Regularity theory for fully nonlinear equations of porous medium-type

TL;DR

This paper proves global $C^{2,eta}$ regularity for nonnegative viscosity solutions to fully nonlinear porous medium-type equations, overcoming limitations of traditional Schauder estimates.

Contribution

It introduces new techniques to establish $C^{2,eta}$ regularity for porous medium-type equations where Schauder estimates are not applicable.

Findings

Established global $C^{2,eta}$-estimates for solutions.

Developed novel methods for regularity in porous medium equations.

Extended regularity theory beyond classical Schauder estimates.

Abstract

In this paper, we establish the regularity results for nonnegative viscosity solutions to fully nonlinear equations of porous medium-type in bounded domains with the zero Dirichlet boundary condition, to be precise, we prove the global $C^{2,\alpha}$-estimates of viscosity solutions. In many PDE problems, the $C^{2,\alpha}$-estimates have been obtained through Schauder-type estimates. However, the Schauder-type estimates are not applicable to the porous medium-type equations. We provide techniques for handling porous medium-type equations so that the global $C^{2,\alpha}$-estimates can be established.