Analysis & PDE Vol. 6, No. 5, 2013

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Some minimization problems in the class of convex functions with prescribed determinant

Nam Q. Le and Ovidiu Savin

Vol. 6 (2013), No. 5, 1025–1050

DOI: 10.2140/apde.2013.6.1025

Abstract

We consider minimizers of linear functionals of the type

L (u) = \int_{\partial Ω} u d σ - \int_{Ω} u d x

in the class of convex functions $u$ with prescribed determinant $det D^{2} u = f$ .

We obtain compactness properties for such minimizers and discuss their regularity in two dimensions.

Keywords

boundary regularity, convex minimizer, fourth-order elliptic equation, prescribed determinant

Mathematical Subject Classification 2010

Primary: 35J96

Secondary: 35J66

Milestones

Received: 5 April 2012

Revised: 12 December 2012

Accepted: 28 February 2013

Published: 3 November 2013

Authors

Nam Q. Le
Department of Mathematics Columbia University New York, NY 10027 United States

Ovidiu Savin
Department of Mathematics Columbia University New York, NY 10027 United States