Monge-Ampère functionals and the second boundary value problem

Albert Chau, Ben Weinkove

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

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≤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.

Original languageEnglish (US)
Pages (from-to)1005-1022
Number of pages18
JournalMathematical Research Letters
Volume22
Issue number4
DOIs
StatePublished - 2015

ASJC Scopus subject areas

  • Mathematics(all)

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