Mathematical Research Letters

Volume 22 (2015)

Number 4

Monge-Ampère functionals and the second boundary value problem

Pages: 1005 – 1022



Albert Chau (Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada)

Ben Weinkove (Department of Mathematics, Northwestern University, Evanston, Illinois, U.S.A.)


We consider a Monge-Ampère 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 $n \geqslant 2$. 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.

Accepted 13 January 2015

Published 24 July 2015