A frame-indifferent rank-one convex function in three dimensions which is not quasiconvex
Abstract
We construct a function W:ℝ3×3→ℝ which is frame-indifferent, W(QF)=W(F) for all Q∈SO(3), rank-one convex, but not quasiconvex. In a neighbourhood of SO(3) the function is an explicit polynomial of degree four in the strain FTF−Id; the failure of quasiconvexity occurs at F=Id through a third-order mechanism of Šverák type, with test functions whose gradients are arbitrarily close to Id. The polynomial is obtained from a geometrically linear example, a rank-one convex, non-quasiconvex quartic polynomial of the symmetric part of the gradient, by evaluating it at the strain 1/2(FTF−Id), adding a uniaxial prestress and a cubic correction, and enlarging the quartic term. The local function is extended to all of ℝ3×3 by adding a frame-indifferent polyconvex function which vanishes near SO(3). This work was carried out with the assistance of the AI model Claude Opus 5.5; the proofs of the main results have been formally verified in Lean 4.
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BibTeX
@misc{conti2026frameindifferent,
title = {{A frame-indifferent rank-one convex function in three dimensions which is not quasiconvex}},
author = {Sergio Conti and Georg Dolzmann and Patrick Dondl},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31886},
eprint = {2610.31886},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31886},
}
RIS
TY - UNPB
TI - A frame-indifferent rank-one convex function in three dimensions which is not quasiconvex
AU - Sergio Conti
AU - Georg Dolzmann
AU - Patrick Dondl
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31886
UR - https://arxiv.org/abs/2610.31886
AB - We construct a function W:ℝ^3×3→ℝ which is frame-indifferent, W(QF)=W(F) for all Q∈SO(3), rank-one convex, but not quasiconvex. In a neighbourhood of SO(3) the function is an explicit polynomial of degree four in the strain F^TF−Id; the failure of quasiconvexity occurs at F=Id through a third-order mechanism of Šverák type, with test functions whose gradients are arbitrarily close to Id. The polynomial is obtained from a geometrically linear example, a rank-one convex, non-quasiconvex quartic polynomial of the symmetric part of the gradient, by evaluating it at the strain 1/2(F^TF−Id), adding a uniaxial prestress and a cubic correction, and enlarging the quartic term. The local function is extended to all of ℝ^3×3 by adding a frame-indifferent polyconvex function which vanishes near SO(3). This work was carried out with the assistance of the AI model Claude Opus 5.5; the proofs of the main results have been formally verified in Lean 4.
ER -