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

TL;DR

This paper establishes Berezin-Li-Yau inequalities for a mixed local-nonlocal Laplacian eigenvalue problem with Dirichlet boundary conditions, unifying classical and fractional bounds and extending to cases with negative coefficients.

Contribution

It introduces a unified inequality framework for mixed local-nonlocal Laplacians, covering positive and certain negative coefficient cases, with explicit dependence on embedding constants.

Findings

Derived lower bounds for eigenvalues in the positive coefficient case.

Extended inequalities to cases with negative coefficients depending on embedding constants.

Unified classical and fractional Berezin-Li-Yau inequalities.

Abstract

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 $C_E$.