Berezin-Li-Yau inequality for mixed local-nonlocal Dirichlet-Laplacian

In this paper, we consider an eigenvalue problem for mixed local-nonlocal Laplacian $${\mathcal{L}}_{\text{Om}}^{a,b}:=-a\mathrm{\Delta }+b(-\mathrm{\Delta }{)}^s,a>0,b\in \mathbb{R},s\in (0,1),$$ with Dirichlet boundary conditions. First, the case $a>0$ and $b>0$ is considered and the Berezin-Li-Yau inequality (lower bounds of the sum of eigenvalues) is established. This inequality is characterised as the maximum of the classical and fractional versions of the Berezin-Li-Yau inequality, and, in particular, yields both the classical and fractional forms of the Berezin-Li-Yau inequality. Next, we consider the case $a>0$ and $-\frac{a}{C_E}<b<0$, where $C_E\ge 1$ is the constant of the continuous embedding $H_0^1(\text{Om})\subset H_0^s(\text{Om})$. In this setting, we also derive the Berezin-Li-Yau inequality, which explicitly depends on the constant $C_E$.