Title: The second boundary value problem of the prescribed affine mean curvature equation and related linearized Monge-Ampère equation,
Abstract: These lecture notes are concerned with the solvability of the second boundary
value problem of the prescribed affine mean curvature equation and related
regularity theory of the Monge-Ampère and linearized Monge-Ampère
equations. The prescribed affine mean curvature equation is a fully nonlinear,
fourth order, geometric partial differential equation of the following form
$$\sum_{i, j=1}^n U^{ij}\frac{\partial^2}{\partial
{x_i}\partial{x_j}}\left[(\det D^2 u)^{-\frac{n+1}{n+2}}\right]=f$$ where
$(U^{ij})$ is the cofactor matrix of the Hessian matrix $D^2 u$ of a locally
uniformly convex function $u$. Its variant is related to the problem of finding
Kähler metrics of constant scalar curvature in complex geometry. We first
introduce the background of the prescribed affine mean curvature equation which
can be viewed as a coupled system of Monge-Ampère and linearized
Monge-Ampère equations. Then we state key open problems and present the
solution of the second boundary value problem that prescribes the boundary
values of the solution $u$ and its Hessian determinant $\det D^2 u$. Its proof
uses important tools from the boundary regularity theory of the Monge-Ampère
and linearized Monge-Ampère equations that we will present in the lecture
notes.