Math Conjectures

Morrey's 2×22\times2 Problem

OPENLandmarkOpen problemProposed 1952 · Standard version

Canonical statement

Is every finite-valued rank-one convex function f:R2×2→Rf:\mathbb R^{2\times2}\to\mathbb R quasiconvex? Here rank-one convexity means convexity of t↦f(A+tB)t\mapsto f(A+tB) whenever rank⁡B=1\operatorname{rank}B=1, while quasiconvexity means f(A)≤∣Ω∣−1∫Ωf(A+Dφ(x)) dxf(A)\le |\Omega|^{-1}\int_\Omega f(A+D\varphi(x))\,dx for every bounded domain Ω⊂R2\Omega\subset\mathbb R^2 and every φ∈Cc∞(Ω;R2)\varphi\in C_c^\infty(\Omega;\mathbb R^2).
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Is every finite-valued rank-one convex function \(f:\mathbb R^{2\times2}\to\mathbb R\) quasiconvex? Here rank-one convexity means convexity of \(t\mapsto f(A+tB)\) whenever \(\operatorname{rank}B=1\), while quasiconvexity means \(f(A)\le |\Omega|^{-1}\int_\Omega f(A+D\varphi(x))\,dx\) for every bounded domain \(\Omega\subset\mathbb R^2\) and every \(\varphi\in C_c^\infty(\Omega;\mathbb R^2)\).
ANAL-CVAR-022:[Morrey1952Quasiconvexity][Sverak1992RankOne][Pedregal2026RankOne]
Analysis and functional analysisPartial differential equations

Notes

Morrey introduced quasiconvexity as the variational condition governing weak lower semicontinuity of multiple integrals and asked how it relates to rank-one convexity [Morrey1952Quasiconvexity]. Every quasiconvex integrand is rank-one convex. The 2×22\times2 problem asks for the converse for finite-valued functions on real 2×22\times2 matrices; its often-used smooth formulation is equivalent by mollification and passage to local limits.

The dimensional boundary is genuine: Šverák constructed a rank-one convex function that is not quasiconvex for 3×23\times2 matrices [Sverak1992RankOne], but the construction does not descend to 2×22\times2. Many positive results require extra symmetry, homogeneity, or restricted classes of matrices, leaving the unrestricted finite-valued case untouched.

A June 2026 revision by Pedregal claims a proof of the 2×22\times2 implication [Pedregal2026RankOne]. Because an earlier version of that manuscript was withdrawn after a gap and the revised argument has not yet received independent end-to-end verification, the claim is tracked as pending rather than as a settled theorem. Operationally, the canonical 2×22\times2 problem remains open.

Proof-claim watch (1)

Morrey 2x2 rank-one convexity problem

Grade C

A renewed full affirmative claim is available, but the 2x2 problem is not independently established.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.