Harnack inequality for the linearized parabolic Monge-Ampère equation

Huang, Qingbo

doi:10.1090/S0002-9947-99-02142-X

Harnack inequality for the linearized parabolic Monge-Ampère equation
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Trans. Amer. Math. Soc. 351 (1999), 2025-2054 Request permission

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 ${\mathbf {R}}^{n}$.

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