Some isoperimetric inequalities with respect to monomial weights

Angelo Alvino; Anna Mercaldo; Francesco Chiacchio; Friedemann Brock; Maria Rosaria Posteraro

arxiv: 1907.03659 · v1 · pith:RZRSJUUQnew · submitted 2019-07-08 · 🧮 math.AP

Some isoperimetric inequalities with respect to monomial weights

Angelo Alvino , Friedemann Brock , Francesco Chiacchio , Anna Mercaldo , Maria Rosaria Posteraro This is my paper

Pith reviewed 2026-05-25 00:58 UTC · model grok-4.3

classification 🧮 math.AP

keywords isoperimetric inequalitymonomial weightsweighted perimeterupper half-planeCheeger constanteigenvalue bound

0 comments

The pith

Among smooth sets in the upper half-plane with fixed y^β measure, the y^α weighted perimeter is minimized by an explicit y-axis symmetric set.

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

The paper establishes an isoperimetric inequality on the upper half-plane using monomial weights that are powers of the vertical coordinate. For sets with a fixed value of the weighted area integral involving y to the power β, it identifies the sets that minimize the weighted perimeter integral involving y to the power α. The minimizing sets are proved to be smooth and symmetric with respect to the y-axis under explicit restrictions on the two exponents. The same construction supplies a bound on the associated weighted Cheeger constant and a lower bound on the first eigenvalue of related nonlinear eigenvalue problems.

Core claim

We show that, among all smooth sets Ω in R²₊ with fixed weighted measure ∬_Ω y^β dx dy, the weighted perimeter ∫_{∂Ω} y^α ds achieves its minimum for a smooth set which is symmetric w.r.t. the y-axis, and is explicitly given.

What carries the argument

The explicit y-axis symmetric minimizer of the weighted perimeter under the weighted area constraint.

If this is right

The minimizer is smooth and symmetric with respect to the y-axis.
The construction yields an estimate for the weighted Cheeger constant.
A lower bound follows for the first eigenvalue of a class of nonlinear problems.

Where Pith is reading between the lines

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

The explicit form of the minimizer permits direct calculation of the optimal isoperimetric ratio under the stated exponent conditions.
The symmetry conclusion may indicate how the same weighted perimeter behaves when the domain is restricted further or when the weights are applied in related variational problems.

Load-bearing premise

The result requires the sets to be smooth and the exponents to satisfy 0 ≤ α < β + 1 together with β ≤ 2α.

What would settle it

A smooth set in the upper half-plane whose weighted area equals that of the claimed minimizer but whose weighted perimeter is strictly smaller would falsify the minimality statement.

Figures

Figures reproduced from arXiv: 1907.03659 by Angelo Alvino, Anna Mercaldo, Francesco Chiacchio, Friedemann Brock, Maria Rosaria Posteraro.

**Figure 1.** Figure 1: Isoperimetric sets for different values of α and β 2. Isoperimetric inequality in the upper half plane Let R 2 + := {(x, y) ∈ R 2 : y > 0}. Throughout this paper, we assume that α, β ∈ R and (2.1) β + 1 > 0 and α ≥ 0. If Ω ⊂ R 2 + is measurable, we set (2.2) Ω(y) := {x ∈ R : (x, y) ∈ Ω}, (y ∈ R+), Ω 0 (2.3) := {y ∈ R+ : Ω(y) 6= ∅}. Further, we define the weighted area of Ω by Aβ(Ω) := Z Z Ω y β dxdy, and t… view at source ↗

read the original abstract

We solve a class of isoperimetric problems on $\mathbb{R}^2_+ :=\left\{ (x,y)\in \mathbb{R} ^2 : y>0 \right\}$ with respect to monomial weights. Let $\alpha $ and $\beta $ be real numbers such that $0\le \alpha <\beta+1$, $\beta\le 2 \alpha$. We show that, among all smooth sets $\Omega$ in $\mathbb{R} ^2_+$ with fixed weighted measure $\iint_{\Omega } y^{\beta} dxdy$, the weighted perimeter $\int_{\partial \Omega } y^\alpha \, ds$ achieves its minimum for a smooth set which is symmetric w.r.t. to the $y$--axis, and is explicitly given. Our results also imply an estimate of a weighted Cheeger constant and a lower bound for the first eigenvalue of a class of nonlinear problems.

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

This paper gives an explicit symmetric minimizer for a monomial-weighted isoperimetric problem in the half-plane under the stated parameter restrictions.

read the letter

The main point is that the authors solve the weighted isoperimetric problem in the upper half-plane for monomial weights y^α on the perimeter and y^β on the measure. For 0 ≤ α < β+1 and β ≤ 2α they show that among smooth sets of fixed weighted measure the perimeter is minimized by an explicitly described set symmetric across the y-axis. They also extract a bound on the weighted Cheeger constant and a lower bound on the first eigenvalue of some nonlinear problem. That is the concrete deliverable. The parameter window is chosen so the associated weighted curvature equation has a smooth closed curve that can serve as the candidate, which is a standard way to get an explicit answer. The restriction to smooth sets is stated up front, so the claim does not overreach. The result looks new for this exact combination of weights and domain. The abstract is clear about what is proved and what is left out. One soft spot is that the abstract supplies no proof outline or comparison with earlier work on weighted isoperimetric problems, so it is impossible to judge how much of the argument is routine rearrangement versus a new calculation. Extending the result past smooth sets would require additional work on existence and regularity in BV, but the paper does not claim that extension. The eigenvalue and Cheeger consequences are stated as implications rather than developed in detail. This is a focused contribution in geometric analysis and weighted PDEs. Readers who care about explicit solutions in half-spaces or monomial weights will get something usable from it. The claim is self-contained enough and the conditions are explicit enough that it deserves a serious referee rather than a desk rejection. I would send it out for review.

Referee Report

0 major / 2 minor

Summary. The manuscript solves a class of weighted isoperimetric problems in the upper half-plane R²₊. For real parameters satisfying 0 ≤ α < β + 1 and β ≤ 2α, it asserts that among all smooth sets Ω ⊂ R²₊ with fixed weighted measure ∬_Ω y^β dx dy, the weighted perimeter ∫_{∂Ω} y^α ds attains its minimum at an explicitly described smooth set that is symmetric with respect to the y-axis. The results are also used to bound the weighted Cheeger constant and to obtain a lower bound on the first eigenvalue of an associated class of nonlinear eigenvalue problems.

Significance. If the central claim holds, the explicit identification of the minimizer supplies a concrete, verifiable example of equality in a weighted isoperimetric inequality under monomial weights. The parameter restrictions are presented as the natural regime in which a smooth closed comparison curve exists, and the consequences for the Cheeger constant and nonlinear eigenvalues extend the utility of the result beyond the pure isoperimetric statement.

minor comments (2)

The abstract states that the minimizer is 'explicitly given' but does not display the explicit form or the associated curvature equation; adding the explicit expression (or at least the ODE it satisfies) to the abstract or the first paragraph of the introduction would improve immediate readability.
The restriction to smooth sets is stated clearly, yet the introduction could briefly indicate whether the same explicit set remains a minimizer (or at least a stationary point) in a larger class such as sets of finite weighted perimeter.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. The referee's summary correctly reflects the scope and consequences of the results. No major comments appear in the report, so we have no points requiring rebuttal or clarification at this stage. Any minor suggestions will be incorporated in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper proves a direct minimization theorem: among smooth sets of fixed weighted measure ∬_Ω y^β dx dy, the weighted perimeter ∫_{∂Ω} y^α ds is minimized by an explicitly described y-axis symmetric set, under the stated restrictions 0 ≤ α < β+1 and β ≤ 2α. This is a self-contained existence and characterization result in the calculus of variations; the minimizer is constructed independently (via the associated weighted curvature problem) rather than being defined in terms of the perimeter itself. No steps reduce by construction to fitted inputs, self-citations, or ansatzes smuggled from prior work by the same authors. The result stands as an independent theorem with explicit assumptions and does not rely on renaming known patterns or load-bearing self-references.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard existence and regularity results from the calculus of variations for isoperimetric problems but introduces no free parameters, no new entities, and no ad-hoc axioms beyond the usual background of geometric measure theory.

axioms (1)

domain assumption Existence of minimizers in the class of smooth sets for the weighted isoperimetric problem under the given parameter restrictions.
The claim is stated only for smooth sets and relies on the parameters ensuring the problem is well-posed.

pith-pipeline@v0.9.0 · 5696 in / 1299 out tokens · 32050 ms · 2026-05-25T00:58:18.186069+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · 2 internal anchors

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E. Abreu, L.G. Fernandes , On existence and nonexistence of isoperimetric inequalities with diﬀerent monomial weights. arXiv:1904.01441v2, 11 Apr 2019

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

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D. Bucur, I. Fragal `a, A Faber-Krahn inequality for the Cheeger constant of N-gons. J. Geom. Anal. 26 (2016), 88–117

work page 2016
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Cabre, X

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Cabre, X

X. Cabre, X. Ros-Oton, J. Serra, Euclidean balls solve some isoperimetric problems with nonradial weights. C. R. Math. Acad. Sci. Paris 350 (2012), 945–947

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A. Ca˜nete, M. Miranda Jr., D. Vittone , Some isoperimetric problems in planes with density. J. Geom. Anal. 20 (2010), no.2, 243–290

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T. Carroll, A. Jacob, C. Quinn, R. Walters, The isoperimetric problem on planes with density. Bull. Aust. Math. Soc. 78 (2008), no.2, 177–197. SOME ISOPERIMETRIC INEQUALITIES WITH RESPECT TO MONOMIAL WEIGHTS 29

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V. Caselles, M. Miranda jr, M. Novaga , Total variation and Cheeger sets in Gauss space. J. Funct. Anal. 259 (2010), 1491–1516

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H. Castro, Hardy-Sobolev inequalities with monomial weights. Ann. Mat. Pura Appl. (4) 196 (2017), no. 2, 579–598

work page 2017
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G. R. Chambers, Proof of the Log-Convex Density Conjecture, arXiv:1311.4012v3, to appear in: Journal of the European Mathematics Society

work page internal anchor Pith review Pith/arXiv arXiv
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Csat´o, An isoperimetric problem with density and the Hardy Sobolev inequality in R2, Diﬀerential Integral Equations 28 (2015), no

G. Csat´o, An isoperimetric problem with density and the Hardy Sobolev inequality in R2, Diﬀerential Integral Equations 28 (2015), no. 9-10, 971-988

work page 2015
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J. Dahlberg, A. Dubbs, E. Newkirk, H. Tran , Isoperimetric regions in the plane with densityrp, New York J. Math. 16 (2010), 31-51

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G. De Philippis, G. Franzina, A. Pratelli, Existence of isoperimetric sets with densities ”converging from below” on RN. J. Geom. Anal. 27 (2017), 1086–1105

work page 2017
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Balls Isoperimetric in $\mathbb{R}^n$ with Volume and Perimeter Densities $r^m$ and $r^k$

L. Di Giosia, J. Habib, L. Kenigsberg, D. Pittman, W. Zhu , Balls Isoperimetric in Rn with Volume and Perimeter Densities rm and rk. (2019), arXiv:1610.05830v2

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A. Diaz, N. Harman, S. Howe, D. Thompson Isoperimetric problems in sectors with density. Adv. Geom. 12 (2012), 589-619

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P. Gurka, B. Opic , Continuous and compact imbeddings of weighted Sobolev spaces II , Czechoslovak Math. J. 39 (113) (1989), no. 1, 78–94

work page 1989
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Harman, S

N. Harman, S. Howe, F. Morgan , Steiner and Schwarz symmetrization in warped prod- ucts and ﬁber bundles with density. Rev. Mat. Iberoamericana 27 (2011), no. 3, 909–918

work page 2011
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Howe, The Log-Convex Density Conjecture and vertical surface area in warped products

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Morgan, Manifolds with density

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Morgan, The Log-Convex Density Conjecture

F. Morgan, The Log-Convex Density Conjecture. Contemporary Mathematics 545 (2011), 209–211

work page 2011
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Morgan, A

F. Morgan, A. Pratelli , Existence of isoperimetric regions in RN with density. Ann. Global Anal. Geom. 43 (2013), 331–365

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[1] [1]

Abreu, L.G

E. Abreu, L.G. Fernandes , On existence and nonexistence of isoperimetric inequalities with diﬀerent monomial weights. arXiv:1904.01441v2, 11 Apr 2019

work page arXiv 1904

[2] [2]

Alvino, F

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

work page 2017

[3] [3]

Alvino, F

A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo, M.R. Posteraro , On weighted isoperimetric inequalities with non-radial densities (2018), Appl. Anal. , to appear doi.org/10.1080/00036811.2018.1506106

work page doi:10.1080/00036811.2018.1506106 2018

[4] [4]

Alvino, F

A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo, M.R. Posteraro, The isoperimet- ric problem for a class of non-radial weights and applications (2018), J. Diﬀerential Equations, to appear

work page 2018

[5] [5]

Bayle, A

V. Bayle, A. Ca ˜nete, F. Morgan, C. Rosales , On the isoperimetric problem in Eu- clidean space with density. Calc. Var. PDE 31 (2008), 27–46

work page 2008

[6] [6]

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

[7] [7]

Betta, F

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

work page 2008

[8] [8]

Boyer, B

W. Boyer, B. Brown, G. Chambers, A. Loving, S. Tammen , Isoperimetric regions in Rn with density rp, Anal. Geom. Metr. Spaces 4 (2016), 236–265

work page 2016

[9] [9]

Brandolini, F

B. Brandolini, F. Della Pietra, C. Nitsch, C. Trombetti , Symmetry breaking in a constrained Cheeger type isoperimetric inequality. (English summary) ESAIM Control Optim. Calc. Var. 21 (2015), 359–371

work page 2015

[10] [10]

Brock, F

F. Brock, F. Chiacchio, A. Mercaldo, A weighted isoperimetric inequality in an orthant. Potential Anal. 41 (2012), 171–186

work page 2012

[11] [11]

Brock, A

F. Brock, A. Mercaldo, M.R. Posteraro, On isoperimetric inequalities with respect to inﬁnite measures. Revista Matem´ atica Iberoamericana29 (2013), 665–690

work page 2013

[12] [12]

Bucur, I

D. Bucur, I. Fragal `a, Proof of the honeycomb asymptotics for optimal Cheeger clusters. Adv. Math. 350 (2019), 97–129

work page 2019

[13] [13]

Bucur, I

D. Bucur, I. Fragal `a, A Faber-Krahn inequality for the Cheeger constant of N-gons. J. Geom. Anal. 26 (2016), 88–117

work page 2016

[14] [14]

Cabre, X

X. Cabre, X. Ros-Oton, Sobolev and isoperimetric inequalities with monomial weights. J. Diﬀerential Equations 255 (2013), 4312–4336

work page 2013

[15] [15]

Cabre, X

X. Cabre, X. Ros-Oton, J. Serra, Euclidean balls solve some isoperimetric problems with nonradial weights. C. R. Math. Acad. Sci. Paris 350 (2012), 945–947

work page 2012

[16] [16]

Ca˜nete, M

A. Ca˜nete, M. Miranda Jr., D. Vittone , Some isoperimetric problems in planes with density. J. Geom. Anal. 20 (2010), no.2, 243–290

work page 2010

[17] [17]

Carroll, A

T. Carroll, A. Jacob, C. Quinn, R. Walters, The isoperimetric problem on planes with density. Bull. Aust. Math. Soc. 78 (2008), no.2, 177–197. SOME ISOPERIMETRIC INEQUALITIES WITH RESPECT TO MONOMIAL WEIGHTS 29

work page 2008

[18] [18]

Caselles, M

V. Caselles, M. Miranda jr, M. Novaga , Total variation and Cheeger sets in Gauss space. J. Funct. Anal. 259 (2010), 1491–1516

work page 2010

[19] [19]

Castro, Hardy-Sobolev inequalities with monomial weights

H. Castro, Hardy-Sobolev inequalities with monomial weights. Ann. Mat. Pura Appl. (4) 196 (2017), no. 2, 579–598

work page 2017

[20] [20]

G. R. Chambers, Proof of the Log-Convex Density Conjecture, arXiv:1311.4012v3, to appear in: Journal of the European Mathematics Society

work page internal anchor Pith review Pith/arXiv arXiv

[21] [21]

Cheeger, A lower bound for the smallest eigenvalue of the Laplacian,Problems in analysis: A symposium in honor of Salomon Bochner (1970), 195–199

J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian,Problems in analysis: A symposium in honor of Salomon Bochner (1970), 195–199

work page 1970

[22] [22]

Csat´o, An isoperimetric problem with density and the Hardy Sobolev inequality in R2, Diﬀerential Integral Equations 28 (2015), no

G. Csat´o, An isoperimetric problem with density and the Hardy Sobolev inequality in R2, Diﬀerential Integral Equations 28 (2015), no. 9-10, 971-988

work page 2015

[23] [23]

Dahlberg, A

J. Dahlberg, A. Dubbs, E. Newkirk, H. Tran , Isoperimetric regions in the plane with densityrp, New York J. Math. 16 (2010), 31-51

work page 2010

[24] [24]

De Philippis, G

G. De Philippis, G. Franzina, A. Pratelli, Existence of isoperimetric sets with densities ”converging from below” on RN. J. Geom. Anal. 27 (2017), 1086–1105

work page 2017

[25] [25]

Balls Isoperimetric in $\mathbb{R}^n$ with Volume and Perimeter Densities $r^m$ and $r^k$

L. Di Giosia, J. Habib, L. Kenigsberg, D. Pittman, W. Zhu , Balls Isoperimetric in Rn with Volume and Perimeter Densities rm and rk. (2019), arXiv:1610.05830v2

work page internal anchor Pith review Pith/arXiv arXiv 2019

[26] [26]

A. Diaz, N. Harman, S. Howe, D. Thompson Isoperimetric problems in sectors with density. Adv. Geom. 12 (2012), 589-619

work page 2012

[27] [27]

Gurka, B

P. Gurka, B. Opic , Continuous and compact imbeddings of weighted Sobolev spaces II , Czechoslovak Math. J. 39 (113) (1989), no. 1, 78–94

work page 1989

[28] [28]

Harman, S

N. Harman, S. Howe, F. Morgan , Steiner and Schwarz symmetrization in warped prod- ucts and ﬁber bundles with density. Rev. Mat. Iberoamericana 27 (2011), no. 3, 909–918

work page 2011

[29] [29]

Howe, The Log-Convex Density Conjecture and vertical surface area in warped products

S. Howe, The Log-Convex Density Conjecture and vertical surface area in warped products. Adv. Geom. 15 (2015), 455-468

work page 2015

[30] [30]

Ionescu, T

I.R. Ionescu, T. Lachand-Robert , Generalized Cheeger sets related to landslides. Calc. Var. Partial Diﬀerential Equations 23 (2005), 227–249

work page 2005

[31] [31]

Kawohl, V

B. Kawohl, V. Fridman , Isoperimetric estimates for the ﬁrst eigenvalue of the p-Laplace operator and the Cheeger constant. Comment. Math. Univ. Carolin. 44 (2003), no. 4, 659–667

work page 2003

[32] [32]

Kolesnikov, R.I

A.V. Kolesnikov, R.I. Zhdanov , On isoperimetric sets of radially symmetric measures. Concentration, functional inequalities and isoperimetry, 123-154, Contemp. Math. 545, Amer. Math. Soc., Providence, RI, 2011

work page 2011

[33] [33]

Maderna, S

C. Maderna, S. Salsa , Sharp estimates for solutions to a certain type of singular elliptic boundary value problems in two dimensions. Applicable Analysis 12 (1981), no.4, 307–321

work page 1981

[34] [34]

Maz’ja, Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces

V. Maz’ja, Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces. Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), 307– 340, Contemp. Math. 338, Amer. Math. Soc., Providence, RI, 2003

work page 2002

[35] [35]

Maz’ja, T

V. Maz’ja, T. Shaposhnikova, A collection of sharp dilation invariant integral inequalities for diﬀerentiable functions, Sobolev spaces in mathematics I , 223-247. Int. Math. Ser. (N. Y.) 8 Springer, 2009

work page 2009

[36] [36]

Morgan, Manifolds with density

F. Morgan, Manifolds with density. Notices Amer. Math. Soc. 52 (2005), no.8, 853–858

work page 2005

[37] [37]

Morgan, The Log-Convex Density Conjecture

F. Morgan, The Log-Convex Density Conjecture. Contemporary Mathematics 545 (2011), 209–211

work page 2011

[38] [38]

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