TY - JOUR

T1 - Monge-Ampère functionals and the second boundary value problem

AU - Chau, Albert

AU - Weinkove, Ben

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

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U2 - 10.4310/MRL.2015.v22.n4.a3

DO - 10.4310/MRL.2015.v22.n4.a3

M3 - Article

AN - SCOPUS:84937843973

SN - 1073-2780

VL - 22

SP - 1005

EP - 1022

JO - Mathematical Research Letters

JF - Mathematical Research Letters

IS - 4

ER -