$C^{1,\al}$ regularity of solutions to parabolic Monge-Amp\'ere equations

TL;DR

This paper establishes interior $C^{1, eta}$ regularity for solutions to the parabolic Monge-Ampère equation under certain conditions on the exponent and initial data, highlighting instant regularization effects.

Contribution

It proves new regularity results for viscosity solutions of the parabolic Monge-Ampère equation with bounded measurable coefficients, including instant $C^{1, eta}$ regularity for subcritical exponents.

Findings

Solutions are $C^{1, eta}$ when $p < 1/(n-2)$.

Regularity holds at points where solutions separate from initial data.

Results extend to all $p > 0$ under specific initial data conditions.

Abstract

We study interior $C^{1, \al}$ regularity of viscosity solutions of the parabolic Monge-Amp\'ere equation $$u_t = b(x,t) \ddua,$$ with exponent $p >0$ and with coefficients $b$ which are bounded and measurable. We show that when $p$ is less than the critical power $\frac{1}{n-2}$ then solutions become instantly $C^{1, \al}$ in the interior. Also, we prove the same result for any power $p>0$ at those points where either the solution separates from the initial data, or where the initial data is $C^{1, \beta}$.