Work

Morrey's problem in $\mathbb{R}^{2 \times 4}$ and $\mathbb{R}^{3 \times 3}_\mathrm{sym}$

Year: 2026

Type: preprint

Abstract: We find an explicit rank-one convex non-quasiconvex integrand in $\mathbb{R}^{2\times 4}$: to falsify the quasiconvexity inequality, we exhibit a map $\mathbb{T}^4\to \mathbb{R}^2$ with $12$ non-zero Fourier modes. In fact, this map is obtained from a scalar potential, so we also find a rank one convex integrand in $\mathbb{R}^{4\times 4}_\text{sym}$ which is not quasiconvex. These examples are obtained by transpositions and restrictions of Grabovsky's example of a rank-one convex, non-quasiconvex integrand in $\mathbb{R}^{8 \times 2}.$ We also modify Šverák's example to construct a rank-one convex non-quasiconvex integrand in $\mathbb{R}^{3\times 3}_\text{sym}$.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2608.12298

Authors: Agazzi, Andrea, Bruno, Giuseppe, Guerra, André, Pasqualotto, Federico

Institutions: University of Bern, Bern University of Applied Sciences, University of Cambridge, University of California San Diego

Topic: Elasticity and Material Modeling

Subfield: Biomedical Engineering

Field: Engineering

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, American Mathematical Society, Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung

OpenAlex ID: W7202350213

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