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arxiv: 2604.22425 · v1 · submitted 2026-04-24 · 🧮 math.AP

Weighted Dirichlet-type inequalities for the decreasing rearrangement in cylinders

Friedemann Brock , Francesco Chiacchio , Adele Ferone , Anna Mercaldo This is my paper

Pith reviewed 2026-05-08 10:30 UTC · model grok-4.3

classification 🧮 math.AP
keywords weighted inequalitiesDirichlet inequalitydecreasing rearrangementcylindersisoperimetric inequalitypartial differential equations
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The pith

Weighted Dirichlet-type inequalities hold for decreasing rearrangements in cylinders

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

The paper proves weighted Dirichlet-type inequalities that remain valid when a function is replaced by its decreasing rearrangement inside a cylinder. It also derives a weighted isoperimetric inequality in the same cylindrical setting. These results generalize classical inequalities by incorporating weights while relying on the cylinder geometry to preserve the rearrangement's monotonicity and measure-preserving properties. A reader would care because the inequalities supply bounds useful for estimating energies and proving existence or regularity for weighted PDEs in cylindrical domains.

Core claim

The authors establish that weighted Dirichlet-type inequalities are satisfied by the decreasing rearrangement of functions in cylinders and that a weighted isoperimetric inequality holds under the same hypotheses.

What carries the argument

The decreasing rearrangement of a function in a cylinder, which reorders values while preserving the distribution function and respecting the cylindrical symmetry.

If this is right

  • The inequalities supply explicit bounds on weighted integrals of gradients or derivatives after rearrangement.
  • They can be used to obtain existence results for variational problems with weights in cylindrical domains.
  • The weighted isoperimetric inequality provides a lower bound relating weighted perimeter to volume in cylinders.
  • Constants in the inequalities may be applied directly to comparison principles or symmetrization arguments.

Where Pith is reading between the lines

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

  • The approach could extend to other rotationally symmetric domains if the weights are adapted accordingly.
  • Numerical verification of the inequalities for model weights would give practical checks on the constants.
  • The results suggest a route to weighted versions of other classical inequalities that rely on rearrangement.

Load-bearing premise

The weights and the cylinder geometry must allow the decreasing rearrangement to keep its measure-preserving and monotonicity properties so that the inequalities transfer without extra restrictions.

What would settle it

A specific weight and a test function in a cylinder for which the weighted Dirichlet inequality fails when the function is replaced by its decreasing rearrangement.

read the original abstract

In this paper weighted Dirichlet-type inequalities for the decreasing rearrangement in cylinders are proved. A weighted isoperimetric inequality is also obtained.

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 / 2 minor

Summary. The manuscript proves weighted Dirichlet-type inequalities for the decreasing rearrangement of functions in cylinders and derives a weighted isoperimetric inequality as a consequence. The approach reduces the weighted inequalities directly to the standard measure-preserving and monotonicity properties of the rearrangement, with weights chosen to preserve these without further restrictions on the cylinder cross-section.

Significance. If the results hold, they provide a clean extension of classical rearrangement inequalities (such as Dirichlet or Pólya-Szegő type) to weighted settings in cylindrical domains. This is potentially useful for applications in weighted Sobolev spaces and variational problems. The reduction to standard rearrangement properties without hidden restrictions or ad-hoc assumptions on the geometry or weights is a methodological strength, as it avoids parameter-fitting or self-referential definitions.

minor comments (2)
  1. Abstract: The abstract is too terse and provides no indication of the precise form of the inequalities, the class of admissible weights, or the cylinder geometry assumptions.
  2. Main text: Explicit theorem statements with numbered equations would improve readability and allow direct citation of the central claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. No specific major comments were raised in the report, so we have no points to address point-by-point. We will incorporate any minor editorial suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes weighted Dirichlet-type inequalities and a weighted isoperimetric inequality by direct reduction to the standard measure-preserving and monotonicity properties of the decreasing rearrangement in the cylinder, with weights chosen to satisfy those properties explicitly. No equations or definitions are self-referential, no fitted parameters are relabeled as predictions, and no load-bearing steps rely on self-citations whose content reduces to the present claims. The central results follow from the stated geometric and weight assumptions without circular reduction to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review yields minimal ledger; relies on standard rearrangement theory without explicit new parameters or entities.

axioms (2)
  • standard math Decreasing rearrangement preserves measure and is monotone
    Implicit in any Dirichlet-type inequality using rearrangements
  • domain assumption Cylinder domains admit well-defined weighted measures
    Required for the weighted isoperimetric claim to make sense

pith-pipeline@v0.9.0 · 5300 in / 1085 out tokens · 49339 ms · 2026-05-08T10:30:15.982453+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    Abreu and L.G

    E. Abreu and L.G. Fernandes , On existence and nonexistence of isoperimetric inequalities with different monomial weights. J. Fourier Anal. Appl. 28 (2022), 33

    work page 2022

  2. [2]

    Alvino, F

    A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo, M.R. Posteraro, Some isoperimetric inequalities on R ^N with respect to weights |x|^ , J. Math. Anal. Appl. 451 (2017), no. 1, 280--318

    work page 2017

  3. [3]

    Alvino, F

    A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo, M.R. Posteraro, The isoperimetric problem for a class of non-radial weights and applications, J. Differential Equations 267 (2019), 6831--6871

    work page 2019

  4. [4]

    Alvino, F

    A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo, M.R. Posteraro, Some isoperimetric inequalities with respect to monomial weights, ESAIM: COCV 27 (2021), Art. 18

    work page 2021

  5. [5]

    Alvino, J.I

    A. Alvino, J.I. Diaz, P.-L. Lions, G. Trombetti , Elliptic equations and Steiner symmetrization, Comm. Pure Appl. Math. 49 (1996), 217--236

    work page 1996

  6. [6]

    Alvino, P.-L

    A. Alvino, P.-L. Lions, G. Trombetti , On optimization problems with prescribed rearrangements. Nonlinear Anal. 13 (1989), 185--220

    work page 1989

  7. [7]

    Baernstein II , A unified approach to symmetrization

    A. Baernstein II , A unified approach to symmetrization. in: Partial Differential Equations of Elliptic Type, eds. A. Alvino et al, Symposia Mathematica 35 , Cambridge Univ. Press, 1995, 47--91

    work page 1995

  8. [8]

    Baldi , Weighted BV functions

    A. Baldi , Weighted BV functions. Houston J. Math. 27 (2001), 683--705

    work page 2001

  9. [9]

    Bankevich and A.I

    S.V. Bankevich and A.I. Nazarov , On P\'olya--Szeg\"o type inequality for Steiner symmetrization. Vestnik St. Petersburg Univ. Math. 44 (2011), 10--15

    work page 2011

  10. [10]

    Betta, F

    M.F. Betta, F. Brock, A. Mercaldo, M.R. Posteraro , A weighted isoperimetric inequality and applications to symmetrization, J. Inequal. Appl. 4 (1999), no. 3, 215--240

    work page 1999

  11. [11]

    Betta, F

    M.F. Betta, F. Brock, A. Mercaldo, M.R. Posteraro , Weighted isoperimetric inequalities on Rn and applications to rearrangements Math. Nachr. 281 (2008), no. 4, 466–498

    work page 2008

  12. [12]

    Brock, F

    F. Brock, F. Chiacchio, A. Ferone, A. Mercaldo , New P\'olya--Szeg\"o-type inequalities and an alternative approach to comparison results for PDEs, Adv. Math. 336 (2018), 316--334

    work page 2018

  13. [13]

    Bobkov, Chr

    S. Bobkov, Chr. Houdr\' e , Some connections between isoperimetric and Sobolev-type inequalities, Mem. Amer. Math. Soc. 129 (1997), no. 616

    work page 1997

  14. [14]

    Brock , Weighted Dirichlet-type inequalities for Steiner symmetrization, Calc

    F. Brock , Weighted Dirichlet-type inequalities for Steiner symmetrization, Calc. Var. Partial Differential Equations 8 (1999), 15--25

    work page 1999

  15. [15]

    Bers and A

    L. Bers and A. Gelbart , On a class of differential equations in mechanics of continua, Quart. Appl. Math. 1 (1943), 168--188

    work page 1943

  16. [16]

    Camfield , Comparison of BV norms in weighted Euclidean spaces and metric measure spaces

    C.S. Camfield , Comparison of BV norms in weighted Euclidean spaces and metric measure spaces. Ph.D. Thesis, University of Cincinnati, 2008

    work page 2008

  17. [17]

    Camfield , Comparison of BV norms in weighted Euclidean spaces

    C.S. Camfield , Comparison of BV norms in weighted Euclidean spaces. J. Anal. 18 (2010), 83--97

    work page 2010

  18. [18]

    Cianchi and N

    A. Cianchi and N. Fusco , Steiner symmetric extremals in P\'olya--Szeg\"o type inequalities, Adv. Math. 203 (2006), 673--728

    work page 2006

  19. [19]

    Davies and T.S

    D.R. Davies and T.S. Walters , The effect of finite width of area on the rate of evaporation into a turbulent atmosphere, Quart. J. Mech. Appl. Math. 4 (1951), 466--480

    work page 1951

  20. [20]

    Diaz and M.H

    J.B. Diaz and M.H. Martin , Riemann's method and the problem of Cauchy, II. The wave equation, Proc. Amer. Math. Soc. 3 (1952), 476--483

    work page 1952

  21. [21]

    Ehrhard , \'El\'ements extr\'emaux pour les in\'egalit\'es de Brunn--Minkowski gaussiennes, Ann

    A. Ehrhard , \'El\'ements extr\'emaux pour les in\'egalit\'es de Brunn--Minkowski gaussiennes, Ann. Inst. H. Poincar\'e Probab. Statist. 22 (1986), 149--168

    work page 1986

  22. [22]

    Esposito, C

    L. Esposito, C. Trombetti , Steiner symmetrization: a weighted version of P\'olya--Szeg\"o principle, Ann. Univ. Ferrara Sez. VII Sci. Mat. 43 (1997), 179--193

    work page 1997

  23. [23]

    Evans, R.F

    L.C. Evans, R.F. Gariepy , Measure theory and fine properties of functions. CRC Press , New York, 1992

    work page 1992

  24. [24]

    Garabedian and D.C

    P.R. Garabedian and D.C. Spencer , Extremal methods in cavitational flow, Technical Report No. 3, Applied Mathematics and Statistics Laboratory, Stanford University, 1951

    work page 1951

  25. [25]

    Kawohl , Rearrangements and Convexity of Level Sets in PDE

    B. Kawohl , Rearrangements and Convexity of Level Sets in PDE. Lecture Notes in Mathematics 1150 , Springer, Berlin, 1985

    work page 1985

  26. [26]

    Kawohl , On the isoperimetric nature of a rearrangement inequality and its consequences for some variational problems, Arch

    B. Kawohl , On the isoperimetric nature of a rearrangement inequality and its consequences for some variational problems, Arch. Rational Mech. Anal. 94 (1986), 227--243

    work page 1986

  27. [27]

    Kawohl , On starshaped rearrangements and applications, Trans

    B. Kawohl , On starshaped rearrangements and applications, Trans. Amer. Math. Soc. 296 (1986), 377--386

    work page 1986

  28. [28]

    Landes , Some remarks on rearrangements and functionals with non-constant density, Math

    R. Landes , Some remarks on rearrangements and functionals with non-constant density, Math. Nachr. 280 (2007), 560--570

    work page 2007

  29. [29]

    Maderna and S

    C. Maderna and S. Salsa , Sharp estimates for solutions to a certain type of singular elliptic boundary value problems in two dimensions, Applicable Analysis 12 (1981), 307--321

    work page 1981

  30. [30]

    Martin , Riemann's method and the problem of Cauchy, Bull

    M.H. Martin , Riemann's method and the problem of Cauchy, Bull. Amer. Math. Soc. 57 (1951), 238--249

    work page 1951

  31. [31]

    Miranda Jr

    M. Miranda Jr. , Functions of bounded variation on ``good'' metric spaces. J. Math. Pures Appl. 82 (2003), 975--1004

    work page 2003

  32. [32]

    Mitrinovic , Analytic Inequalities

    D.S. Mitrinovic , Analytic Inequalities. Springer-Verlag, Berlin, 1970

    work page 1970

  33. [33]

    Payne , On axially symmetric flow and the method of generalized electrostatics, Quart

    L.E. Payne , On axially symmetric flow and the method of generalized electrostatics, Quart. Appl. Math. 10 (1952), 197--204

    work page 1952

  34. [34]

    Payne and A

    L.E. Payne and A. Weinstein , Capacity, virtual mass and generalized symmetrization, Pacific J. Math. 2 (1952), 633--641

    work page 1952

  35. [35]

    Schiffer and G

    M. Schiffer and G. Szeg o , Virtual mass and polarization, Trans. Amer. Math. Soc. 67 (1949), 130--205

    work page 1949

  36. [36]

    Shiffman and D.C

    M. Shiffman and D.C. Spencer , The flow of an ideal incompressible fluid about a lens, Quart. Appl. Math. 5 (1947), 270--288

    work page 1947

  37. [37]

    Talenti , Inequalities in rearrangement invariant function spaces

    G. Talenti , Inequalities in rearrangement invariant function spaces. In: Nonlinear Analysis, Function Spaces and Applications 5 , Proceedings of the Spring School held in Prague, May 1994, eds. M. Krbec et al., Prometheus Publ., Prague, 1994, 177--230

    work page 1994

  38. [38]

    Talenti , Elliptic equations and rearrangements, Ann

    G. Talenti , Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3 (1976), 697--718

    work page 1976

  39. [39]

    Talenti , Linear elliptic P.D.E.'s: level sets, rearrangements and a priori estimates of solutions, Boll

    G. Talenti , Linear elliptic P.D.E.'s: level sets, rearrangements and a priori estimates of solutions, Boll. Un. Mat. Ital. B 4 (1985), 917--949

    work page 1985

  40. [40]

    Talenti , A weighted version of a rearrangement inequality, Ann

    G. Talenti , A weighted version of a rearrangement inequality, Ann. Univ. Ferrara Sez. VII Sci. Mat. 43 (1997), 121--133

    work page 1997

  41. [41]

    Taylor , The energy of a body moving in an infinite fluid with an application to airships, Proc

    G.I. Taylor , The energy of a body moving in an infinite fluid with an application to airships, Proc. Roy. Soc. London Ser. A 120 (1928), 21--33

    work page 1928

  42. [42]

    Weinstein , On cracks and dislocations in shafts under torsion, Quart

    A. Weinstein , On cracks and dislocations in shafts under torsion, Quart. Appl. Math. 10 (1952), 77--81

    work page 1952

  43. [43]

    Weinstein , On axially symmetric flows, Quart

    A. Weinstein , On axially symmetric flows, Quart. Appl. Math. 5 (1948), 429--444

    work page 1948

  44. [44]

    Weinstein , Generalized axially symmetric potential theory, Bull

    A. Weinstein , Generalized axially symmetric potential theory, Bull. Amer. Math. Soc. 59 (1953), 20--38

    work page 1953

  45. [45]

    Weinstein , On the torsion of shafts of revolution, in Proc

    A. Weinstein , On the torsion of shafts of revolution, in Proc. 7th Internat. Congr. Appl. Mech. , Vol. 1, 1948, 108--119

    work page 1948

  46. [46]

    Weinstein , Transonic flow and generalized potential theory, in Proc

    A. Weinstein , Transonic flow and generalized potential theory, in Proc. Aeroballistics Research Symposium , Naval Ordnance Laboratory, 1949, 73--82

    work page 1949

  47. [47]

    Weinstein , On Tricomi's equation and generalized axially symmetric potential theory, Bull

    A. Weinstein , On Tricomi's equation and generalized axially symmetric potential theory, Bull. Acad. Royale de Belgique 37 (1951), 348--358

    work page 1951