  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
$L^2$ 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 F\"oppl-von-K\'arm\'an energy.
