Monge Amp\`ere functionals and the second boundary value problem

TL;DR

This paper extends the solvability of a Monge-Ampère boundary value problem to arbitrary right hand sides in higher dimensions, enabling new geometric and functional analysis applications.

Contribution

It removes previous restrictions on the right hand side, solving the second boundary value problem for Monge-Ampère functionals in dimensions n ≥ 2.

Findings

Solved the second boundary value problem with arbitrary right hand side

Connected the problem to properness of a convex functional

Extended results to the case n=1

Abstract

We consider a Monge-Amp\`ere functional and its corresponding second boundary value problem, a nonlinear fourth order PDE with two Dirichlet boundary conditions. This problem was solved by Trudinger-Wang and Le under the assumption that the right hand side of the equation is nonpositive. We remove this assumption, to settle the case of the second boundary value problem with arbitrary right hand side, in dimensions $n \ge 2$. In particular, this shows that one can prescribe the affine mean curvature of the graph of a convex function with Dirichlet boundary conditions on the function and the determinant of its Hessian. We relate our results, and the case of $n=1$, to a notion of properness for a certain functional on the set of convex functions.