A short proof of the $C^{0,\alpha}$--regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications

TL;DR

This paper provides a simplified proof of the H"older continuity of viscosity subsolutions for superquadratic elliptic equations, extending previous results and exploring their implications in nonlinear PDE analysis.

Contribution

It offers a new, streamlined proof of existing regularity results and discusses extensions and applications in the context of superquadratic viscosity solutions.

Findings

Viscosity subsolutions are locally H"older continuous.

The regularity results extend to certain boundary conditions.

Applications include improved understanding of nonlinear elliptic equations.

Abstract

Recently I. Capuzzo Dolcetta, F. Leoni and A. Porretta obtain a very surprising regularity result for fully nonlinear, superquadratic, elliptic equations by showing that viscosity subsolutions of such equations are locally H\"older continuous, and even globally if the boundary of the domain is regular enough. The aim of this paper is to provide a simplified proof of their results, together with an interpretation of the regularity phenomena, some extensions and various applications.