Variational competition between 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 $L^2$ 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 Föppl-von-Kármán energy.

Figures (5)

• Figure 1: The elastic sheets in its reference configuration.
• Figure 2: The conically deformed sheet, satisfying the right boundary conditions with zero membrane and infinite bending energy.
• Figure 3: The approximately one-dimensional gradient image $\nabla v(S)$.
• Figure 4: The preimage of $(0,\infty)e_\varphi$ in $\tilde{S}$ under $\nabla v$ contains one connected component that starts at $e_\varphi$ and exits the domain at $y_\varphi$.
• Figure 5: If there are two preimages of angles $l_\varphi,l_{\tilde{\varphi}}$ such that a) they do not deviate much from the rays connecting the origin with $e_\varphi,e_{\tilde{\varphi}}$ respectively for radii larger than $\rho$ and b) $|\nabla v|$ is close to 1 on these curves, then the bending energy density $|\nabla^2 v|^2$ at radius $\rho$ is bounded from below, since $\nabla v\approx e_\varphi$ on $l_\varphi$ and $\nabla v\approx e_{\tilde{\varphi}}$ on $l_{\tilde{\varphi}}$.

Theorems & Definitions (19)

• Theorem 1
• Remark 1
• Corollary 1
• proof
• Remark 2
• Lemma 1
• proof
• Definition 1
• Lemma 2
• proof