Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 11, pp. 1-8.
Title: Aleksandrov-type estimates for a parabolic Monge-Ampere equation
Author: David Hartenstine (Western Washington Univ., Bellingham, WA, USA)
Abstract:
A classical result of Aleksandrov allows us to estimate the size of
a convex function $u$ at a point $x$ in a bounded domain $\Omega$
in terms of the distance from $x$ to the boundary of $\Omega$ if
$\int_{\Omega} \det D^{2}u \, dx < \infty$. This estimate plays a
prominent role in the existence and regularity theory of the
Monge-Ampere equation. Jerison proved an extension of Aleksandrov's
result that provides a similar estimate, in some cases for which this
integral is infinite. Gutierrez and Huang proved a variant of
the Aleksandrov estimate, relevant to solutions of a parabolic
Monge-Ampere equation. In this paper, we prove Jerison-like
extensions to this parabolic estimate.
Submitted January 12, 2005. Published January 27, 2005.
Math Subject Classifications: 35K55, 35B45, 35D99.
Key Words: Parabolic Monge-Ampere measure; pointwise estimates.