# ${\cal C}^{1, \gamma} $ regularity for singular or degenerate fully nonlinear operators and applications

## TL;DR

This paper establishes $ ext{C}^{1,eta}$ regularity for solutions to certain degenerate fully nonlinear elliptic equations with specific Hamiltonian growth conditions, and applies these results to ergodic problems, proving uniqueness of solutions.

## Contribution

It provides new regularity results for degenerate elliptic equations with superlinear and subquadratic Hamiltonians, extending previous work on ergodic problems.

## Key findings

- Proves $ ext{C}^{1,eta}$ regularity for solutions.
- Completes the analysis of ergodic problems related to these equations.
- Shows uniqueness of the ergodic function up to additive constants.

## Abstract

In this note, we prove $\mathcal{C}^{1,\gamma}$ regularity for solutions of some fully nonlinear degenerate elliptic equations with "superlinear" and "subquadratic " Hamiltonian terms. As an application, we complete the results of \cite{BDL1} concerning the associated ergodic problem, proving, among other facts, the uniqueness, up to constants, of the ergodic function.