Interpolated time-Hölder regularity of solutions of fully nonlinear parabolic equations

TL;DR

The paper establishes an Evans-Krylov-type interior regularity result for fully nonlinear parabolic Isaacs equations of the form $F(D^2u)-\partial_t u=0$, where $F(M)=\min\{F^{\cap}(M),F^{\cup}(M)\}$ with $F^{\cap}$ concave and $F^{\cup}$ convex and both uniformly elliptic. Using a maximum-principle approach, it combines Krylov-Safonov parabolic Hölder estimates to obtain $\partial_t u$ and $Du$ in $C^{\alpha,\alpha/2}$, freezes time, applies stationary elliptic $C^{2,\alpha}$ estimates for the nonhomogeneous problem, and then uses interpolation to upgrade to $D^2u\in C^{\alpha}_x$ and $\partial_t u\in C^{\alpha/2}_t$, yielding $u\in C^{2+\alpha,1+\alpha/2}$ with a universal exponent $\tilde{\alpha}$. The results extend parabolic regularity theory to nonconvex/nonconcave operators, with corollaries for nonhomogeneous equations with Hölder data and a general abstract framework linking elliptic $C^{2,\alpha}$ estimates to parabolic $C^{2+\alpha,1+\alpha/2}$ regularity.

Abstract

We show interior Schauder estimates for a special class of fully nonlinear parabolic Isaacs equations by the maximum principle, providing an Evans-Krylov result for the model equation $\min\{\inf_βL_βu,\sup_γL_γu\}-\partial_t u=0$, where $L_β,L_γ$ are linear operators with possibly variable Hölder coefficients. We also give a proof of the Evans-Krylov theorem for fully nonlinear uniformly parabolic equations for which a regularity theory of the stationary non-homogeneous equation is available.

Theorems & Definitions (16)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 3.1
• Remark 3.2
• Theorem 3.3: Krylov-Safonov parabolic Hölder regularity
• Theorem 3.4: First-order space-time Hölder regularity
• Theorem 3.5
• Remark 3.6
• Proposition 3.7