Smoothness and stability in the Alt-Phillips problem

Giorgio Tortone; Matteo Carducci

REVIEW 3 major objections 6 minor 52 references

Smoothness and stability in the Alt-Phillips problem

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that regular free-boundary points in the Alt-Phillips problem are smooth for every exponent $\gamma\in(-2,2)$, and derives a stability inequality that makes axially symmetric cones one-dimensional in low dimensions.

desk verdict Serious paper that proves a long-open smoothness result, with a real but likely fixable gap in the reduction from minimizers to regular solutions and an overclaim in the abstract. read the letter →

arxiv 2507.10336 v1 pith:M75BIFAW submitted 2025-07-14 math.AP

Matteo Carducci , Giorgio Tortone This is my paper · ORCID

classification math.AP MSC 35R3535B6535J6135B0749Q05

keywords Alt-PhillipsfreeboundaryhigherregularityhodographtransformdegeneratequasilinearequationsSchauderestimatesstablesolutionsminimalcones

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

At regular free-boundary points, the Alt-Phillips interface is smooth for every exponent $\gamma\in(-2,2)$; the paper closes the negative-exponent gap and gives one unified proof. This matters because the Alt-Phillips functional interpolates between the obstacle problem ($\gamma=1$), the Alt-Caffarelli/Bernoulli problem ($\gamma=0$), and minimal surfaces as $\gamma\to-2$, so a common smoothness theorem anchors the whole family. The proof sends the solution through a hodograph transformation and reduces smoothness to Schauder estimates for a degenerate quasilinear PDE with a Neumann condition. In the negative-exponent regime the same regularity is used to compute the second variation and to show that stable axially symmetric cones are one-dimensional in low dimensions.

What carries the argument

The load-bearing object is the hodograph transform $h$: the inverse of $\Phi(x',x_d)=(x',w(x',x_d))$, whose trace on $\{x_d=0\}$ is a local parametrization of the free boundary. The key identity is that the Alt-Phillips equations become the degenerate quasilinear system $\operatorname{div}(x_d^s DF(\nabla h))=0$ in the half-ball, with $F(p)=(|p|^2+1)/p_d$ and boundary condition $\lim_{x_d\to0^+} x_d^s DF(\nabla h)\cdot e_d=0$. Theorem 1.2, a Schauder estimate for uniformly convex $F$ in this weighted setting, upgrades $C^{1,\alpha}$ to $C^{k,\alpha}$ and then $C^\infty$. On the stability side, the central identity is the second inner variation along the normal field $\xi=(\nabla w/|\nabla w|)f$, which evaluates to $\int_{\Omega_w} w^s|\nabla w|^2(|\nabla f|^2-A_w^2 f^2)\,dx$ with $A_w^2=|\nabla^2 w|^2/|\nabla w|^2 - |\nabla^2 w\nabla w|^2/|\nabla w|^4$.

What would settle it

A concrete check would be to construct a solution of $\operatorname{div}(x_d^s DF(\nabla v))=0$ with $F$ uniformly convex and $F\in C^{3,\alpha}$ that is $C^{1,\alpha}$ but not $C^{2,\alpha}$ in $B^+_{1/2}$; such a counterexample would falsify Theorem 1.2 and with it the smoothness claim. On the stability side, evaluating inequality (4.2) on any explicit nontrivial axially symmetric 1-homogeneous cone in dimension 6 would settle Theorem 1.4.

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Extended reading notes

Core claim

The central claim, Theorem 1.1, is that if $u$ is a local minimizer of $J_\gamma$ in $B_1$ and $x_0\in\partial\Omega_u$ is a regular free-boundary point, then $\partial\Omega_u$ is locally the graph of a smooth function and both $w=\beta u^{1/\beta}$ and $u/\operatorname{dist}(\cdot,\partial\Omega_u)^\beta$ are smooth in $\Omega_u$ near $x_0$. The proof treats all $\gamma\in(-2,2)$ at once: with $s=\beta\gamma$, the hodograph transform $\Phi(x',x_d)=(x',w(x',x_d))$ has an inverse $h$ whose trace parameterizes the free boundary, and $h$ solves the degenerate quasilinear equation $\operatorname{div}(x_d^s DF(\nabla h))=0$ with $F(p)=(|p|^2+1)/p_d$ and a Neumann boundary condition. Theorem 1.2 supplies Schauder estimates for such equations under uniform convexity of $F$, and iteration gives $C^\infty$. For $\gamma\in(-2,0)$, Theorem 1.3 derives the stability inequality $\int_{\Omega_w} w^s|\nabla w|^2(|\nabla f|^2-A_w^2 f^2)\,dx\ge 0$, where $A_w^2$ encodes the second fundamental form of level sets, and Theorem 1.4 uses it to prove that axially symmetric minimizing cones are one-dimensional for $d\le 6$, or for $d=7$ and $\gamma<(10-8\sqrt5)/11$.

Load-bearing premise

The load-bearing premise is Lemma 2.3: near a regular free-boundary point every local minimizer is a regular solution in the sense of Definition 2.1, meaning it has Hölder regularity with $\alpha>-s$ and satisfies the weighted boundary condition pointwise; the proof of this lemma is a compressed linearization-and-iteration argument that cites [20, Prop. 7.2] instead of carrying out the full bootstrap.

Editorial extensions

If this is right

Regular free-boundary points are $C^\infty$: all derivatives of the interface exist, so curvature and other higher-order geometric quantities are well defined there.
The stability inequality supplies one second-variation criterion for the whole negative-exponent regime and reduces to the known Alt-Caffarelli and minimal-surface criteria at the endpoints.
Axially symmetric stable or minimizing cones are one-dimensional in low dimensions ($d\le 6$, and $d=7$ for $\gamma$ below about $-0.7171$), so any singular minimizer there must be non-axial.
The eigenvalue criterion $\lambda_s(\Sigma_w)\ge -((d+s-2)/2)^2$ gives an explicit spectral test for homogeneous cones and converges to the minimal-surface stability criterion as $\gamma\to-2$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper's claims, the hodograph-Schauder route looks portable to other degenerate one-phase free boundary problems whose boundary condition is encoded by a power weight; the authors do not pursue this.
Beyond the paper's claims, the stability inequality can be used as a spectral test on candidate cones, since $A_w^2$ is computable from the geometry of level sets; the paper does not run such tests.
Beyond the paper's claims, removing the uniform $C^{2,\alpha}$ compactness assumption in the $\gamma\to-2$ limit would turn the convergence result into a statement about all stable solutions; that remains open here.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a serious piece of work. It proves C∞ smoothness of the regular free boundary for negative Alt-Phillips exponents, which was open, and gives a unified proof for the full range γ∈(−2,2). The quasilinear Schauder theory in Theorem 1.2 is genuinely new and useful on its own; the stability inequality and the axially symmetric cone rigidity are solid contributions. The hodograph transformation and the bootstrap are detailed and internally coherent. I checked the main identities and they hold up. The paper is honest about what is proved and what is not, and the reliance on the authors' prior linear theory in [47] is legitimate — that theory is published and parameter-free, and the quasilinear extension is a real step beyond it.

The weak link is Lemma 2.3. It is load-bearing: Theorem 1.1 for local minimizers rests on it, and the proof is a sketch. The stress-test concern has teeth. The improvement-of-flatness step needs a radius that is uniform for the specific α > −s required by Definition 2.1, and neither the paper nor the cited [20, Prop. 7.2] shows that uniformity. Near γ→−2, α can be arbitrarily close to 1, and if the linearized estimates only give a fixed exponent below that threshold, the iteration caps out. This is not an internal contradiction, and the gap is probably fillable, but the paper should either supply the full iteration or state Theorem 1.1 for regular solutions and only conjecture the minimizer step.

Second issue: the abstract says the stability criterion recovers the minimal-surface criterion as γ→−2, but assumption (5.6) — uniform C^{2,α} regularity — is explicitly unproven and the paper says so in Section 5. That claim should be qualified in the abstract.

Who this is for: anyone working on free boundary regularity or degenerate PDEs. Theorem 1.2 alone is worth the read, and the stability computation in Section 3 is careful and useful. I would send this to a serious referee. The right expectation is a major revision that closes or clearly restricts Lemma 2.3 and tones down the abstract. I'd cite it for the Schauder theorem even if the minimizer-to-regular reduction stays open.

Referee Report

3 major / 6 minor

Summary. The paper studies the one-phase Alt–Phillips functional J_γ for γ∈(-2,2), with emphasis on the negative-exponent range γ∈(-2,0). For regular free boundary points it proves C^∞ smoothness of the free boundary and of u/dist(·,∂Ω_u)^β (Theorem 1.1), by a hodograph transform that reduces the problem to a degenerate quasilinear PDE div(x_d^s DF(∇h))=0 with a Neumann boundary condition, for which the authors establish a Schauder theory (Theorem 1.2). A preliminary step, Lemma 2.3, asserts that local minimizers are regular solutions near regular points. The paper then computes the second inner variation of E_s, derives a Sternberg–Zumbrun-type stability inequality (Theorem 1.3), uses it to prove that axially symmetric β-homogeneous minimizers are one-dimensional in low dimensions (Theorem 1.4), and reformulates the stability inequality as an eigenvalue bound on S^{d-1} (Proposition 5.1). A final section discusses the limit γ→-2 and claims convergence of the stability criterion to the minimal-surface criterion, under an explicit unproved uniform C^{2,α} compactness assumption (5.6).

Significance. If the results are correct, Theorem 1.1 completes the higher-regularity program for the Alt–Phillips problem by covering negative exponents, and the unified hodograph/Schauder approach is a genuine methodological advance. The Schauder estimates for degenerate quasilinear operators in Theorem 1.2 are of independent interest and are derived with parameter-free arguments and no fitted constants. The stability inequality and the axial-symmetry rigidity are new in the negative-exponent regime and establish a bridge to the Stability theory of minimal surfaces. The main caveat is that the advertised recovery of the minimal-surface stability criterion in the singular limit γ→-2 is conditional on assumption (5.6), which the authors explicitly state is not proved; the text is honest about this, but the abstract is not.

major comments (3)

[Section 2, Lemma 2.3] The proof of Lemma 2.3 is a two-paragraph sketch that is load-bearing for Theorems 1.1, 3.3, and 1.3. It asserts an improvement-of-flatness estimate valid for every α∈(0,1) with a radius ρ uniform in the flatness parameter ε, citing [20, Prop. 7.2] and [45, Thm. 1.1], then concludes by a 'standard iteration'. The manuscript should provide the full iteration: (i) justify that the radius ρ can be chosen independently of ε and uniformly for the required α > -s, in particular for α arbitrarily close to 1 as γ→-2; (ii) show explicitly that the pointwise weighted free-boundary condition lim_{t→0} w^s(|∇w|^2-1)=0 is preserved through the iteration; (iii) either reproduce the argument or give a precise reference for the claimed C^{1,α} upgrade with α > -s. Without this, Proposition 2.2 applies only to regular solutions, and the reduction from local minimizers to that class is incomplete.
[Abstract and Section 5.2] The abstract states that the variational criterion 'recovers the one for minimal surfaces in the singular limit as γ→-2', but the only result in this direction, Section 5.2, is conditional on assumption (5.6), which the authors explicitly do not prove ('The proof of assumption (5.6) is rather involved and would require a more detailed refinement of our regularity theory, which goes beyond the scope of this paper'). The abstract and introduction should either present this as a conditional result or incorporate assumption (5.6) into the advertised statement.
[Section 5.1, Proposition 5.1] The statement says that stability in R^d is equivalent to the eigenvalue lower bound (5.2), but the proof only establishes equivalence with the scalar inequality (1.4) for test functions of the separated form f(r,θ)=g(r)φ(θ). The converse direction, that the scalar inequality (1.4) implies the full inner-variation stability of Definition 3.1, is not shown. Please either restrict Proposition 5.1 to the scalar stability condition or supply the missing implication.

minor comments (6)

[Abstract] The sentence 'Such method provide a unified proof' should be corrected to 'This method provides a unified proof' or 'Such a method provides a unified proof'.
[Section 4, proof of Proposition 4.1, Step 3] The phrase 'ζε isfirstchosenasanaxiallysymmetriccut-offfunction, andsubsequentlyreplaced by a radial one' is missing spaces due to LaTeX formatting; it should read 'ζ_ε is first chosen as an axially symmetric cut-off function, and subsequently replaced by a radial one'.
[Theorem 1.4] The threshold 'γ <10 −8√5 11 ≈ −0.7171' is ambiguous; it should be written as γ < (10 − 8√5)/11 ≈ −0.7171.
[Section 5.2] The notation 'M := lim_{k→∞} ∂Σ_{w_k}' uses a limit of sets without specifying the notion of convergence; the subsequent argument suggests Hausdorff or varifold convergence, so the mode of convergence should be stated.
[References] Reference [39] (Pacati, Tortone, Velichkov) appears in the reference list but is not cited in the text; either add a citation or remove it.
[Section 5.1, Proposition 5.1] The proposition is stated in terms of w but says 'u is stable in R^d if and only if λ_s(Σ_w) ≥ ...'; since u and w are equivalent, the statement should clarify that stability of u is equivalent to the eigenvalue bound for w, to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main regularity theorem, the quasilinear Schauder theory, and the stability/cone results are genuine extensions built on external inputs; the overlapping-author citation [47] is an independent parameter-free theorem and Lemma 2.3, though compressed, is a derivation rather than an assumed conclusion.

full rationale

I walked the paper's derivation chain and found no step in which a claimed output is equivalent by construction to an input. Theorem 1.1 is proved through Proposition 2.2 for regular solutions, together with Lemma 2.3, which upgrades the known C^{1,\delta} regularity of local minimizers near regular free boundary points (from [20, Theorem 2.3]) to the C^{1,\alpha}, \alpha>-s, regularity required by Definition 2.1. The asserted target regularity is not assumed in Lemma 2.3; it is derived from the linearization and a stated iteration, so this is not self-definitional. The quasilinear Schauder estimates in Theorem 1.2 and Proposition 2.8 genuinely extend the linear theory of [47], and although [47] has an overlapping author, it is a published parameter-free theorem whose stated assumptions do not include the Alt-Phillips Euler-Lagrange system or the desired conclusion; hence the citation is independent support rather than a self-referential loop. The stability condition of Theorem 1.3 is obtained by computing the second inner variation (Lemmas 3.5 and 3.6) using the Euler-Lagrange equation and the free boundary condition that were already established for regular solutions; no fitted parameter is renamed as a prediction. The axial-symmetry result Theorem 1.4 and the spectral criterion Proposition 5.1 are algebraic consequences of that stability inequality together with Hardy-type inequalities. The asymptotic discussion in Section 5 explicitly declares the uniform regularity assumption (5.6) to be unproved and beyond the scope of the paper; that is an admitted hypothesis, not a disguised reuse of the target result. The weakest point is Lemma 2.3, whose proof is compressed and cites [20, Prop. 7.2] and [45, Theorem 1.1] before asserting a uniform improvement-of-flatness estimate and a standard iteration; this is a completeness or rigor risk in the exposition, but it is not circularity, since the cited inputs do not contain the conclusion being proved.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no invented entities. The central claims rest on standard mathematical tools, on the published C^1,α theory of De Silva and Savin, on the published linear Schauder theory of [47], and on the explicitly stated uniform regularity assumption (5.6) for the asymptotic analysis. The lack of fitted parameters and invented entities keeps the axiomatic load low; the main caveat is the unproven compactness assumption.

assumptions (5)

domain assumption Regularity theory of De Silva and Savin [20]: regular points of local minimizers have C^1,α free boundaries and the solution has the optimal Hölder exponent β; used as input in Lemma 2.3.
Invoked in Lemma 2.3 and throughout the paper; not reproved, and the paper relies on it to connect local minimizers to regular solutions.
standard math Linear degenerate Schauder theory from [47] (coauthored by G. Tortone): solutions of div(x_d^s ∇v)=0 with Neumann boundary condition are smooth up to the boundary; used in Lemma 2.7 and Proposition 2.8.
Published result, parameter-free; the present quasilinear Theorem 1.2 extends it, so the dependency is not circular, but it is the main external input for the smoothness proof.
standard math One-dimensional weighted Hardy inequality: inf ∫ r^{s+d-1}(g')² dr / ∫ r^{s+d-3}g² dr equals ((d+s-2)/2)²; used in Proposition 5.1.
Standard Hardy inequality for the weight r^{s+d-1}; the value is computed classically and is not proved in the paper.
ad hoc to paper Uniform C^2,α regularity and convergence of the interfaces as s_k→-1, stated as assumption (5.6) in Section 5.2; explicitly not proved in the paper.
Load-bearing for the claimed recovery of the minimal-surface stability criterion in the γ→-2 limit; the authors state this assumption is beyond the scope and would require a refinement of their regularity theory.
domain assumption Axial symmetry and regularity outside the origin for cones in Theorem 1.4: the theorem assumes ∂Ω_u is C^1,α-regular outside the origin and u is homogeneous and globally minimizing (or globally stable).
The proof of Proposition 4.1 requires w∈C∞(Ω_w∩(R^d\{0})) and a smooth free boundary outside the origin, plus the axial symmetry and homogeneity structure.

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Pith. "Pith review of Smoothness and stability in the Alt-Phillips problem." pith.science (2026). https://pith.science/paper/M75BIFAW

@misc{pith2026250710336,
  author       = {Pith},
  title        = {Pith review of: Smoothness and stability in the Alt-Phillips problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M75BIFAW}},
  note         = {Machine review of arXiv:2507.10336}
}

abstract

We study the one-phase Alt-Phillips free boundary problem, focusing on the case of negative exponents $\gamma \in (-2,0)$. The goal of this paper is twofold. On the one hand, we prove smoothness of $C^{1,\alpha}$-regular free boundaries by reducing the problem to a class of degenerate quasilinear PDEs, for which we establish Schauder estimates. Such method provide a unified proof of the smoothness for general exponents. On the other hand, by exploiting the higher regularity of solutions, we derive a new stability condition for the Alt-Phillips problem in the negative exponent regime, ruling out the existence of nontrivial axially symmetric stable cones in low dimensions. Finally, we provide a variational criterion for the stability of cones in the Alt-Phillips problem, which recovers the one for minimal surfaces in the singular limit as $\gamma \to -2$.

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Pith tools

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