Non-polyconvex $Q$-integrands with lower semicontinuous energies
Daniele De Gennaro, Antonio De Rosa
TL;DR
The work addresses ellipticity in the $Q$-valued variational setting by constructing a geometric obstruction: a measure on positively oriented $2$-vectors in $ eals^4$ with a simple barycenter that cannot be realized as a weak-* limit of Gaussian images of Lipschitz $Q$-graphs at fixed $Q$. This obstruction sharpens the De Rosa–Lei–Young density result, showing that allowing all $Q$ is necessary for density, and it extends to higher dimensions $(m,n)$. Leveraging this, the authors build a non-polyconvex $Q$-integrand $A$ whose associated energy is weakly lower semicontinuous in $W^{1,p}$ for every $p\ge 2$, revealing that lower semicontinuity does not imply polyconvexity in the $Q$-valued setting. The paper develops a robust framework combining currents, multivalued maps, and an envelope construction for $Q$-integrands, and provides an almost-polyhedral approximation result to underpin the semicontinuity analysis. Collectively, these contributions offer new insight into the ellipticity landscape of anisotropic variational problems for $Q$-valued graphs and broaden the understanding of when lower semicontinuity can be achieved without polyconvexity.
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].