  The second boundary value problem of the prescribed affine mean curvature
equation is a nonlinear, fourth order, geometric partial differential equation.
It was introduced by Trudinger and Wang in 2005 in their investigation of the
affine Plateau problem in affine geometry. The previous works of
Trudinger-Wang, Chau-Weinkove and the author solved this global problem in
$W^{4,p}$ under some restrictions on the sign or integrability of the affine
mean curvature. We remove these restrictions in this paper and obtain $W^{4,p}$
solution to the second boundary value problem when the affine mean curvature
belongs to $L^p$ with $p$ greater than the dimension. Our self-contained
analysis also covers the case of Abreu's equation.
