On the symmetric Morrey problem in three dimensions

Abstract

We study the relations between seven generalized notions of convexity for continuous integrands in the space of symmetric 3× 3 matrices. Here, symmetric quasiconvexity is defined through symmetrized gradients, and symmetric rank-one convexity through symmetrized rank-one lines. We construct a symmetric quasiconvex integrand that is not polyconvex at symmetric matrices, and an integrand that is both symmetric rank-one convex and Hessian quasiconvex but not symmetric quasiconvex. We also establish a transfer principle that converts smooth rank-one convex functions that lack Hessian quasiconvexity into symmetric rank-one convex integrands while preserving this lack of of Hessian quasiconvexity. Applying this scheme to already existing counterexamples yields the existence of symmetric rank-one convex integrands that are not Hessian quasiconvex. Together with known examples, these results determine all pairwise implications among the seven notions considered here.

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BibTeX RIS

BibTeX

@misc{gammer2026symmetric,
  title = {{On the symmetric Morrey problem in three dimensions}},
  author = {Florian Gammer},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35216},
  eprint = {2610.35216},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35216},
}

RIS

TY  - UNPB
TI  - On the symmetric Morrey problem in three dimensions
AU  - Florian Gammer
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35216
UR  - https://arxiv.org/abs/2610.35216
AB  - We study the relations between seven generalized notions of convexity for continuous integrands in the space of symmetric 3× 3 matrices. Here, symmetric quasiconvexity is defined through symmetrized gradients, and symmetric rank-one convexity through symmetrized rank-one lines. We construct a symmetric quasiconvex integrand that is not polyconvex at symmetric matrices, and an integrand that is both symmetric rank-one convex and Hessian quasiconvex but not symmetric quasiconvex. We also establish a transfer principle that converts smooth rank-one convex functions that lack Hessian quasiconvexity into symmetric rank-one convex integrands while preserving this lack of of Hessian quasiconvexity. Applying this scheme to already existing counterexamples yields the existence of symmetric rank-one convex integrands that are not Hessian quasiconvex. Together with known examples, these results determine all pairwise implications among the seven notions considered here.
ER  -

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