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.

Lecture notes, to appear in Springer Lecture Notes in Mathematics; v2: fixed typos and changed format of the lecture notes

References in corpus (2)

• Interior estimates for solutions of Abreu's equation
• Geometric properties of boundary sections of solutions to the Monge--Ampère equation and applications