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

## TL;DR

This paper establishes bounds for a variational functional involving the full Hessian and its determinant for convex functions on a sector, motivated by nonlinear elasticity, with implications for F"oppl-von-Kármán energy.

## Contribution

It introduces a novel variational framework balancing the full Hessian and its determinant, with bounds relevant to elasticity problems.

## Key findings

- Derived upper and lower bounds for the functional.
- Connected the variational problem to nonlinear elasticity models.
- Provided insights into the behavior of convex functions under these bounds.

## 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 $L^2$ 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.