$W^{4, p}$ solution to the second boundary value problem of the prescribed affine mean curvature and Abreu's equations

Abstract

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.