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Morrey's problem in $\mathbb{R}^{2 \times 4}$ and $\mathbb{R}^{3 \times 3}_\mathrm{sym}$

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper constructs explicit rank-one convex integrands in $\mathbb{R}^{2\times4}$, $\mathbb{R}^{4\times4}_{\mathrm{sym}}$, and $\mathbb{R}^{3\times3}_{\mathrm{sym}}$ that fail quasiconvexity, giving the first Hessian counterexamples on…

desk verdict First explicit rank-one convex non-quasiconvex integrand in R^{2x4} and first Hessian counterexamples in symmetric spaces—genuinely new, but one unverified computer-algebra identity carries the whole construction. read the letter →

arxiv 2608.12298 v1 pith:3MA2TODA submitted 2026-08-12 math.AP

Andrea Agazzi , Giuseppe Bruno , André Guerra , Federico Pasqualotto This is my paper · ORCID
classification math.AP MSC 49J4549J1026B25
keywords rank-oneconvexityquasiconvexitycalculusofvariationsHessiansymmetricmatricesFouriermodesexplicitcounterexamplescalarpotential
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper constructs explicit functions where convexity along rank-one directions holds, yet the stronger quasiconvexity inequality—the one that underlies lower semicontinuity in the calculus of variations—fails. The main new example is an integrand on $\mathbb{R}^{2\times4}$, obtained by restricting and transposing a known rank-one convex non-quasiconvex integrand on $\mathbb{R}^{8\times2}$; a test map with 12 Fourier modes falsifies quasiconvexity at the origin. Because that map is the gradient of a scalar potential, the same mechanism yields integrands on symmetric $4\times4$ and $3\times3$ matrices that are convex along symmetric rank-one directions but not Hessian-quasiconvex. These are the first such explicit examples on symmetric matrices, while the original $2\times2$ square case remains open.

What carries the argument

The base object is the integrand $G(\xi,\eta)=\sqrt{\|\xi\|^{2}\|\eta\|^{2}-|\langle\xi,\eta\rangle|^{2}}$ on pairs of quaternionic vectors, identified with $\mathbb{R}^{8\times2}$; it is homogeneous, rank-one convex, and non-quasiconvex. The paper restricts it to purely imaginary quaternionic entries and transposes it to $\mathbb{R}^{2\times6}$. The central new device is a scalar potential $\psi_{\tau}$ on $\mathbb{T}^{4}$ with 12 cosine modes, chosen so that, with $\varphi_{\tau}=(\partial_{z_1}\psi_{\tau},-\partial_{y_1}\psi_{\tau})$, the expansion takes the form $G_{\mathrm{im}}^{T}(M+\varepsilon D\varphi_{\tau})^{2}=(4-\varepsilon^{2}S_{\tau})^{2}+\varepsilon^{4}E_{\tau}$. The $\varepsilon^{2}$ coefficient $S_{\tau}$ is a null Lagrangian—a function whose integral has zero average over the torus—while the $\varepsilon^{4}$ coefficient $E_{\tau}$ has torus average $96\tau(17\tau-2)$, verified symbolically; at $\tau=1/12$ this average is negative, so the fourth-order term drives the total integral down. For the symmetric $3\times3$ case, the machinery changes: four rank-one matrices $\xi_j\otimes\xi_j$ span a subspace where the integrand $-r_1r_2r_3r_4$ is convex along symmetric rank-one lines, and a four-mode trigonometric test map gives the negative variation.

What would settle it

Take the explicit 12-mode potential $\psi_{1/12}$ from (3.6), form $E_{1/12}$ from (3.4), and compute its average over $\mathbb{T}^4$ with an independent computer algebra system or high-precision quadrature; it must equal $-14/3$. Equivalently, evaluate $\int G_{\mathrm{im}}^{T}(M+\varepsilon D\varphi_{1/12})\,dx$ for several small $\varepsilon$ and check that the $\varepsilon^{4}$ coefficient approaches $-7/12$; any other value would break the strict inequality.

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Extended reading notes

Core claim

The paper claims that the gap between rank-one convexity and quasiconvexity occurs in matrix spaces that were previously unresolved. Specifically, $F_{2,4}(A)=G_{\mathrm{im}}^{T}(M+\iota(A))$ on $\mathbb{R}^{2\times4}$ is rank-one convex but not quasiconvex at $0$: for the explicit 12-mode map $\tilde{\varphi}_{1/12}$ the averaged value of $F_{2,4}(\varepsilon D\tilde{\varphi}_{1/12})$ is strictly below $F_{2,4}(0)$ for all sufficiently small nonzero $\varepsilon$. The same construction, read through a scalar potential, gives $H$ on $\mathbb{R}^{4\times4}_{\mathrm{sym}}$ that is convex along every line $A+t(a\otimes a)$ but fails Hessian quasiconvexity, and a quartic polynomial on $\mathbb{R}^{3\times3}_{\mathrm{sym}}$ with a four-mode test map does the same in dimension three. Along the way, the paper shows that its starting eight-by-two quaternionic example contains the classical three-mode example as a leading-order slice.

Load-bearing premise

The argument stands on a long symbolic computation, reported but not reproduced in the text, that the torus average of $E_{\tau}$ is $96\tau(17\tau-2)$; at $\tau=1/12$ its negative sign is exactly what makes the averaged integrand decrease, so if that identity were wrong the counterexample would fail.

Editorial extensions

If this is right

  • The explicit $\mathbb{R}^{2\times4}$ integrand shows the rank-one convex/quasiconvex gap does not require a very large number of columns; range dimension two is not by itself enough to force equivalence.
  • The scalar-potential construction converts the gradient counterexample into Hessian counterexamples on symmetric $4\times4$ and $3\times3$ matrices, so for scalar second-order variational problems convexity along symmetric rank-one directions does not imply Hessian quasiconvexity.
  • The sign of the fourth-order coefficient $96\tau(17\tau-2)$ is the deciding quantity; the choice $\tau=1/12$ is an explicit nondegenerate direction, and a computer-algebra verification is part of the argument.
  • The construction does not settle the $2\times2$ or $2\times3$ cases; transposition is essential to the low-dimensional examples, and quasiconvexity is not invariant under transposition, so the same route cannot be applied directly to the square case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The 12-mode potential is a perturbation of a 9-mode profile that lies in the kernel of the fourth-order variation. A natural extension is to search for similar nondegenerate perturbations in $\mathbb{R}^{2\times3}$ or $\mathbb{R}^{2\times2}$, where lower-order cancellations might still be arranged while a higher-order term controls the sign.
  • The descent pattern—restrict a high-dimensional quaternionic example to an algebraic subspace, transpose, then find a scalar potential—could be tested on other norm-based integrands; one concrete check is to enumerate Fourier modes satisfying the determinant-zero constraint $\det(k',k'')=0$ and look for negative fourth-order averages in lower dimensions.
  • An independent proof of the 84-coefficient torus average could expose why the coefficient $17\tau-2$ has this particular form and may suggest optimal perturbations for other target spaces.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The manuscript constructs explicit rank-one convex, non-quasiconvex integrands in R^{2×4} and in the spaces of symmetric matrices R^{4×4}_sym and R^{3×3}_sym. The construction is based on Grabovsky's example in R^{8×2}; the authors identify a relationship with Šverák's example, then restrict and transpose to lower dimensions. For the R^{2×4} example, a twelve-mode map derived from a scalar potential produces a negative fourth-order variation whose sign is fixed by a computer-algebra identity. A modified Šverák-type construction yields the symmetric 3×3 example.

Significance. If the computations are correct, these are the first explicit counterexamples to 'rank-one convex implies quasiconvex' in R^{2×4} and in symmetric matrix spaces, resolving open cases in Morrey's problem. The examples are fully explicit and the key algebraic identity is backed by a public Mathematica notebook. The observed relation between Grabovsky's and Šverák's constructions is interesting in its own right and may guide further work on lower-dimensional and symmetric cases.

major comments (2)
  1. [Section 3, Eq. (3.7)] The proof of non-quasiconvexity of G^T_im, and hence of Corollaries 1.3 and 1.4, depends on the identity ∫ Eτ dx = 96τ(17τ−2). The text states that this is obtained with a computer algebra system and refers to a notebook, but does not reproduce the computation or even the structure of the Fourier-coefficient summation. Since the sign of this average is exactly what makes the variation negative, the central claim is not self-contained. Please either include a human-readable derivation, or move the notebook into the submission as supplementary material and add a short verification protocol, for example by listing the 84 scalar coefficients or by showing a symmetry reduction that makes the calculation transparent.
  2. [Section 4, Theorem 1.5] The extension from the subspace L to all of R^{3×3}_sym is dismissed with the phrase 'follows exactly as in [23]'. In the symmetric setting the relevant rank-one directions are matrices a⊗a, and the orthogonal projection of a⊗a onto L is not generally a rank-one matrix; for instance, with a=(1,2,3) the projection has full rank. The argument in [23] may still apply, but the authors should confirm that the same choice of α,λ,μ works in this setting, or give a reference to a symmetric-matrix version of the extension lemma. As written, this is a gap in the proof of Theorem 1.5.
minor comments (4)
  1. [Corollary 1.3 proof] The notation for the restricted map alternates between φ1/12 and φ~1/12 without consistency; please add tildes consistently so the reader can follow which domain is meant.
  2. [Section 2 heading] The heading 'Grabovsky's and Šveráks's examples' contains a typo; it should read 'Šverák's example'.
  3. [Section 3, Eq. (3.7)] It would be helpful to remark that the sign of 96τ(17τ−2) changes at τ=2/17 and that the choice τ=1/12 is one admissible value; a short sentence on the admissible range would aid reproducibility.
  4. [Introduction, AI disclosure] The AI disclosure is transparent, but it would fit more naturally in the acknowledgements than in the main text; this is purely a presentational suggestion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction is explicit, the load-bearing prior art is external, and the one self-cited artifact is a reproducibility notebook rather than an imported theorem.

full rationale

The paper's central claim is a constructive counterexample, not a fitted prediction. Rank-one convexity of the new integrands follows by linear changes of variables from Grabovsky's externally published example [13] and from Sverak's example [23]; non-quasiconvexity is established by explicit Fourier test maps and the displayed inequality (3.9). The parameter tau=1/12 is not a fitted value: equation (3.7) gives the fourth-order average as 96*tau*(17*tau-2), which is negative for every tau in the open interval (0,2/17), so the choice is an open-condition selection, not an optimized fit. The main gap is that identity (3.7) is asserted as a computer-algebra computation and delegated to the authors' Mathematica notebook [1]; this is a verification and reproducibility issue, not circularity, because (3.7) is an independent arithmetic identity that does not assume the conclusion of non-quasiconvexity. The self-citation [1] is not used as an unverified theorem; it is an artifact supplied for checking. No uniqueness theorem from the authors is invoked to force a choice, and no known result is renamed as a new one. Thus there are no circular steps in the derivation chain.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No hidden empirical parameters enter the construction. The only hand-chosen constants are tau and the Sverak perturbation coefficients. All mathematical objects are explicit maps on finite-dimensional matrix spaces, so no new physical or formal entities are postulated.

free parameters (2)
  • tau = 1/12
    Introduced in the scalar potential psi_tau in (3.6) to control the fourth-order Fourier term. The average of E_tau equals 96 tau (17 tau - 2), so tau=1/12 makes the variation negative; any tau in (0,2/17) would work. This is a construction parameter, not a fit to data.
  • alpha, lambda, mu = not specified, chosen as in Sverak [23]
    Used in the extension of f from the 4-dimensional subspace L to full R^{3x3}_sym in Theorem 1.5. The paper does not give explicit values and defers to the parameter choice in Sverak's proof.
assumptions (4)
  • standard math Grabovsky's theorem: G:H^2 x H^2 -> [0,infinity) defined in (1.2) is rank-one convex and non-quasiconvex at I_2.
    Used to establish rank-one convexity of the related integrands F, G_im, and G^T_im in Section 2 and Section 3.
  • standard math Sverak's extension argument [23]: the perturbed functional f(PZ)+alpha|PZ|^4+lambda|PZ|^2|QZ|^2+mu|QZ|^4 is rank-one convex for suitable positive constants.
    Invoked verbatim in Section 4 to extend from the subspace L to all of R^{3x3}_sym; the inequalities are not reproduced in this paper.
  • standard math Determinants are null Lagrangians, so the average of S_tau over T^6 vanishes as stated in (3.5).
    Essential to cancel the second-order term in the quasiconvexity expansion in the proof of Theorem 1.2.
  • standard math Fourier orthogonality on the torus and Jensen's inequality.
    Used in Lemma 2.1 and throughout the Fourier computations to evaluate integrals of products of cosine modes.

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Cite this review

Pith. "Pith review of Morrey's problem in $\mathbb{R}^{2 \times 4}$ and $\mathbb{R}^{3 \times 3}_\mathrm{sym}$." pith.science (2026). https://pith.science/paper/3MA2TODA

@misc{pith2026260812298,
  author       = {Pith},
  title        = {Pith review of: Morrey's problem in $\mathbbR^2 \times 4$ and $\mathbbR^3 \times 3_\mathrmsym$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3MA2TODA}},
  note         = {Machine review of arXiv:2608.12298}
}
abstract

We find an explicit rank-one convex non-quasiconvex integrand in $\mathbb{R}^{2\times 4}$: to falsify the quasiconvexity inequality, we exhibit a map $\mathbb{T}^4\to \mathbb{R}^2$ with $12$ non-zero Fourier modes. In fact, this map is obtained from a scalar potential, so we also find a rank one convex integrand in $\mathbb{R}^{4\times 4}_\text{sym}$ which is not quasiconvex. These examples are obtained by transpositions and restrictions of Grabovsky's example of a rank-one convex, non-quasiconvex integrand in $\mathbb{R}^{8 \times 2}.$ We also modify \v{S}ver\'{a}k's example to construct a rank-one convex non-quasiconvex integrand in $\mathbb{R}^{3\times 3}_\text{sym}$.

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Reference graph

Works this paper leans on

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