Suppose $(M,g)$ is a smooth compact Riemannian manifold of dimension $n\geq 2$ with smooth boundary and denote by $\{\phi_k,\lambda_k\}_{k\in \mathbb N}$ its Dirihclet spectral decomposition for the Laplace-Beltrami operator $-\Delta_g$.
Is there a way to use somewhat elementary arguments to prove the convergence
$$ \sum_{k=1}^{\infty} \frac{1}{\lambda_k^p} <\infty,$$
for sufficiently large $p$ depending on $n$, without using the Weyl's law?
