A family of nonlocal degenerate operators: maximum principles and related properties

TL;DR

This paper studies a class of nonlinear nonlocal degenerate operators related to the fractional Laplacian, exploring maximum principles, eigenvalues, and solution existence for associated boundary value problems.

Contribution

It introduces a new class of operators, analyzes their maximum principles, eigenvalues, and provides existence results for Dirichlet problems.

Findings

Maximum and minimum principles established for the operators

Existence of solutions for Dirichlet problems demonstrated

Representation formulas derived in specific cases

Abstract

We consider a class of fully nonlinear nonlocal degenerate elliptic operators which are modeled on the fractional Laplacian and converge to the truncated Laplacians. We investigate the validity of (strong) maximum and minimum principles, and their relation with suitably defined principal eigenvalues. We also show a Hopf type Lemma, the existence of solutions for the corresponding Dirichlet problem, and representation formulas in some particular cases.