In this paper, we consider an eigenvalue problem for mixed local-nonlocal Laplacian $$\mathcal{ L}^{ a,b}_{ \Om}:=-a\Delta+b(-\Delta)^s,\,a>0,\,b\in\mathbb{ R},\,s\in (0,1),$$ 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 $C_E\geq 1$ 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 $C_E$.
Categories: math.AP math.SP