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 \[ u t − tr ( ( D 2 ϕ ( x ) ) − 1 D 2 u ) = 0 u_{t}-\text {tr}((D^{2}\phi (x))^{-1}D^{2}u)=0 \] on parabolic sections associated with ϕ ( x ) \phi (x) , under the assumption that the Monge-Ampère measure generated by ϕ \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 A T ( n ) × A T ( 1 ) AT(n)\times AT(1) , where A T ( n ) AT(n) denotes the group of all invertible affine transformations on R n {\mathbf {R}}^{n} .