pith. sign in

arxiv: 2604.13895 · v1 · submitted 2026-04-15 · 🧮 math.AP

Existence and Regularity in the Small-Mass Regime for a Hartree--Ohta-Kawasaki Shape Optimization Problem

Dario Mazzoleni , Riccardo Moraschi , Berardo Ruffini This is my paper

Pith reviewed 2026-05-10 12:26 UTC · model grok-4.3

classification 🧮 math.AP
keywords shape optimizationHartree energyOhta-Kawasaki functionalsmall mass regimefree boundary regularitynonlocal repulsionCoulomb interactionGamma-convergence
0
0 comments X

The pith

In the small-mass regime, volume-constrained minimizers of the hybrid local-nonlocal energy exist and are C^{2,α} perturbations of a ball.

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

The paper proves existence and regularity for minimizers of a shape optimization problem that blends local confinement with nonlocal Coulomb repulsion in three-dimensional space. It restricts attention to the regime of small prescribed mass, where the optimal domains turn out to be smooth and nearly spherical. A reader would care because the model captures competing effects that appear in physical systems such as charged droplets or quantum many-body problems, and knowing when the preferred shape remains round informs predictions of stability. The argument proceeds by surgery on the domains, Gamma-convergence to a limiting local problem, and one-phase free-boundary regularity theory. A distinctive feature is that the nonlocal term, lacking a priori sign constraints on the optimizers, simultaneously scatters and homogenizes the mass distribution.

Core claim

We show that in the small mass regime, volume-constrained minimizers of this geometric functional exist and are C^{2,α} perturbations of a ball. The proof relies on a combination of surgery techniques, Γ-convergence, elliptic PDE theory, and one-phase free boundary regularity. The Coulombic repulsive term acts both as a scattering and an homogenizing force because the optimal functions lack sign constraints.

What carries the argument

The small-mass scaling regime that renders the nonlocal Coulomb repulsion perturbative relative to the local term, allowing Gamma-convergence to the ball and subsequent regularity via free-boundary theory.

If this is right

  • Existence of minimizers is guaranteed once the mass drops below some threshold.
  • The minimizing domains are C^{2,α} regular and converge to a ball in the zero-mass limit.
  • The absence of sign constraints on the density forces the nonlocal term to play both scattering and averaging roles simultaneously.
  • Standard elliptic and free-boundary techniques suffice once the nonlocal contribution is controlled.

Where Pith is reading between the lines

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

  • The result supplies a baseline against which one can test when non-spherical shapes appear at larger masses.
  • Analogous perturbative arguments may apply to other weak nonlocal kernels provided they remain dominated by the local term at small scale.
  • The regularity obtained opens the door to studying second-variation stability or slow dynamics around the spherical state.

Load-bearing premise

The mass is small enough that the nonlocal repulsion remains a small perturbation of the local confinement term, so the ball stays the unique limiting shape.

What would settle it

Numerical computation of the energy minimizer for a sequence of successively smaller masses, followed by measurement of the Hausdorff distance or curvature deviation from the sphere of equal volume.

Figures

Figures reproduced from arXiv: 2604.13895 by Berardo Ruffini, Dario Mazzoleni, Riccardo Moraschi.

Figure 1
Figure 1. Figure 1: represents the tree of the steps of the proof detailed here above. min  Eq(Ω) : Ω ⊆ R 3 , |Ω| = |B1| [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

We consider a shape optimization problem for a hybrid energy combining local confinement and nonlocal Coulomb repulsion. Specifically, for any open set $\Omega \subseteq \mathbb{R}^3$ of prescribed volume, we consider the ground state energy of an $L^2$-normalized function supported in $\Omega$, defined as a linear combination of its homogeneous $\dot{H}^1$ and $\dot{H}^{-1}$ seminorms. We show that in the small mass regime, volume-constrained minimizers of this geometric functional exist and are $C^{2,\alpha}$ perturbations of a ball. The proof relies on a combination of surgery techniques, $\Gamma$-convergence, elliptic PDE theory, and one-phase free boundary regularity. A key novelty of this paper lies in the treatment of the Coulombic repulsive term: unlike standard competitive models, the lack of (a priori) sign constraints on the optimal functions forces the nonlocal term to exhibit two natures: it acts both as a scattering and an homogenizing force.

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

1 major / 2 minor

Summary. The manuscript studies a volume-constrained shape optimization problem in R^3 whose energy is a linear combination of the homogeneous Sobolev seminorm Ḣ¹ (local confinement) and the Coulomb repulsion term in Ḣ^{-1} for L²-normalized densities supported in the domain. It claims that, in the small-mass regime, minimizers exist and are C^{2,α} perturbations of the ball. The proof combines surgery techniques, Γ-convergence, elliptic PDE theory, and one-phase free-boundary regularity, with emphasis on the sign-indefinite nonlocal term acting simultaneously as a scattering and homogenizing force.

Significance. If the central claims hold, the work supplies a regularity theory for hybrid local-nonlocal geometric functionals without a priori sign constraints on the optimal densities. The adaptation of established tools (Γ-convergence, one-phase free-boundary theory) to this setting, together with the explicit treatment of the indefinite Coulomb term, constitutes a genuine technical contribution to the analysis of competing energies.

major comments (1)
  1. Abstract and statement of main results: the small-mass regime is invoked as the setting in which the nonlocal repulsion remains perturbative relative to the local term, yet no explicit threshold m₀ (depending on the coefficients of the linear combination of seminorms) is supplied. This renders the existence and C^{2,α}-regularity statements non-quantitative; without a concrete bound the Γ-convergence and free-boundary arguments cannot be verified to close for any specific m, which is load-bearing for the limiting-shape claim.
minor comments (2)
  1. The abstract and introduction would benefit from an explicit display of the energy functional, including the precise coefficients multiplying the Ḣ¹ and Ḣ^{-1} seminorms.
  2. Clarify how the surgery construction is modified to accommodate the sign-indefinite character of the nonlocal term; a short paragraph contrasting the argument with the sign-definite case would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: Abstract and statement of main results: the small-mass regime is invoked as the setting in which the nonlocal repulsion remains perturbative relative to the local term, yet no explicit threshold m₀ (depending on the coefficients of the linear combination of seminorms) is supplied. This renders the existence and C^{2,α}-regularity statements non-quantitative; without a concrete bound the Γ-convergence and free-boundary arguments cannot be verified to close for any specific m, which is load-bearing for the limiting-shape claim.

    Authors: We agree that the result is non-quantitative in the sense that no explicit numerical value or closed-form expression for the threshold m₀ is provided. Our proof establishes the existence of some m₀ > 0 (depending on the coefficients in the linear combination of seminorms) via abstract compactness and Γ-convergence arguments, combined with quantitative estimates from elliptic regularity and one-phase free-boundary theory that hold for all sufficiently small masses. These techniques do not yield explicit constants, which is standard in such limiting arguments. We have revised the introduction and the statement of the main theorem to explicitly note the existence of this m₀ without computing its value, and we have added a brief remark clarifying that the Γ-convergence and regularity arguments close for all m < m₀. We disagree that the arguments cannot be verified for any specific m; the proofs in Sections 3–5 are self-contained and apply once the mass is small enough for the perturbative regime to hold. revision: partial

Circularity Check

0 steps flagged

No circularity: central claims rest on external analytic tools applied to a new functional

full rationale

The derivation invokes surgery techniques, Γ-convergence, elliptic PDE theory, and one-phase free-boundary regularity to establish existence and C^{2,α} regularity of volume-constrained minimizers in the small-mass regime. These are standard, independently verifiable tools from the literature; the small-mass hypothesis is an explicit scaling assumption under which the nonlocal term is perturbative, not a self-definitional tautology or fitted input renamed as prediction. No load-bearing step reduces to a self-citation chain, ansatz smuggled via prior work by the same authors, or renaming of a known empirical pattern. The argument is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background results from elliptic regularity and free-boundary theory rather than introducing new axioms or entities.

axioms (1)
  • standard math Standard results from one-phase free-boundary regularity and Gamma-convergence for Sobolev seminorms
    Invoked to obtain C^{2,α} regularity and passage to the limit in the small-mass regime.

pith-pipeline@v0.9.0 · 5484 in / 1288 out tokens · 45356 ms · 2026-05-10T12:26:30.402345+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

  1. [1]

    Aguilera, H

    N. Aguilera, H. W. Alt, and L. A. Caffarelli. An optimization problem with volume constraint.SIAM J. Control Optim., 24(2):191–198, 1986

    work page 1986

  2. [2]

    Alberti, R

    G. Alberti, R. Choksi, and F. Otto. Uniform energy distribution for an isoperimetric problem with long-range interactions.J. Amer. Math. Soc., 22(2):569–605, 2009

    work page 2009

  3. [3]

    H. W. Alt and L. A. Caffarelli. Existence and regularity for a minimum problem with free boundary.J. Reine Angew. Math., 325:105–144, 1981

    work page 1981

  4. [4]

    R. D. Benguria and M. C. Pereira. Remarks on the spectrum of a non-local Dirichlet problem.Bull. Lond. Math. Soc., 53(6):1898–1915, 2021

    work page 1915

  5. [5]

    wrong sign

    S. Biagi, S. Dipierro, E. Valdinoci, and E. Vecchi. On a sobolev critical problem for the superposition of a local and nonlocal operator with the “wrong sign”, preprint, 2026

    work page 2026

  6. [6]

    Brasco, G

    L. Brasco, G. De Philippis, and B. Velichkov. Faber–krahn inequalities in sharp quantitative form.Duke Math. J., 164(9):1777– 1831, 2015

    work page 2015

  7. [7]

    Brian¸ con, M

    T. Brian¸ con, M. Hayouni, and M. Pierre. Lipschitz continuity of state functions in some optimal shaping.Calc. Var. Partial Differential Equations, 23(1):13–32, 2005

    work page 2005

  8. [8]

    D. Bucur. Minimization of the k-th eigenvalue of the Dirichlet Laplacian.Arch. Ration. Mech. Anal., 206(3):1073–1083, 2012

    work page 2012

  9. [9]

    Bucur and G

    D. Bucur and G. Buttazzo. On the characterization of the compact embedding of sobolev spaces.Calc. Var. Partial Differential Equations, 44:455–475, 2011

    work page 2011

  10. [10]

    Bucur and D

    D. Bucur and D. Mazzoleni. A surgery result for the spectrum of the dirichlet laplacian.SIAM J. Math. Anal., 47(6):4451–4466, 2015

    work page 2015

  11. [11]

    Bucur, D

    D. Bucur, D. Mazzoleni, A. Pratelli, and B. Velichkov. Lipschitz regularity of the eigenfunctions on optimal domains.Arch. Ration. Mech. Anal., 216(1):117–151, 2014

    work page 2014

  12. [12]

    Choksi and M

    R. Choksi and M. A. Peletier. Small volume fraction limit of the diblock copolymer problem: I. Sharp-interface functional. SIAM J. Math. Anal., 42(3):1334–1370, 2010

    work page 2010

  13. [13]

    Cicalese and G

    M. Cicalese and G. P. Leonardi. A selection principle for the sharp quantitative isoperimetric inequality.Arch. Ration. Mech. Anal., 206(2):617–643, 2012

    work page 2012

  14. [14]

    Cicalese and E

    M. Cicalese and E. Spadaro. Droplet minimizers of an isoperimetric problem with long-range interactions.Comm. Pure Appl. Math., 66(8):1298–1333, 2013

    work page 2013

  15. [15]

    Dal Maso.An introduction toΓ-convergence, volume 8 ofProgress in Nonlinear Differential Equations and their Applications

    G. Dal Maso.An introduction toΓ-convergence, volume 8 ofProgress in Nonlinear Differential Equations and their Applications. Birkh¨ auser Boston Inc. Boston, MA, 1993

    work page 1993

  16. [16]

    Daneri and E

    S. Daneri and E. Runa. A rigorous approach to pattern formation for isotropic isoperimetric problems with competing nonlocal interactions, preprint, 2024

    work page 2024

  17. [17]

    De Philippis, M

    G. De Philippis, M. Marini, and E. Mukoseeva. The sharp quantitative isocapacitary inequality.Rev. Mat. Iberoam., 37(6):2191–2228, 2021

    work page 2021

  18. [18]

    Figalli, F

    A. Figalli, F. Maggi, and A. Pratelli. A mass transportation approach to quantitative isoperimetric inequalities.Invent. Math., 182(1):167–211, 2010

    work page 2010

  19. [19]

    Figalli and Y

    A. Figalli and Y. R.-Y. Zhang. Serrin’s overdetermined problem in rough domains.J. Eur. Math. Soc., in press, 2025

    work page 2025

  20. [20]

    Fusco, F

    N. Fusco, F. Maggi, and A. Pratelli. The sharp quantitative isoperimetric inequality.Ann. of Math. (2), pages 941–980, 2008

    work page 2008

  21. [21]

    Fusco and A

    N. Fusco and A. Pratelli. Sharp stability for the Riesz potential.ESAIM Control Optim. Calc. Var., 26:Paper No. 113, 24, 2020

    work page 2020

  22. [22]

    Giaquinta.Multiple Integrals in the Calculus of Variations and non linear Elliptic Systems

    M. Giaquinta.Multiple Integrals in the Calculus of Variations and non linear Elliptic Systems. Princeton University Press, 1983

    work page 1983

  23. [23]

    Giaquinta and E

    M. Giaquinta and E. Giusti. On the regularity of the minima of variational integrals.Acta Math., 148:31–46, 1982

    work page 1982

  24. [24]

    Giaquinta and L

    M. Giaquinta and L. Martinazzi.An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, volume 11 ofAppunti. Scuola Normale Superiore di Pisa (Nuova Serie). Edizioni della Normale, Pisa, 2012

    work page 2012

  25. [25]

    Gilbarg and N

    D. Gilbarg and N. Trudinger.Elliptic Partial Differential Equations of Seconds Order, volume 224. Springer Berlin, Heidelberg, 2001

    work page 2001

  26. [26]

    Goldman, M

    M. Goldman, M. Novaga, and B. Ruffini. Existence and stability for a non-local isoperimetric model of charged liquid drops. Arch. Ration. Mech. Anal., 217(1):1–36, 2015

    work page 2015

  27. [27]

    Goldman, M

    M. Goldman, M. Novaga, and B. Ruffini. Rigidity of the ball for an isoperimetric problem with strong capacitary repulsion.J. Eur. Math. Soc., 27(10):4159–4199, 2025

    work page 2025

  28. [28]

    Gustafsson and H

    B. Gustafsson and H. Shahgholian. Existence and geometric properties of solutions of a free boundary problem in potential theory.J. Reine Angew. Math., 1996(473):137–180, 1996

    work page 1996

  29. [29]

    Henrot and M

    A. Henrot and M. Pierre.Shape variation and optimization: A geometrical analysis, volume 28 ofEMS Tracts in Mathematics. European Mathematical Society (EMS), Z¨ urich, 2018

    work page 2018

  30. [30]

    Kn¨ upfer and C

    H. Kn¨ upfer and C. B. Muratov. On an isoperimetric problem with a competing nonlocal term I: The planar case.Comm. Pure Appl. Math., 66(7):1129–1162, 2013

    work page 2013

  31. [31]

    Kn¨ upfer and C

    H. Kn¨ upfer and C. B. Muratov. On an isoperimetric problem with a competing nonlocal term II: The general case.Comm. Pure Appl. Math., 67(12):1974–1994, 2014

    work page 1974

  32. [32]

    Lieb and M

    E. Lieb and M. Loss.Analysis. American Mathematical Society, 2001

    work page 2001

  33. [33]

    Maggi.Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, volume 135 ofCambridge Studies in Advanced Mathematics

    F. Maggi.Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, volume 135 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2012

    work page 2012

  34. [34]

    Maz’ya.Sobolev Spaces: with Applications to Elliptic Partial Differential Equations, volume 342

    V. Maz’ya.Sobolev Spaces: with Applications to Elliptic Partial Differential Equations, volume 342. Springer Berlin Heidelberg, 2011

    work page 2011

  35. [35]

    Mazzoleni, C

    D. Mazzoleni, C. B. Muratov, and B. Ruffini. An optimal design problem for a charge qubit.Comm. Partial Differential Equations, 50:1029–1073, 2025

    work page 2025

  36. [36]

    Mazzoleni and A

    D. Mazzoleni and A. Pratelli. Existence of minimizers for spectral problems.J. Math. Pures Appl. (9), 100(3):433–453, 2013. HARTREE–OHTA-KAWASAKI IN THE SMALL MASS REGIME 29

    work page 2013

  37. [37]

    Mazzoleni and B

    D. Mazzoleni and B. Ruffini. A spectral shape optimization problem with a nonlocal competing term.Calc. Var. Partial Differential Equations, 60(3):114, 2021

    work page 2021

  38. [38]

    Mazzoleni, S

    D. Mazzoleni, S. Terracini, and B. Velichkov. Regularity of the optimal sets for some spectral functionals.Geom. Funct. Anal., 27(2):373–426, 2017

    work page 2017

  39. [39]

    J. Serrin. A symmetry problem in potential theory.Arch. Ration. Mech. Anal., 43(4):304–318, 1971

    work page 1971

  40. [40]

    Velichkov.Regularity of the One-phase Free Boundaries

    B. Velichkov.Regularity of the One-phase Free Boundaries. Springer International Publishing, 2023

    work page 2023

  41. [41]

    H. F. Weinberger. Remark on the preceding paper of Serrin.Arch. Rational Mech. Anal., 43:319–320, 1971. Dipartimento di Matematica F. Casorati, Universit`a di Pavia, Via Ferrata 5, 27100 Pavia, Italy Email address:dario.mazzoleni@unipv.it Dipartimento di Matematica F. Casorati, Universit`a di Pavia, Via Ferrata 5, 27100 Pavia, Italy Email address:r.morasc...

    work page 1971