On $C^{1+\alpha}$ regularity of solutions of Isaacs parabolic equations with VMO coefficients

TL;DR

This paper establishes $C^{1+eta}$ regularity for solutions of fully nonlinear parabolic equations with VMO coefficients, expanding understanding of regularity under minimal regularity assumptions on coefficients.

Contribution

It proves $C^{1+eta}$ regularity for solutions of fully nonlinear parabolic equations with VMO coefficients, a significant extension of regularity theory.

Findings

Solutions are in $C^{1+eta}$ for some $eta ext{ in }(0,1)$.

Boundary value problems admit $L_{p}$-viscosity solutions with this regularity.

The equations have a structure with a positive homogeneous second-order part and VMO regularity in space.

Abstract

We prove that boundary value problems for fully nonlinear second-order parabolic equations admit $L_{p}$-viscosity solutions, which are in $C^{1+\alpha}$ for an $\alpha\in(0,1)$. The equations have a special structure that the "main" part containing only second-order derivatives is given by a positive homogeneous function of second-order derivatives and as a function of independent variables it is measurable in the time variable and, so to speak, VMO in spatial variables.