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Harnack inequality for the linearized parabolic Monge-Ampère equation

Author(s): Qingbo Huang
Journal: Trans. Amer. Math. Soc. 351 (1999), 2025-2054.
MSC (1991): Primary 35K10; Secondary 35B45
Posted: January 27, 1999
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Abstract: In this paper we prove the Harnack inequality for nonnegative solutions of the linearized parabolic Monge-Ampère equation

\begin{displaymath}u_{t}-\text{tr}((D^{2}\phi (x))^{-1}D^{2}u)=0\end{displaymath}

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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Additional Information:

Qingbo Huang
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
Address at time of publication: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
Email: qhuang@math.utexas.edu

DOI: 10.1090/S0002-9947-99-02142-X
PII: S 0002-9947(99)02142-X
Keywords: Harnack inequality, affine invariant, linear parabolic Monge-Amp\`{e}re equation, section
Received by editor(s): December 15, 1996
Received by editor(s) in revised form: May 13, 1997
Posted: January 27, 1999
Copyright of article: Copyright 1999, American Mathematical Society

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