$C^{1+\alpha}$-regularity of viscosity solutions of general nonlinear parabolic equations

TL;DR

This paper proves that solutions to a broad class of nonlinear parabolic equations are locally $C^{1+eta}$ regular, even with minimal regularity assumptions on the equation's coefficients, extending previous regularity results.

Contribution

It establishes $C^{1+eta}$ regularity for viscosity solutions under very general conditions, allowing for measurable time dependence and small spatial discontinuities in the principal part.

Findings

Solutions are in $C^{1+eta}_{loc}$ under minimal assumptions.

No Lipschitz continuity in $v,Dv$ needed for $H$.

Regularity holds despite discontinuities in the principal part.

Abstract

We investigate the $C^{1+\alpha}$-regularity of solutions of parabolic equations $\partial_{t}v+H(v,Dv,D^{2}v,t,x)=0$. Our main result says that under rather general assumptions there exist $C$-viscosity and $L_{p}$-viscosity solutions which are in $C^{1+\alpha}_{loc}$. We allow $H$ to be just measurable in $t$ and for its principal part to have sufficiently small discontinuities as a function of~$x$. No Lipschitz continuity of $H$ with respect to $v,Dv$ is required.