PreprintCC BY 4.0arXiv
A solution to Morrey's problem in ℝ2× m
Abstract
We construct, for any exponent p∈(1,∞), p-homogeneous rank-one convex integrands F: ℝ2× m→ ℝ that are nowhere quasiconvex when m is large. When p is sufficiently close to $4$, such examples can be constructed on ℝ2× 3: for p=4, our example is an explicit quartic polynomial. Related constructions give, for every p, conjugation- and transposition-invariant examples on ℝd× d, and examples on ℝ4× 2 for every p≠ 2.
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BibTeX
@misc{cassese2026solution,
title = {{A solution to Morrey's problem in $\mathbbR^2\times m$}},
author = {Gabriele Cassese},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.03488},
eprint = {2608.03488},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.03488},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A solution to Morrey's problem in ℝ^2× m
AU - Gabriele Cassese
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2608.03488
UR - https://arxiv.org/abs/2608.03488
AB - We construct, for any exponent p∈(1,∞), p-homogeneous rank-one convex integrands F: ℝ^2× m→ ℝ that are nowhere quasiconvex when m is large. When p is sufficiently close to $4$, such examples can be constructed on ℝ^2× 3: for p=4, our example is an explicit quartic polynomial. Related constructions give, for every p, conjugation- and transposition-invariant examples on ℝ^d× d, and examples on ℝ^4× 2 for every p≠ 2.
ER -