The Neumann problem for singular fully nonlinear operators

TL;DR

This paper studies the Neumann boundary value problem for a class of singular, fully nonlinear elliptic operators, establishing principal eigenvalues, regularity, and existence results in smooth bounded domains.

Contribution

It introduces a new concept of principal eigenvalue for these operators and characterizes it via the maximum principle, along with proving regularity and existence of solutions.

Findings

Principal eigenvalue characterized through maximum principle

Lipschitz regularity of solutions established

Existence and uniqueness results proven

Abstract

We consider the Neumann problem in $C^2$ bounded domains for fully nonlinear second order operators which are elliptic, homogenous with lower order terms. Inspired by \cite{bnv}, we define the concept of principal eigenvalue and we characterize it through the maximum principle. Moreover, Lipschitz regularity, uniqueness and existence results for solutions of the Neumann problem are given.