Harnack inequality for the linearized parabolic Monge-Ampère equation

Abstract:

In this paper we prove the Harnack inequality for nonnegative solutions of the linearized parabolic Monge-Ampère equation \[ u_{t}-\text {tr}((D^{2}\phi (x))^{-1}D^{2}u)=0\] on parabolic sections associated with $\phi (x)$, under the assumption that the Monge-Ampère measure generated by $\phi$ satisfies the doubling condition on sections and the uniform continuity condition with respect to Lebesgue measure. The theory established is invariant under the group $AT(n)\times AT(1)$, where $AT(n)$ denotes the group of all invertible affine transformations on ${\mathbf {R}}^{n}$.