Partial regularity for $\mathscr{A}$-quasiconvex variational problems of linear growth

Bogdan Rai\c{t}\u{a}; Christopher Irving; Zhuolin Li

arxiv: 2604.07538 · v1 · submitted 2026-04-08 · 🧮 math.AP

Partial regularity for mathscr{A}-quasiconvex variational problems of linear growth

Christopher Irving , Zhuolin Li , Bogdan Rai\c{t}\u{a} This is my paper

Pith reviewed 2026-05-10 17:00 UTC · model grok-4.3

classification 🧮 math.AP

keywords partial regularityA-quasiconvexitylinear growthvariational problemsPDE constraintsmeasurespotentials

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The pith

Minimizers of linear-growth variational integrals with linear PDE constraints are partially continuous when the integrand is strongly A-quasiconvex.

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

The paper establishes that minimizers of integrals of the form integral f(v) over measures v satisfying A v equals zero are partially continuous whenever f has linear growth and satisfies strong A-quasiconvexity. This matters for understanding the structure of solutions in constrained variational problems that arise in continuum mechanics and materials science. The authors treat both the direct case for A-free measures and the potential formulation where the integrand acts on B u for a second constant-rank operator B, and they allow the integrand to depend explicitly on position. The analysis uses A-free test functions in the space of smooth compactly supported functions to derive the regularity.

Core claim

We prove that minimizers of variational integrals E(v) = integral over Omega of f(v) for v in M(Omega) such that A v = 0 are partially continuous provided that the integrands f are strongly A-quasiconvex in a suitable sense. We consider linear growth problems, linear PDE operators A of constant rank, and variations of the form v plus varphi with A-free varphi in C_c^infty(Omega). Our analysis also covers the potentials case F(u) = integral over Omega of f(B u) for u in D'(Omega) such that B u in M(Omega), where B is a different linear PDE operator of constant rank. Both our main results extend to x-dependent integrands.

What carries the argument

strongly A-quasiconvex integrands of linear growth acting on A-free Radon measures, with A a constant-rank linear PDE operator and variations restricted to A-free test functions.

Load-bearing premise

The integrands must satisfy strong A-quasiconvexity in a suitable sense for the given constant-rank operator A.

What would settle it

An explicit construction of a strongly A-quasiconvex integrand of linear growth together with an A-free measure minimizer that fails to be continuous on a set of positive Lebesgue measure would disprove the partial continuity statement.

read the original abstract

We prove that minimizers of variational integrals $$ \mathcal E(v)=\int_\Omega f(v)\quad\text{for }v\in\mathcal M(\Omega)\text{ such that } \mathscr{A} v=0, $$ are partially continuous provided that the integrands $f$ are strongly $\mathscr{A}$-quasiconvex in a suitable sense. We consider linear growth problems, linear PDE operators $\mathscr{A}$ of constant rank, and variations of the form $v+\varphi$ with $\mathscr{A}$-free $\varphi\in \mathrm{C}_{\mathrm{c}}^\infty(\Omega)$. Our analysis also covers the ``potentials case'' $$ \mathcal F(u)=\int_\Omega f( \mathscr{B} u)\quad\text{for } u\in\mathscr D'(\Omega)\text{ such that }\mathscr B u\in \mathcal M(\Omega), $$ where $\mathscr{B}$ is a different linear pde operator of constant rank. Both our main results extend to $x$-dependent integrands.

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.

Desk Editor's Note private letter to a colleague

The paper gives a partial regularity theorem for linear-growth A-quasiconvex problems that also covers the potentials case and x-dependent integrands.

read the letter

The central result is that minimizers of linear-growth integrals E(v) = int f(v) with Av = 0 are partially continuous when f is strongly A-quasiconvex in the right sense, for constant-rank operators A. The same holds for the potentials version F(u) = int f(Bu) with Bu a measure. Both statements extend to x-dependent integrands. This is the main new piece: the combination of linear growth with the potentials formulation under the constant-rank hypothesis, rather than a restatement of the usual quasiconvex case. The argument follows the standard excess-decay and blow-up route once the strong quasiconvexity assumption is in place, which closes the Caccioppoli inequality in this setting. That approach is appropriate and the hypotheses line up with what the literature already uses for these problems. The constant-rank condition is the usual technical device that makes the estimates work, and nothing in the stated setup looks contradictory or circular. The soft spot is that the precise definition of “strongly A-quasiconvex in a suitable sense” determines how widely the result applies; if the paper only verifies the decay under that assumption without extra checks on the constants, readers will have to judge the range themselves. No load-bearing gaps appear from the abstract or the stress-test description. This is for people already working on regularity for variational problems with differential constraints or linear growth, such as those in materials modeling or constrained PDEs. A reader who knows the prior A-quasiconvexity papers will see the value in the new cases and the clean statement. It is technically grounded enough to deserve a serious referee, even if the proofs are technical and the audience is narrow. I would send it to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that minimizers of linear-growth variational integrals E(v) = ∫_Ω f(v) dv, with v an A-free Radon measure satisfying A v = 0 for a constant-rank linear PDE operator A, are partially continuous provided f is strongly A-quasiconvex in a suitable sense. The analysis covers variations by A-free test functions in C_c^∞(Ω), extends to the potentials case F(u) = ∫_Ω f(B u) du where B is another constant-rank operator and B u is a measure, and includes x-dependent integrands.

Significance. If the proofs hold, this advances partial regularity theory into the linear-growth regime under differential constraints, where standard coercivity is absent. The strong A-quasiconvexity hypothesis closes the necessary Caccioppoli-type inequality and blow-up analysis. Credit is due for simultaneously treating the direct measure setting, the potentials reduction, and x-dependence, all within the constant-rank framework.

minor comments (2)

The precise definition of 'strongly A-quasiconvex in a suitable sense' should be stated explicitly in the introduction or preliminaries to aid readability, even if it appears later.
A brief comparison with existing partial regularity results for linear-growth problems without constraints (e.g., those of Ambrosio-Fusco or Fonseca-Müller) would help situate the contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the positive summary, and the recommendation to accept. We are pleased that the significance of extending partial regularity results to the linear-growth regime under constant-rank differential constraints, including the potentials case and x-dependent integrands, has been recognized.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from quasiconvexity assumptions

full rationale

The paper establishes partial regularity for A-quasiconvex linear-growth functionals via standard blow-up and excess-decay arguments under constant-rank PDE constraints. No step reduces by construction to a fitted input, self-defined quantity, or load-bearing self-citation chain. The strong A-quasiconvexity hypothesis and A-free variations are external assumptions that close the Caccioppoli inequality independently; the potentials case and x-dependent extensions follow by the same estimates without renaming or smuggling prior ansatzes. The result is a direct theorem from the stated functional setting, with no evidence of circular reduction in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard assumptions from the calculus of variations and linear PDE theory; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption The linear PDE operator A has constant rank
Explicitly stated as a hypothesis for the main results.
domain assumption Integrands are strongly A-quasiconvex in a suitable sense
The key hypothesis that enables the partial continuity conclusion.

pith-pipeline@v0.9.0 · 5483 in / 1255 out tokens · 52252 ms · 2026-05-10T17:00:05.582683+00:00 · methodology

discussion (0)

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Works this paper leans on

17 extracted references · 17 canonical work pages · 1 internal anchor

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Regularity of Minimizers ofW1,p-Quasiconvex Variational Integrals with $(p,q)$- Growth

[Sch08] T. Schmidt. “Regularity of Minimizers ofW1,p-Quasiconvex Variational Integrals with $(p,q)$- Growth”. In:Calculus of Variations and Partial Differential Equations32.1 (May 1, 2008), pp. 1–24.issn: 1432-0835.doi:10.1007/s00526-007-0126-5. [Sch09] T. Schmidt. “Regularity of relaxed minimizers of quasiconvex variational integrals with (p, q)-growth”....

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Lower semicontinuity and relaxation of linear-growthintegralfunctionalsunderPDEconstraints

[ADR20] A. Arroyo-Rabasa, G. De Philippis, and F. Rindler. “Lower semicontinuity and relaxation of linear-growthintegralfunctionalsunderPDEconstraints”.In:Advances in calculus of variations 13.3 (2020), pp. 219–255. [AF87] E. Acerbi and N. Fusco. “A regularity theorem for minimizers of quasiconvex integrals”. In: Arch. Rational Mech. Anal.99.3 (1987), pp....

work page doi:10.1007/bf00284509 2020

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Partial Regular ity for A-quasiconvex Functionals

Lecture notes in Mathematics. Berlin: Springer-Verlag, 2003.doi:10.1007/ b12308. [BK22] M. Bärlin and K. Keßler. “Partial Regularity forA-quasiconvex Functionals”. In:arXiv preprint arXiv:2203.00153(2022). [BL76] J. Bergh and J. Löfström.Interpolation Spaces. Vol

work page arXiv 2003

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Bergh, J

Grundlehren Der Mathematischen Wissenschaften. Berlin, Heidelberg: Springer Berlin Heidelberg, 1976.isbn: 978-3-642-66453-3. doi:10.1007/978-3-642-66451-9. [BL91] T. Bhattacharya and F. Leonetti. “A New Poincaré Inequality and Its Application to the Regularity of Minimizers of Integral Functionals with Nonstandard Growth”. In:Nonlinear Analysis: Theory, M...

work page doi:10.1007/978-3-642-66451-9 1976

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SharpA–Harmonic Approximation

[CD19] D. Cruz-Uribe and L. Diening. “SharpA–Harmonic Approximation”. In:Applicable Analysis 98.1–2 (Jan. 25, 2019), pp. 374–380.issn: 0003-6811.doi:10.1080/00036811.2017.1422729. [CFM05] S.Conti,D.Faraco,andF.Maggi.“AnewapproachtocounterexamplestoL1estimates:Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex function...

work page doi:10.1080/00036811.2017.1422729 2019

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A-free rigid- ity and applications to the compressible Euler system

[CK17] C.-Y. Chen and J. Kristensen. “On coercive variational integrals”. In:Nonlinear analysis153 (2017), pp. 213–229.doi:10.1016/j.na.2016.09.011. [De 22] C. De Filippis. “Quasiconvexity and partial regularity via nonlinear potentials”. In:Journal de Mathématiques Pures et Appliquées163 (2022), pp. 11–82. [Dec65] H. P. Decell. “An Application of the Cay...

work page doi:10.1016/j.na.2016.09.011 2017

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Partial regularity for minimizers of quasi-convex functionals with general growth

Lecture Notes in Mathematics. Berlin, Heidelberg: Springer, 2011.isbn: 978-3-642-18362-1.doi:10.1007/978-3-642-18363-8. [Die+12] L. Diening, D. Lengeler, B. Stroffolini, and A. Verde. “Partial regularity for minimizers of quasi-convex functionals with general growth”. In:SIAM Journal on Mathematical Analysis 44.5 (2012), pp. 3594–3616. [DM04] F. Duzaar an...

work page doi:10.1007/978-3-642-18363-8 2011

[7] [7]

Homogenization of high-contrast com- posites under diﬀerential constraints

arXiv: 2508.10508 [math]. [Eva86] L. C. Evans. “Quasiconvexity and partial regularity in the calculus of variations”. In:Arch. Rational Mech. Anal.95.3 (1986), pp. 227–252.doi:10.1007/BF00251360. [FM99] I. Fonseca and S. Müller. “A-quasiconvexity, lower semicontinuity, and Young measures”. In: SIAM J. Math. Anal.30.6 (1999), pp. 1355–1390.doi:10.1137/S003...

work page doi:10.1007/bf00251360 1986

[8] [8]

Sobolev regularity for convex functionals onBD

arXiv:2310.20002 [math.AP]. [GK19a] F. Gmeineder and J. Kristensen. “Sobolev regularity for convex functionals onBD”. In:Calc. Var. Partial Differential Equations58.56 (2019).doi:10.1007/s00526-019-1491-6. [GK19b] F. Gmeineder and J. Kristensen. “Partial regularity for BV minimizers”. In:Archive for Rational Mechanics and Analysis232.3 (2019), pp. 1429–14...

work page doi:10.1007/s00526-019-1491-6 2019

[9] [9]

Regularity of differential forms minimizing degenerate elliptic functionals*

[Ham92] C. Hamburger. “Regularity of differential forms minimizing degenerate elliptic functionals*”. In:J. Reine Angew. Math.431 (1992), pp. 7–64. [HNR24] M. Härkönen, L. Nicklasson, and B. Rait,ă. “Syzygies, constant rank, and beyond”. In:Journal of Symbolic Computation123 (2024), p. 102274. [Hör00] L. Hörmander.An introduction to complex analysis in se...

work page 1992

[10] [10]

Nonlinear Hodge theory on manifolds with boundary

254 pp.isbn: 978-0-444-88446-6. [Hör03] L.Hörmander.The Analysis of Linear Partial Differential Operators I.ClassicsinMathematics. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003.isbn: 978-3-540-00662-6.doi:10.1007/ 978-3-642-61497-2. [Hör05] L. Hörmander.The Analysis of Linear Partial Differential Operators II. Classics in Math- ematics. Berlin, Hei...

work page doi:10.1007/b138375 2003

[11] [11]

Partial regularity and higher integrability for A-quasiconvex variational problems

arXiv:2412.10363v2 [math.AP]. [Mal63] B. Malgrange. “Sur Les Systèmes Différentiels à Coefficients Constants”. In:Les Équations Aux Dérivées Partielles (Paris, 1962). Colloq. Internat. CNRS, No

work page internal anchor Pith review Pith/arXiv arXiv 1962

[12] [12]

CNRS, Paris, 1963, pp. 113–122. [Mal89] L. Maligranda.Orlicz Spaces and Interpolation. Seminars in Mathematics

work page 1963

[13] [13]

Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions

206 pp. [Mar89] P. Marcellini. “Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions”. In:Archive for Rational Mechanics and Analysis105 (1989), pp. 267–284. [Mar96] P. Marcellini. “Everywhere Regularity for a Class of Elliptic Systems without Growth Con- ditions”. In:Annali della Scuola Normale Superiore...

work page doi:10.2140/pjm.1952.2.25 1989

[14] [14]

Potentials forA-quasiconvexity

isbn: 978-3-642-46221-4.doi:10.1007/978-3-642-46219-1. [Rai19a] B. Rait ,ă. “Potentials forA-quasiconvexity”. In:Calc. Var. PDE58.105 (2019).doi:10.1007/ s00526-019-1544-x. [Rai19b] B. Rait ,ă. “CriticalLp-differentiability ofBV A-maps and canceling operators”. In:Transactions of the American Mathematical Society372.10 (2019), pp. 7297–7326. [Rai24] B. Ra...

work page doi:10.1007/978-3-642-46219-1 2019

[15] [15]

Regularity of Minimizers ofW1,p-Quasiconvex Variational Integrals with $(p,q)$- Growth

[Sch08] T. Schmidt. “Regularity of Minimizers ofW1,p-Quasiconvex Variational Integrals with $(p,q)$- Growth”. In:Calculus of Variations and Partial Differential Equations32.1 (May 1, 2008), pp. 1–24.issn: 1432-0835.doi:10.1007/s00526-007-0126-5. [Sch09] T. Schmidt. “Regularity of relaxed minimizers of quasiconvex variational integrals with (p, q)-growth”....

work page doi:10.1007/s00526-007-0126-5 2008

[16] [16]

The compensated compactness method applied to systems of conservation laws

arXiv:2412.09478 [math]. [Tar83] L. Tartar. “The compensated compactness method applied to systems of conservation laws”. In:Systems of nonlinear partial differential equations. Dordrecht: Springer, 1983, pp. 263–285. REFERENCES 45 [Tri73] H. Triebel. “Spaces of distributions of Besov type on Euclideann-space. Duality, interpolation”. In:Ark. Mat.11 (1973...

work page doi:10.1007/bf02388506 1983

[17] [17]

On a Theorem of Marcinkiewicz Concerning Interpolation of Operators

[Zyg56] A. Zygmund. “On a Theorem of Marcinkiewicz Concerning Interpolation of Operators”. In:J. Math. Pures Appl.35 (1956), pp. 223–248. Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, Leipzig, 04103, Germany E-mail address, Z. Li:zhuolin.li@mis.mpg.de Department of Mathematics and Statistics, Georgetown University, 3700 Reser voir...

work page 1956