pith. sign in

arxiv: 2606.17015 · v2 · pith:EVUN2QWKnew · submitted 2026-06-15 · 🧮 math.AP

Uniqueness of the blow-up for some Alt-Phillips cones

Matteo Carducci , Giorgio Tortone This is my paper

Pith reviewed 2026-06-27 03:32 UTC · model grok-4.3

classification 🧮 math.AP
keywords Alt-Phillips problemblow-up uniquenessepiperimetric inequalitiesfree boundarysingular conesminimizing coneslogarithmic convergenceradial cone
0
0 comments X

The pith

Blow-up limits are unique for singular minimizing cones in the Alt-Phillips problem when gamma is in (0,2)

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

This paper sets out to prove that blow-ups at free boundary points are unique for several classes of singular minimizing cones in the Alt-Phillips problem, for gamma between 0 and 2. The uniqueness comes with sharp rates of convergence that are either polynomial or logarithmic depending on an integrability condition. A reader would care because this determines the local structure of the free boundary, giving uniqueness everywhere in dimensions 2, 3 and 4 for gamma in (1,2) and in higher dimensions for gamma in (1,3/2). The work also shows when the radial cone is minimizing using a calibration argument and identifies when logarithmic convergence is sharp even in dimension two.

Core claim

We establish uniqueness of blow-ups, with sharp quantitative convergence, for several classes of singular minimizing cones in the Alt-Phillips problem, in the range γ∈(0,2). As a consequence, we obtain uniqueness at every free boundary point in dimensions d=2,3,4 for γ∈(1,2), and in dimensions d≥5 for γ∈(1,3/2). The proof of uniqueness is based on three new logarithmic epiperimetric inequalities. The sharp distinction between polynomial and logarithmic convergence is governed by a finite-dimensional integrability condition for the spherical linearized problem. We prove this sharpness for the radial cone and its cylindrical extensions through an explicit integrability and bifurcation analysis

What carries the argument

Three new logarithmic epiperimetric inequalities that control the rate of convergence to the cones, together with a finite-dimensional integrability condition on the spherical linearized problem that determines whether the convergence rate is polynomial or logarithmic.

If this is right

  • Uniqueness holds at every free boundary point in dimensions 2, 3, 4 for γ in (1,2)
  • Uniqueness holds at every free boundary point in dimensions 5 and higher for γ in (1, 3/2)
  • Logarithmic convergence is sharp for the radial cone and its cylindrical extensions, even in dimension two
  • The one-dimensional cone has polynomial convergence despite the integrability condition failing
  • The radial cone is minimizing only in certain regimes of dimension d and parameter γ

Where Pith is reading between the lines

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

  • The logarithmic epiperimetric inequalities may extend to establish uniqueness for singular cones in other free boundary problems.
  • The exceptional polynomial convergence of the one-dimensional cone suggests that some cones converge faster than the integrability condition alone would predict.
  • Where uniqueness holds, the free boundary near those points is expected to inherit regularity properties from the unique cone.
  • The one-dimensional calibration argument for minimality could be tested on additional cylindrical extensions of the radial cone.

Load-bearing premise

The three new logarithmic epiperimetric inequalities hold and the finite-dimensional integrability condition on the spherical linearized problem correctly predicts the type of convergence rate.

What would settle it

An explicit construction or numerical solution in dimension three with gamma=1.5 where the same free boundary point admits two distinct blow-up cones would show the uniqueness claim fails.

read the original abstract

We establish uniqueness of blow-ups, with sharp quantitative convergence, for several classes of singular minimizing cones in the Alt-Phillips problem, in the range $\gamma\in(0,2)$. As a consequence, we obtain uniqueness at every free boundary point in dimensions $d=2,3,4$ for $\gamma\in(1,2)$, and in dimensions $d\geq 5$ for $\gamma\in\left(1,\frac32\right)$. The proof of uniqueness is based on three new logarithmic epiperimetric inequalities. The sharp distinction between polynomial and logarithmic convergence is governed by a finite-dimensional integrability condition (sub-integrability) for the spherical linearized problem. We prove this sharpness for the radial cone and its cylindrical extensions through an explicit integrability and bifurcation analysis, showing in particular that logarithmic convergence may be sharp even in dimension two. In contrast, we show that the one-dimensional cone is exceptional: although the integrability condition fails, the convergence is polynomial. Finally, we characterize the minimality of the radial cone in terms of $d$ and $\gamma$ by means of a one-dimensional calibration argument, exhibiting in dimension $d\geq6$ a nontrivial regime in which the radial cone is stable but not minimizing.

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 establishes uniqueness of blow-ups, with sharp quantitative convergence, for several classes of singular minimizing cones in the Alt-Phillips problem for γ ∈ (0,2). The proof relies on three new logarithmic epiperimetric inequalities. The distinction between polynomial and logarithmic convergence is determined by a finite-dimensional integrability condition on the spherical linearized problem. The authors provide an explicit integrability and bifurcation analysis for the radial cone and its cylindrical extensions, demonstrate that the one-dimensional cone is exceptional with polynomial convergence despite the integrability condition failing, and characterize the minimality of the radial cone using a one-dimensional calibration argument, identifying a regime in d ≥ 6 where the radial cone is stable but not minimizing. As a consequence, uniqueness holds at every free boundary point in d = 2,3,4 for γ ∈ (1,2) and in d ≥ 5 for γ ∈ (1, 3/2).

Significance. If the new logarithmic epiperimetric inequalities hold, the work advances free-boundary regularity theory for the Alt-Phillips problem by delivering uniqueness of blow-ups together with sharp (polynomial versus logarithmic) convergence rates in dimensions where such results were previously unavailable. Credit is due for the explicit integrability/bifurcation analysis on the radial cone and cylindrical extensions (including the observation that logarithmic convergence can be sharp already in dimension two) and for the one-dimensional calibration that cleanly separates stability from minimality in d ≥ 6. These concrete, falsifiable statements on rates and on the stable-but-not-minimizing regime strengthen the manuscript beyond a pure existence result.

minor comments (2)
  1. The introduction paragraph on the proof strategy states that the three logarithmic epiperimetric inequalities 'control the rate of convergence' but does not indicate whether they are stated for the full range γ ∈ (0,2) or only for the sub-ranges appearing in the uniqueness corollaries; a single clarifying sentence would improve readability.
  2. In the abstract and the final paragraph, the exceptional behavior of the one-dimensional cone is described clearly, yet the precise statement of the finite-dimensional integrability condition (sub-integrability) is not recalled; repeating its definition once in the introduction would help readers who skip the technical sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and detailed summary of our manuscript, as well as for highlighting the significance of the logarithmic epiperimetric inequalities, the explicit integrability/bifurcation analysis, and the one-dimensional calibration argument separating stability from minimality. We appreciate the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on three newly introduced logarithmic epiperimetric inequalities and an explicit integrability/bifurcation analysis of the spherical linearized problem for the radial cone and its extensions. These are independent constructions that control convergence rates and uniqueness; the one-dimensional calibration for minimality is likewise a separate argument. No step reduces by definition to the target result, no fitted parameter is relabeled as a prediction, and no load-bearing premise collapses to a self-citation chain. The paper is self-contained against external benchmarks with independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review yields limited visibility into parameters and background assumptions; the work relies on standard well-posedness of the Alt-Phillips minimization problem and on the cones being minimizers.

axioms (2)
  • domain assumption The Alt-Phillips minimization problem is well-posed for γ in (0,2)
    Required for the problem statement and the range in which uniqueness is claimed.
  • domain assumption The objects under study are singular minimizing cones
    The uniqueness statement applies specifically to minimizing cones.

pith-pipeline@v0.9.1-grok · 5748 in / 1492 out tokens · 60231 ms · 2026-06-27T03:32:10.479921+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

57 extracted references · 1 linked inside Pith

  1. [1]

    Adams and L

    D. Adams and L. Simon. Rates of asymptotic convergence near isolated singularities of geometric extrema. Indiana Univ. Math. J., 37(2):225–254, 1988

    1988

  2. [2]

    W. K. Allard and F. J. Almgren. On the radial behavior of minimal surfaces and the uniqueness of their tangent cones.Ann. of Math. (2), 113(2):215–265, 1981

    1981

  3. [3]

    Allen, D

    M. Allen, D. Kriventsov, and H. Shahgholian. The free boundary for semilinear problems with highly oscillating singular terms.J. Lond. Math. Soc. (2), 111(5):Paper No. e70180, 2025

    2025

  4. [4]

    H. W. Alt and D. Phillips. A free boundary problem for semilinear elliptic equations.J. Reine Angew. Math., 368:63–107, 1986

    1986

  5. [5]

    L. P. Bonorino. Regularity of the free boundary for some elliptic and parabolic problems. II.Comm. Partial Differential Equations, 26(3-4):355–380, 2001

    2001

  6. [6]

    L. P. Bonorino, E. H. M. Brietzke, J. P. Lukaszczyk, and C. A. Taschetto. Properties of the period function for some Hamiltonian systems and homogeneous solutions of a semilinear elliptic equation.J. Differential Equations, 214(1):156–175, 2005

    2005

  7. [7]

    L. A. Caffarelli. The obstacle problem revisited.J. Fourier Anal. Appl., 4(4-5):383–402, 1998

    1998

  8. [8]

    Caniato and D

    R. Caniato and D. Parise. Symmetric log-epiperimetric inequality for harmonic maps with analytic target and applications.Preprint arXiv:2505.08152, 2025

    arXiv 2025

  9. [9]

    Carducci and G

    M. Carducci and G. Tortone. Smoothness and stability in the Alt-Phillips problem.Math. Ann., 394(3):Pa- per No. 75, 43, 2026

    2026

  10. [10]

    M.CarducciandB.Velichkov.Anepiperimetricinequalityforoddfrequenciesinthethinobstacleproblem. J. Funct. Anal., 289(10):Paper No. 111115, 31, 2025. 74 M. CARDUCCI AND G. TORTONE

    2025

  11. [11]

    T. H. Colding and W. P. Minicozzi, II. On uniqueness of tangent cones for Einstein manifolds.Invent. Math., 196(3):515–588, 2014

    2014

  12. [12]

    T. H. Colding and W. P. Minicozzi, II. Uniqueness of blowups and Łojasiewicz inequalities.Ann. of Math. (2), 182(1):221–285, 2015

    2015

  13. [13]

    M.Colombo, L.Spolaor, andB.Velichkov.Alogarithmicepiperimetricinequalityfortheobstacleproblem. Geom. Funct. Anal., 28(4):1029–1061, 2018

    2018

  14. [14]

    Colombo, L

    M. Colombo, L. Spolaor, and B. Velichkov. Direct epiperimetric inequalities for the thin obstacle problem and applications.Comm. Pure Appl. Math., 73(2):384–420, 2020

    2020

  15. [15]

    De Lellis

    C. De Lellis. The regularity theory for the area functional (in geometric measure theory). InICM— International Congress of Mathematicians. Vol. 2. Plenary lectures, pages 872–913. EMS Press, Berlin,

  16. [16]

    De Philippis, L

    G. De Philippis, L. Spolaor, and B. Velichkov. Regularity of the free boundary for the two-phase Bernoulli problem. Invent. Math., 225(2):347–394, 2021

    2021

  17. [17]

    De Silva and O

    D. De Silva and O. Savin. On certain degenerate one-phase free boundary problems.SIAM J. Math. Anal., 53(1):649–680, 2021

    2021

  18. [18]

    De Silva and O

    D. De Silva and O. Savin. The Alt-Phillips functional for negative powers. Bull. Lond. Math. Soc., 55(6):2749–2777, 2023

    2023

  19. [19]

    De Silva and O

    D. De Silva and O. Savin. Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents. Adv. Nonlinear Stud., 23(1):Paper No. 20220055, 19, 2023

    2023

  20. [20]

    De Silva and O

    D. De Silva and O. Savin. Uniform density estimates andΓ-convergence for the Alt-Phillips functional of negative powers.Math. Eng., 5(5), 2023

    2023

  21. [21]

    Engelstein, D

    M. Engelstein, D. Restrepo, and Z. Zhao. On the asymptotic properties of solutions to one-phase free boundary problems.Preprint arXiv:2511.08393, 2025

    arXiv 2025

  22. [22]

    Engelstein, L

    M. Engelstein, L. Spolaor, and B. Velichkov. (Log-)epiperimetric inequality and regularity over smooth cones for almost area-minimizing currents.Geom. Topol., 23(1):513–540, 2019

    2019

  23. [23]

    Engelstein, L

    M. Engelstein, L. Spolaor, and B. Velichkov. Uniqueness of the blowup at isolated singularities for the Alt-Caffarelli functional.Duke Math. J., 169(8):1541–1601, 2020

    2020

  24. [24]

    Fefferman

    C. Fefferman. Extension ofC m,ω-smooth functions by linear operators.Rev. Mat. Iberoam., 25(1):1–48, 2009

    2009

  25. [25]

    Fernández-Real

    X. Fernández-Real. Classification of blow-ups for the Alt–Phillips problem in three dimensions.Preprint arXiv:2606.00538, 2026

    Pith/arXiv arXiv 2026

  26. [26]

    Fernández-Real and H

    X. Fernández-Real and H. Yu. Generic properties in free boundary problems.Amer. J. Math., To appear, 2026

    2026

  27. [27]

    Figalli, X

    A. Figalli, X. Ros-Oton, and J. Serra. Generic regularity of free boundaries for the obstacle problem.Publ. Math. Inst. Hautes Études Sci., 132:181–292, 2020

    2020

  28. [28]

    Figalli and J

    A. Figalli and J. Serra. On the fine structure of the free boundary for the classical obstacle problem. Invent. Math., 215(1):311–366, 2019

    2019

  29. [29]

    Firester, R

    B. Firester, R. Tsiamis, and Y. Wang. Uniqueness of cylindrical tangent conesCp,q × R. Calc. Var. Partial Differential Equations, 65(5):12, 2026. Id/No 165

    2026

  30. [30]

    Focardi and E

    M. Focardi and E. Spadaro. An epiperimetric inequality for the thin obstacle problem.Adv. Differential Equations, 21(1-2):153–200, 2016

    2016

  31. [31]

    Fotouhi and H

    M. Fotouhi and H. Koch. Higher regularity of the free boundary in a semilinear system.Math. Ann., 388(4):3897–3939, 2024

    2024

  32. [32]

    Garofalo and A

    N. Garofalo and A. Petrosyan. Some new monotonicity formulas and the singular set in the lower dimen- sional obstacle problem.Invent. Math., 177:415–461, 2009

    2009

  33. [33]

    Garofalo, A

    N. Garofalo, A. Petrosyan, and M. Smit Vega Garcia. An epiperimetric inequality approach to the regu- larity of the free boundary in the Signorini problem with variable coefficients.J. Math. Pures Appl. (9), 105(6):745–787, 2016

    2016

  34. [34]

    Q. Han, N. Nadirashvili, and Y. Yuan. Linearity of homogeneous order-one solutions to elliptic equations in dimension three.Comm. Pure Appl. Math., 56(4):425–432, 2003

    2003

  35. [35]

    Karakhanyan and T

    A. Karakhanyan and T. Sanz-Perela. Stable cones in the Alt-Phillips free boundary problem.Calc. Var. Partial Differential Equations, 65(3):Paper No. 83, 29, 2026

    2026

  36. [36]

    Kielhöfer

    H. Kielhöfer. Bifurcation Theory: An Introduction with Applications to Partial Differential Equations. Springer, 2 edition, 2012. UNIQUENESS OF THE BLOW-UP FOR SOME ALT-PHILLIPS CONES 75

    2012

  37. [37]

    Łojasiewicz

    S. Łojasiewicz. Une propriété topologique des sous-ensembles analytiques réels. InLes Équations aux Dérivées Partielles (Paris, 1962), volume No. 117 ofColloq. Internat. CNRS, pages 87–89. CNRS, Paris, 1963

    1962

  38. [38]

    T. Nagura. On the Jacobi differential operators associated to minimal isometric immersions of symmetric spaces into spheres. III.Osaka Math. J., 19(2):241–281, 1982

    1982

  39. [39]

    Phillips

    D. Phillips. A minimization problem and the regularity of solutions in the presence of a free boundary. Indiana Univ. Math. J., 32(1):1–17, 1983

    1983

  40. [40]

    E. R. Reifenberg. An epiperimetric inequality related to the analyticity of minimal surfaces.Ann. of Math. (2), 80:1–14, 1964

    1964

  41. [41]

    Restrepo and X

    D. Restrepo and X. Ros-Oton. C ∞ regularity in semilinear free boundary problems. Math. Ann., 392(3):3397–3446, 2025

    2025

  42. [42]

    Sánchez-Ruiz

    J. Sánchez-Ruiz. Linearization and connection formulae involving squares of Gegenbauer polynomials. Appl. Math. Lett., 14(3):261–267, 2001

    2001

  43. [43]

    Savin and H

    O. Savin and H. Yu. Stable and minimizing cones in the Alt-Phillips problem.Preprint arXiv:2502.18192, 2025

    arXiv 2025

  44. [44]

    Savin and H

    O. Savin and H. Yu. Concentration of cones in the Alt-Phillips problem.Arch. Ration. Mech. Anal., 250(3):Paper No. 35, 13, 2026

    2026

  45. [45]

    L. Simon. Asymptotics for a class of nonlinear evolution equations, with applications to geometric prob- lems. Ann. of Math. (2), 118(3):525–571, 1983

    1983

  46. [46]

    L. Simon. Uniqueness of some cylindrical tangent cones.Comm. Anal. Geom., 2(1):1–33, 1994

    1994

  47. [47]

    Y. Sire, S. Terracini, and S. Vita. Liouville type theorems and regularity of solutions to degenerate or singular problems. I: Even solutions.Comm. Partial Differential Equations, 46(2):310–361, 2021

    2021

  48. [48]

    Pure Appl

    L.SpolaorandB.Velichkov.Anepiperimetricinequalityfortheregularityofsomefreeboundaryproblems: the 2-dimensional case.Comm. Pure Appl. Math., 72(2):375–421, 2019

    2019

  49. [49]

    Spolaor and B

    L. Spolaor and B. Velichkov. On the logarithmic epiperimetric inequality for the obstacle problem.Math. Eng., 3(1):Paper No. 4, 42, 2021

    2021

  50. [50]

    Székelyhidi

    G. Székelyhidi. Uniqueness of certain cylindrical tangent cones.Preprint arXiv:2012.02065, 2020

    arXiv 2012

  51. [51]

    J. E. Taylor. The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces.Ann. of Math. (2), 103(3):489–539, 1976

    1976

  52. [52]

    J. E. Taylor. The structure of singularities in solutions to ellipsoidal variational problems with constraints in R3. Ann. of Math. (2), 103(3):541–546, 1976

    1976

  53. [53]

    Velichkov

    B. Velichkov. Regularity of the one-phase free boundaries, volume 28 of Lecture Notes of the Unione Matematica Italiana. Springer, Cham, 2023

    2023

  54. [54]

    G. S. Weiss. Partial regularity for weak solutions of an elliptic free boundary problem.Comm. Partial Differential Equations, 23(3-4):439–455, 1998

    1998

  55. [55]

    G. S. Weiss. A homogeneity improvement approach to the obstacle problem.Invent. Math., 138(1):23–50, 1999

    1999

  56. [56]

    G. S. Weiss. The free boundary of a thermal wave in a strongly absorbing medium.J. Differential Equa- tions, 160(2):357–388, 2000

    2000

  57. [57]

    Giuseppe Peano

    B. White. Tangent cones to two-dimensional area-minimizing integral currents are unique.Duke Math. J., 50(1):143–160, 1983. Matteo Carducci Classe di Scienze, Scuola Normale Superiore Piazza dei Cavalieri 7, 56126 Pisa - ITALY Email address: matteo.carducci@sns.it Giorgio Tortone Dipartimento di Matematica “Giuseppe Peano”, Università di Torino Via Carlo ...

    1983