# Regularity properties for a class of non-uniformly elliptic Isaacs operators

## TL;DR

This paper studies a degenerate elliptic Isaacs operator formed by the sum of the minimum and maximum eigenvalues of the Hessian, establishing key regularity and qualitative properties despite its nonlinearity and degeneracy.

## Contribution

It proves maximum principles, comparison principles, ABP and Harnack inequalities, Liouville theorems, and existence and uniqueness results for a class of non-uniformly elliptic Isaacs operators.

## Key findings

- Operator satisfies maximum and comparison principles.
- Establishes ABP and Harnack inequalities for solutions.
- Proves existence, uniqueness, and Hölder regularity of solutions.

## Abstract

We consider the elliptic differential operator defined as the sum of the minimum and the maximum eigenvalue of the Hessian matrix, which can be viewed as a degenerate elliptic Isaacs operator, in dimension larger than two. Despite of nonlinearity, degeneracy, non-concavity and non-convexity, such operator generally enjoys the qualitative properties of the Laplace operator, as for instance maximum and comparison principles, ABP and Harnack inequalities, Liouville theorems for subsolutions or supersolutions. Existence and uniqueness for the Dirichlet problem are also proved as well as local and global Holder estimates for viscosity solutions. All results are discussed for a more general class of weighted partial trace operators.