H\"older regularity for non-variational porous media type equations

TL;DR

This paper develops a new approach using Krylov-Safonov theory to establish H"older regularity for viscosity solutions of non-variational porous media equations, addressing the challenge of either elliptic or eikonal regimes.

Contribution

It introduces a novel method combining sliding paraboloids and ABP-type estimates to prove regularity in non-variational porous media equations.

Findings

Established H"older continuity for viscosity solutions.

Unified treatment of elliptic and eikonal regimes.

New measure estimate technique for regularity analysis.

Abstract

We present a Krylov-Safonov theory approach for the H\"older regularity of viscosity solutions to non-variational porous media type equations. We explore the peculiarity of this type of problem: either the equation falls in a uniformly elliptic regime or the eikonal mechanism takes care of the regularity. Our techniques are based on sliding paraboloids resulting in an ABP-type measure estimate. By combining such estimates, a diminishing of oscillation property is available, resulting in a regularity control in H\"older spaces.