Non-polyconvex $Q$-integrands with lower semicontinuous energies

TL;DR

This paper constructs specific measures demonstrating the limits of approximation by Gaussian images of Lipschitz graphs, leading to the discovery of non-polyconvex integrands with lower semicontinuous energies in Sobolev spaces.

Contribution

It introduces geometric obstructions to approximation by Gaussian images, proving the sharpness of previous density results and constructing non-polyconvex integrands with lower semicontinuous energies.

Findings

Existence of measures not approximable by Gaussian images for any fixed Q.

Extension of geometric obstruction to higher dimensions.

Construction of non-polyconvex integrands with weakly lower semicontinuous energies.

Abstract

We construct a positive measure on the space of positively oriented $2$-vectors in $\mathbb{R}^4$, whose barycenter is a simple $2$-vector, yet which cannot be approximated by weighted Gaussian images of Lipschitz $Q$-graphs for any fixed $Q \in \mathbb{N}$. The construction extends to positively oriented $m$-vectors in $\mathbb{R}^n$ whenever $n-2 \ge m\geq 2$. This geometric obstruction implies that the approximation result established in [Arch. Ration. Mech. Anal., 2025] is sharp: all $Q \in \mathbb{N}$ are indeed necessary to ensure the density of weighted Gaussian images of Lipschitz multigraphs in the space of positive measures with simple barycenter. As an application, we prove that for every $Q\geq 1$ and $p\ge 2$ there exists a non-polyconvex $Q$-integrand whose associated energy is weakly lower semicontinuous in $W^{1,p}$. This also provides new insight into the question posed in [Arch. Ration. Mech. Anal., 2025, Remark 1.14].