A frame-indifferent rank-one convex function in three dimensions which is not quasiconvex

Abstract

We construct a function $W:\mathbb{R}^{3\times3}\to\mathbb{R}$ which is frame-indifferent, $W(QF)=W(F)$ for all $Q\in\mathrm{SO}(3)$, rank-one convex, but not quasiconvex. In a neighbourhood of $\mathrm{SO}(3)$ the function is an explicit polynomial of degree four in the strain $F^TF-\mathrm{Id}$; the failure of quasiconvexity occurs at $F=\mathrm{Id}$ through a third-order mechanism of Šverák type, with test functions whose gradients are arbitrarily close to $\mathrm{Id}$. 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 $\frac12(F^TF-\mathrm{Id})$, adding a uniaxial prestress and a cubic correction, and enlarging the quartic term. The local function is extended to all of $\mathbb{R}^{3\times3}$ by adding a frame-indifferent polyconvex function which vanishes near $\mathrm{SO}(3)$. 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.