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A solution to Morrey's problem in $\mathbb{R}^{2\times m}$

This paper constructs homogeneous rank-one convex integrands on $\mathbb{R}^{2\times m}$ that are nowhere quasiconvex for sufficiently large $m$ , thereby providing a solution to Morrey's problem in this specific dimension.

Original authors: Gabriele Cassese

Published 2026-08-05

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CC BY 4.0

Original authors: Gabriele Cassese

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to build the most efficient, stable structure possible out of a pile of strange, flexible materials. In the world of mathematics, this is the job of the Calculus of Variations. It's the branch of math dedicated to finding the "best" shape or path for a system, whether that's a soap bubble minimizing surface tension or a bridge minimizing stress. To do this, mathematicians use a giant formula called an integrand. Think of this integrand as a rulebook that assigns a "cost" or "energy" to every possible way the material can bend or stretch. The goal is to find a shape where the total cost is as low as possible.

But here's the catch: finding that perfect shape is incredibly hard. To guarantee a solution exists, the rulebook (the integrand) needs to follow a very strict, almost magical property called quasiconvexity. This property ensures that if you wiggle the shape slightly, the energy doesn't suddenly drop in a way that breaks the math. However, checking if a rulebook has this property is like trying to taste every single grain of sand on a beach to see if it's salty; it's practically impossible. So, for decades, mathematicians hoped for a shortcut. They wondered if a simpler, easier-to-check property called rank-one convexity (which is like checking if the material is stable when pulled in just one specific direction) was enough to guarantee the magical quasiconvexity. If this were true, it would be a massive time-saver, a "magic key" to unlock solutions for everything from physics to engineering.

This paper, titled "A Solution to Morrey's Problem in R2×m," is the story of how a mathematician named Gabriele Cassese finally tried that magic key and found out it doesn't work. The question, posed back in the 1950s by a genius named Charles Morrey, asked: "If a material is stable in every single straight-line pull, is it automatically stable in every complex, wiggly twist?" For a long time, people thought the answer was "yes." But Cassese proves that for materials with enough dimensions (specifically, when the material gets "wide" enough), the answer is a definitive no. He constructs a specific, bizarre rulebook that passes the simple "straight-line" test perfectly but fails the complex "wiggly" test. It's a counterexample that shatters the hope for a simple shortcut, showing that the universe of these mathematical materials is far more twisted and unpredictable than anyone had hoped.

The Great Shortcut That Wasn't

In the world of these mathematical materials, there are two ways to check if a rulebook is "good." The first, rank-one convexity, is like testing a rubber band by pulling it in a straight line. If it resists snapping and behaves nicely in that one direction, it passes the test. The second, quasiconvexity, is the real boss fight. It asks: "If you take a chunk of this material and wiggle it in every possible chaotic way at once, does it still hold together?"

For decades, mathematicians suspected that passing the straight-line test (rank-one convexity) was enough to guarantee you'd pass the chaotic wiggle test (quasiconvexity). It seemed logical: if it's strong in every straight line, it should be strong everywhere. This was Morrey's Problem. If true, it would mean we could skip the impossible math of checking every wiggle and just check the straight lines.

But in this paper, Cassese builds a "Frankenstein's monster" of a rulebook to prove them wrong. He creates a mathematical object that is perfectly stable when pulled in any straight line (it is rank-one convex) but falls apart the moment you try to wiggle it (it is nowhere quasiconvex).

The "Magic" Construction

How did he do it? Imagine you have a giant, multi-dimensional grid. Cassese didn't just pick a random rulebook; he built one using a clever trick involving martingales (a fancy word for a sequence of random steps, like a drunkard's walk) and Banach spaces (abstract mathematical playgrounds that aren't as nice and round as the ones we usually use).

He constructed a formula that looks like this:
$\text{Cost} = C \times (\text{Part A})^p - (\text{Part B})^p$
Here, "Part A" and "Part B" are two different ways of measuring the shape of the material. The number $C$ is a "tuning knob."

The Straight-Line Test (Rank-One Convexity): Cassese showed that if you pull the material in a straight line, Part A and Part B are locked together by a strict mathematical rule. They move in perfect sync. Because of this, if you pick the knob $C$ high enough, the formula always stays positive. It passes the easy test.
The Wiggle Test (Quasiconvexity): Then, he looked at what happens when you wiggle the material. He found a specific, very wiggly shape (a "test function") where Part B suddenly becomes huge compared to Part A. If the knob $C$ isn't infinitely high, the formula turns negative. The material collapses.

The genius of the paper is finding a "Goldilocks zone" for the knob $C$ . He proved that for materials with a large enough width (specifically, when the dimension $m$ is large enough), there is a sweet spot where the knob is high enough to pass the straight-line test but too low to pass the wiggle test.

The Results: "No" in Many Dimensions

The paper doesn't just find one weird example; it finds a whole family of them.

The Big Dimensions: For any "power" $p$ (which controls how the cost scales), Cassese proves that if the material is wide enough (dimension $m$ is large), you can always build this counterexample. The paper doesn't give a specific number for "large enough" right away, but it proves such a number exists.
The Square Case: He also shows this works for square materials (where the width equals the height, $d \times d$ ) if the square is big enough.
The 4x2 Case: He even manages to squeeze a counterexample into a smaller, 4-by-2 grid, provided the power $p$ isn't exactly 2.

Why This Matters (and Why It Hurts)

This result is a bit of a bummer for anyone hoping for a simple shortcut. It means that rank-one convexity does not imply quasiconvexity. You cannot just check the straight lines and assume the rest is fine. The "magic key" is a fake.

However, this is a huge victory for mathematical truth. It settles a question that has been open since the 1950s. It tells us that the world of these variational integrals is much more complex and "non-linear" than we thought. The paper also touches on a famous unsolved puzzle called the Iwaniec Conjecture, which is about a specific 2-by-2 case. Cassese's work suggests that if a solution to that puzzle exists, it must rely on a very specific, unique feature of that 2-by-2 case, because his general method (which works for larger dimensions) fails there.

In short, Cassese has built a mathematical "impossible object": a material that looks perfectly stable from every straight angle but is secretly a house of cards waiting to collapse. It's a reminder that in the deep world of math, things are rarely as simple as they seem, and sometimes, the only way to know the truth is to build the counterexample yourself.

Technical Summary: A Solution to Morrey's Problem in $\mathbb{R}^{2\times m}$

Problem Statement
The paper addresses Morrey's problem, a central question in the calculus of variations posed in 1952: Does rank-one convexity imply quasiconvexity for integrands $f: \mathbb{R}^{d\times m} \to \mathbb{R}$ ? While Šverák (1992) provided a counterexample in dimensions $d \times m$ where $d \ge 3$ and $m \ge 2$ , the case $d=2$ (specifically $2 \times m$ ) remained open. Furthermore, the paper investigates the "homogeneous Morrey problem," asking whether rank-one convexity implies quasiconvexity at the origin for $p$ -homogeneous integrands ( $f(tA) = |t|^p f(A)$ ). Previous results, such as Kirchheim–Kristensen's theorem, established that for $p=1$ , rank-one convexity implies quasiconvexity at 0, leaving the case $p \in (1, \infty)$ unresolved.

Methodology
The author constructs counterexamples using a "threshold-separation scheme" based on integrands of the form:
$f_{C,p}(A) = C^p \|P(A)\|_X^p - \|Q(A)\|_X^p$
where $P$ and $Q$ are linear maps from $\mathbb{R}^{d\times m}$ to a Banach space $X$ , and $C$ is a constant. The strategy involves separating the threshold for rank-one convexity ( $C_{rc}$ ) from the threshold for quasiconvexity ( $C_{qc}$ ) at the origin.

Rank-One Convexity ( $C_{rc}$ ): The author utilizes the correspondence between rank-one convexity and martingale transforms. Specifically, rank-one convexity at 0 is equivalent to the condition that $f(0) \le \mathbb{E}[f(M_n)]$ for all rank-one Rademacher martingales $M_n$ . The bound $C_{rc}$ is controlled using UMD (Unconditional Martingale Difference) space theory and Burkholder's inequalities. The construction exploits the non-Hilbertian geometry of the target space $X$ (specifically $\ell_q$ spaces with $q \neq 2$ ) to keep $C_{rc}$ bounded.
Quasiconvexity ( $C_{qc}$ ): The bound $C_{qc}$ is estimated from below by constructing explicit test maps (vector fields) $u$ on the torus or cube. The author employs specific oscillatory test functions (e.g., products of complex conjugates or trigonometric sums) to demonstrate that the ratio $\|Q(\nabla u)\| / \|P(\nabla u)\|$ grows with the dimension $m$ (or $d$ ), eventually exceeding the bound for $C_{rc}$ .
Envelope Construction: Once a constant $C$ is chosen such that $C_{rc} < C < C_{qc}$ , the integrand is not quasiconvex at 0. To ensure the final integrand is rank-one convex everywhere, the author takes the rank-one convex envelope of the constructed function. Proposition 2.3 guarantees that for homogeneous integrands, if the envelope is not quasiconvex at 0, the function is nowhere quasiconvex.

Key Contributions and Results

The paper provides negative solutions to both the original and homogeneous Morrey problems in high dimensions:

Theorem 1.2 (The $2 \times m$ case): For any $p \in (1, \infty)$ , there exists a dimension $m_0(p)$ such that for all $m \ge m_0(p)$ , there exists a $p$ -homogeneous integrand $f: \mathbb{R}^{2\times m} \to \mathbb{R}$ that is rank-one convex but nowhere quasiconvex. The construction relies on the "blockwise Wirtinger pair" and the non-Hilbertian structure of $\ell_q^n$ .
Theorem 1.3 (The square case $d \times d$ ): For any $p \in (1, \infty)$ , there exists $d_0(p)$ such that for all $d \ge d_0(p)$ , there exists a conjugation- and transposition-invariant $p$ -homogeneous integrand on $\mathbb{R}^{d\times d}$ that is rank-one convex but nowhere quasiconvex. This is achieved using the difference between the symmetric and skew-symmetric parts of the gradient in Schatten $q$ -norms.
Theorem 1.4 (The $4 \times 2$ case): For any $p \in (1, \infty) \setminus \{2\}$ , there exists a $p$ -homogeneous integrand on $\mathbb{R}^{4\times 2}$ that is rank-one convex but nowhere quasiconvex. This result is obtained via a specific construction involving linear maps related to the Beurling-Ahlfors transform.

Explicit Constructions
The paper provides explicit examples for specific parameters:

For $p=3$ , an elementary proof is given for the rank-one convexity of a specific integrand involving $\ell_6$ norms.
For the square case with $p=q=4$ , the author constructs a globally rank-one convex homogeneous polynomial of degree 4 that fails to be quasiconvex at 0 for sufficiently large $d$ .
For $4 \times 2$ with $p=4$ , a degree-4 homogeneous polynomial counterexample is explicitly constructed.

Significance and Claims
The paper claims to settle the homogeneous Morrey problem negatively in large dimensions ( $2 \times m$ , $d \times d$ , and $4 \times 2$ ).

Implications: A positive answer to the homogeneous Morrey problem would imply Iwaniec's conjecture regarding the quasiconvexity of a specific integrand related to conformal and anticonformal parts. By providing counterexamples, the paper demonstrates that Iwaniec's conjecture cannot be proven via a general implication from rank-one convexity to quasiconvexity at 0; any proof must rely on specific features of the Iwaniec integrand absent in these counterexamples.
Non-locality: The results imply that quasiconvexity is a non-local condition in $2 \times m$ for large $m$ , as the constructed integrands satisfy the uniform Legendre-Hadamard inequality (strong ellipticity) but fail global quasiconvexity.
Symmetry: The counterexamples in the square case are constructed to be invariant under conjugation and transposition, showing that even strong symmetry assumptions do not force quasiconvexity in high dimensions.

Limitations and Open Questions
The author explicitly notes that the results do not resolve the $2 \times 2$ case. The construction relies on the dimension $m$ being large enough to separate the UMD constants from the quasiconvexity thresholds. The paper conjectures that the homogeneous Morrey problem likely has a positive answer in $2 \times 2$ , and the current method cannot be pushed to disprove it without significantly refined estimates of UMD constants and quasiconvexity thresholds. Additionally, the paper leaves open the conjecture that isotropic (rotationally invariant) rank-one convex integrands might still be quasiconvex at 0.

The Great Shortcut That Wasn't

The "Magic" Construction

The Results: "No" in Many Dimensions

Why This Matters (and Why It Hurts)

Technical Summary: A Solution to Morrey's Problem in R2×m\mathbb{R}^{2\times m}R2×m

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Technical Summary: A Solution to Morrey's Problem in $\mathbb{R}^{2\times m}$