The second boundary value problem of the prescribed affine mean curvature equation and related linearized Monge-Amp\`ere 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-Amp\`ere and linearized Monge-Amp\`ere 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 K\"ahler 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-Amp\`ere and linearized Monge-Amp\`ere 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-Amp\`ere and linearized Monge-Amp\`ere equations that we will present in the lecture notes.