Variational competition between the full Hessian and its determinant for convex functions

Abstract

We prove upper and lower bounds for a variational functional for convex functions satisfying certain boundary conditions on a sector of the unit ball in two dimensions. The functional contains two terms: The full Hessian and its determinant, where the former is treated as a small perturbation in the space L2 and the latter as the leading-order term, in the negative Sobolev space W−2,2. We point out how this setting is motivated by problems in nonlinear elasticity, and obtain a corollary for a variational problem based on the so-called Föppl-von Kármán energy.