In this paper we extend previous results on the regularity of solutions of integro-differential parabolic equations. The kernels are non-necessarily symmetric which could be interpreted as a non-local drift with the same order as the diffusion. We provide a growth lemma and a Harnack inequality which can be used to prove higher regularity estimates. [ABSTRACT FROM AUTHOR]
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. [ABSTRACT FROM AUTHOR]