Linear theory for a mixed operator with Neumann conditions

TL;DR

This paper develops a linear theory for a new mixed local and nonlocal operator with Neumann boundary conditions, analyzing spectral properties and bounds for solutions, with applications to biological logistic models.

Contribution

It introduces a novel mixed operator with Neumann conditions, studies its spectral properties, and applies the results to biological logistic equations.

Findings

Spectral properties of the mixed operator are characterized.

A global bound for subsolutions is established.

Applications to biological logistic models are demonstrated.

Abstract

We consider here a new type of mixed local and nonlocal equation under suitable Neumann conditions. We discuss the spectral properties associated to a weighted eigenvalue problem and present a global bound for subsolutions. The Neumann condition that we take into account comprises, as a particular case, the one that has been recently introduced in [S. Dipierro, X. Ros-Oton, E. Valdinoci, Rev. Mat. Iberoam. (2017)]. Also, the results that we present here find a natural application to a logistic equation motivated by biological problems that has been recently considered in [S. Dipierro, E. Proietti Lippi, E. Valdinoci, preprint (2020)].