d-1}{2}}\left(r_{\mathrm{in}}(\Omega)\sqrt{\lambda} \right)^{-\frac{1}{11}} \] is obtained that reproduces correctly the first two terms in the asymptotics.