Regularity and uniqueness of the first eigenfunction for singular fully non linear operators

TL;DR

This paper establishes the regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators, demonstrating that solutions are $C^{1,eta}$ and eigenvalues are simple.

Contribution

It proves regularity and simplicity of the principal eigenvalues for singular fully nonlinear PDEs, extending understanding of their spectral properties.

Findings

Solutions are $C^{1,eta}$ regular.

Principal eigenvalues are simple.

Regularity aids in proving eigenvalue simplicity.

Abstract

In this article we prove that solutions of singular fully nonlinear partial differential equations are $C^{1,\beta}$. We also prove the simplicity of the principal eigenvalues for the Dirichlet Problem associated to these operators using that regularity, a strict comparison principle and Sard's theorem.