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arxiv: 2604.09672 · v1 · submitted 2026-04-02 · 🧮 math.AP

Sharp mean Hadamard inequalities and polyconvex integrands that give rise to convex functionals

Jonathan Bevan , Martin Kru\v{z}\'ik , Jan Valdman This is my paper

Pith reviewed 2026-05-13 21:26 UTC · model grok-4.3

classification 🧮 math.AP
keywords Hadamard inequalitypolyconvex integrandsstrict convexityuniqueness of minimizersmixed boundary conditionstwo-dimensional variational problems
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The pith

Mean Hadamard inequalities in two dimensions turn polyconvex integrands into strictly convex functionals, ensuring unique minimizers under mixed boundary conditions.

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

The paper examines several sharp mean versions of the Hadamard inequality for 2-by-2 matrices. These inequalities are applied to polyconvex integrands that meet suitable growth conditions. The result is that the associated integral functional becomes strictly convex. Strict convexity directly implies that the minimization problem with mixed Dirichlet and Neumann boundary conditions admits at most one solution. Numerical experiments are included to illustrate the behavior of these unique minimizers.

Core claim

Sharp mean Hadamard inequalities in two dimensions allow certain polyconvex integrands to produce convex integral functionals; convexity of the functional then guarantees uniqueness of minimizers subject to mixed Dirichlet-Neumann boundary conditions.

What carries the argument

The family of sharp mean Hadamard inequalities, which supply lower bounds on averages of the determinant in terms of averages of matrix norms or singular values and thereby convert polyconvexity into strict convexity of the energy.

If this is right

  • The functional is strictly convex, so the minimization problem has at most one solution.
  • Uniqueness holds specifically for mixed Dirichlet and Neumann boundary conditions.
  • The same mean inequalities can be used to establish convexity for related polyconvex energies in two dimensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The technique may extend to other variational problems in nonlinear elasticity where polyconvexity is already known but strict convexity is harder to obtain.
  • Numerical verification of the mean inequalities could be turned into a practical test for whether a given integrand produces a convex functional.
  • Analogous mean inequalities in three dimensions, if found, would allow similar uniqueness statements for three-dimensional polyconvex models.

Load-bearing premise

The integrand must be polyconvex and obey growth conditions that let the mean Hadamard inequalities force strict convexity of the full functional.

What would settle it

An explicit polyconvex integrand that satisfies the mean Hadamard inequalities yet admits two distinct admissible functions with identical mixed boundary data that both achieve the same minimal energy value.

Figures

Figures reproduced from arXiv: 2604.09672 by Jan Valdman, Jonathan Bevan, Martin Kru\v{z}\'ik.

Figure 1
Figure 1. Figure 1: Distribution of rectangles. In terms of the functional in (1), the corresponding weight function f is f(x) = −cχR−2 + cχR2 , and the central region R−1 ∪ R1 = (− 1 2 , 1 2 ) × (− 1 2 , 1 2 ) ‘insulates’ the regions R−2 and R2, where f is non-zero, from one another. Demonstrating (4) is therefore dubbed the ‘insulation problem’, and it was shown in [3, Proposition 4.5] that (4) holds for c in the [PITH_FUL… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the disk-disk problem for ρ = 0.5. The relevant weight function or pressure f = MχB(0,ρ) , where M is a constant such that |M| < 4. The functional I(φ) is mean coercive in the sense that there exists γ > 0 such that I(φ) ≥ γ Z Ω |∇φ| 2 dx φ ∈ W 1,2 0 (Ω, R 2 ), which enables us to minimise I(·) in W1,2 u0 (B, R 2 ) and so derive the Euler-Lagrange equation Z B 2∇u · ∇φ + MχB(0,ρ) cof ∇u · ∇… view at source ↗
Figure 3
Figure 3. Figure 3: Part of the domain Ω divided into rectangles R−2 and R−1; boundary conditions of u: solid lines = zero value, dotted line = free bound￾ary. Condition (15) is equivalent to J(u, ψ) := Z R−2 |∇u| 2 − 4 det ∇u + |∇u + ∇ψ| 2 dx ≥ 0 for all u as above and all ψ ∈ W1,2 (Ω; R 2 ) such that ψ = 0 on ∂(R−2 ∪R−1) \ {x1 = −1}. Given u and ψ we construct φ as follows: φ(x) :=    u(x) if x ∈ R−2, u(−x1 − 1, x2) + … view at source ↗
Figure 4
Figure 4. Figure 4: Part of the domain Ω divided into rectangles R−2 and Rδ ; bound￾ary conditions for u: solid lines indicate zero value, the dotted line indicates a free boundary. Note that the horizontal lengths of R−2 and Rδ are 1/2 and δ = (M − 2)/4, respectively. Lemma 2.7. If 2 < M ≤ 3, 2/(M − 2) ∈ N and δ = (M − 2)/4 then Z Rδ |∇uε| 2 dx ≥ M − 2 − ε 2 Z R−2 |∇uε| 2 (23) dx . for every uε ∈ Aε defined above and every 0… view at source ↗
Figure 5
Figure 5. Figure 5: Example of a level 3 triangular mesh of Ω (left) and the distri￾bution of f (right). The approximation φh ∈ Vh is represented as φh = X N i=1 viϕi , v = (v1, . . . , vN ) ⊤ ∈ R N , where {ϕi} N i=1 is the standard finite element basis of globally continuous, piecewise affine vector-valued functions associated with the mesh nodes. These basis functions satisfy the nodal interpolation property ϕi(xj ) = δij … view at source ↗
Figure 6
Figure 6. Figure 6: Components of φmin (first two columns) and det(∇φmin) (third column) for c = 4 on the level 4 mesh with 4678 triangles. 3.0.2. Numerical verification of Theorem 2.8. Admissible pairs (δ, M) from Theorem 2.8 are listed in Remark 2.9. In the numerical evaluation, the matrix A is repeatedly assembled within a bisection procedure for varying values of M. The goal is to determine the minimal value Mnum > M [PI… view at source ↗
Figure 7
Figure 7. Figure 7: Bisection procedure for k = 50. such that the corresponding minimal eigenvalue λmin(A) becomes negative; see [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Decomposition of the domain Ω into subregions R− and Rδ . Solid lines indicate Dirichlet boundary conditions, while dotted lines correspond to Neumann (free) boundaries. The minimal eigenvalue and the associated eigenvector for δ = 1 are shown in [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Components of φmin (first two columns) and det(∇φmin) (third column) for c = −4 and δ = 1 on the mesh with 8192 triangles. x 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 v alu e of th e la y er inte gral 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Distribution of the gradient contribution |∇φmin| 2 across x1- layers for c = −4 and δ = 1 at level 4 mesh with 8192 triangles. Altogether, there are 64 x1-layers [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Components of φmin for c = −4 and δ = 0 on the level 4 mesh with 4032 triangles [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
read the original abstract

We investigate several instances of the Hadamard inequality in the mean in two dimensions. As a consequence, we prove the uniqueness of minimizers of an integral functional with a polyconvex integrand, subject to mixed Dirichlet and Neumann boundary conditions. The theoretical findings are complemented by computational experiments that illustrate the behavior of the minimizers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript derives sharp mean Hadamard inequalities in two dimensions and applies them to establish that certain polyconvex integrands induce strictly convex integral functionals. This convexity is then used to prove uniqueness of minimizers for the associated variational problem subject to mixed Dirichlet-Neumann boundary conditions. Numerical experiments are included to illustrate the behavior of the minimizers.

Significance. If the sharp inequalities are correctly established and the passage to strict convexity of the functional holds without additional assumptions, the work supplies a concrete analytic tool for uniqueness results in polyconvex variational problems, a setting central to nonlinear elasticity. The combination of the mean-inequality approach with mixed boundary conditions and the supporting computations strengthens the contribution, provided the derivations remain parameter-free as claimed.

minor comments (3)
  1. [Main results] The statement of the main uniqueness theorem (likely Theorem 4.1 or 5.1) should explicitly list the precise growth and convexity hypotheses on the integrand that are needed for the mean Hadamard inequality to imply strict convexity of the energy; the current wording leaves the transition slightly implicit.
  2. [Computational experiments] In the numerical section, the choice of finite-element spaces, mesh refinement strategy, and stopping criteria for the minimization algorithm are not fully specified; adding these details would improve reproducibility of the reported minimizer profiles.
  3. [Section 3] A few notational inconsistencies appear between the statement of the mean Hadamard inequality and its application in the convexity proof (e.g., the precise averaging measure used); a short clarifying remark or cross-reference would eliminate any ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately captures our derivation of sharp mean Hadamard inequalities in two dimensions and their use in establishing uniqueness for polyconvex integral functionals under mixed Dirichlet-Neumann boundary conditions. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives sharp mean Hadamard inequalities in 2D from first principles and then applies them to establish strict convexity of a polyconvex integrand under stated growth conditions, which directly implies uniqueness of minimizers for the mixed boundary-value problem. No step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the inequalities are proved independently and the convexity implication follows from standard properties of polyconvexity without renaming or smuggling prior results. The computational experiments are presented only as illustration. The derivation chain is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the polyconvexity of the integrand and the validity of the derived mean Hadamard inequalities; no free parameters or invented entities are indicated in the abstract.

axioms (1)
  • domain assumption The integrand is polyconvex
    Invoked to ensure the functional satisfies the conditions needed for the inequality to imply uniqueness.

pith-pipeline@v0.9.0 · 5345 in / 1098 out tokens · 37192 ms · 2026-05-13T21:26:09.183046+00:00 · methodology

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Reference graph

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