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

In this paper, we consider an eigenvalue problem for mixed local-nonlocal Laplacian $$\mathcal{L}^{a,b}_{\Om}:=-aΔ+b(-Δ)^s,\,a>0,\,b\in\mathbb{R},\,s\in (0,1),$$ 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 $C_E\geq 1$ is the constant of the continuous embedding $H_{0}^{1}(\Om)\subset H_{0}^{s}(\Om)$. In this setting, we also derive the Berezin-Li-Yau inequality, which explicitly depends on the constant $C_E$.