Existence and regularity in the fully nonlinear one-phase free boundary problem

Bozhidar Velichkov; Matteo Carducci

REVIEW 1 major objections 5 minor 2 cited by

Existence and regularity in the fully nonlinear one-phase free boundary problem

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Pith's one-line read Viscosity solutions to fully nonlinear one-phase free boundary problems with a right-hand side and normal-dependent data exist by Perron's method, and flat free boundaries are necessarily $C^{2,\alpha}$, with higher smoothness following…

desk verdict Substantial paper with a real gap in the pointwise iteration: the slope in the Taylor expansion is evaluated at 0 instead of the free boundary point; fixable but Theorem 1.5 is not proven as written. read the letter →

arxiv 2501.12101 v1 pith:LEUEGFW6 submitted 2025-01-21 math.AP

Matteo Carducci , Bozhidar Velichkov This is my paper · ORCID

classification math.AP MSC 35R35

keywords regularityfreeboundaryfullynonlinearimprovementofflatnessone-phaseproblemPerron'smethodviscositysolutionsobliquecondition

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

The paper aims to show that the classical one-phase free boundary regularity story survives the move to general fully nonlinear uniformly elliptic operators with a right-hand side and a free boundary condition that depends on the inward normal. It proves existence of viscosity solutions by Perron's method for any continuous nonnegative boundary datum, and it proves that flat free boundaries are automatically $C^{2,\alpha}$—no variational structure is needed, only a flatness assumption. If the data are smoother, the boundary inherits $C^{k+2,\beta}$, $C^\infty$, or analytic regularity. The interest is that flatness alone forces second-order differentiability of the boundary in a setting where the only tools are viscosity methods and an improvement-of-flatness iteration.

What carries the argument

The central mechanism is the quadratic improvement of flatness, Proposition 4.1, proved by contradiction and compactness. From a solution trapped between $(g(0,e_d)x_d+p(x)-r^{1+\alpha})_+$ and $(g(0,e_d)x_d+p(x)+r^{1+\alpha})_+$, the rescaled linearized sequence $(u_n-g_n(0,e_d)x_d-p_n)/r_n^{1+\alpha}$ satisfies a partial Harnack inequality and converges to a viscosity solution of the linearized problem $\widetilde{F}(D^2\widetilde{u})=0$ in $B^+_{1/2}$, $\nabla\widetilde{u}\cdot\tau=0$ on $B'_{1/2}$. The expansion of this limit, stated as Proposition 2.7 and resting on the external estimate [LZ18, Theorem 1.3], gives a $C^{2,\alpha_0}$ rate that transfers back to the nonlinear scale; the new normal vector and the new quadratic polynomial are read off from $\nabla\widetilde{u}(0)$ and $D^2\widetilde{u}(0)$. Iterating at every free boundary point produces the $C^{2,\alpha}$ boundary in Theorem 1.5. For Corollary 1.6, the hodograph transform of [KN77] rewrites the problem as a nonlinear elliptic equation with oblique boundary condition, to which classical regularity results [ADN59, Mor08] apply.

What would settle it

The decisive test is to inspect the limit operator $\widetilde{F}_\infty$ that appears in the compactness step of Proposition 4.7 and ask whether it really satisfies the concavity/convexity and nondegenerate-oblique hypotheses of [LZ18, Theorem 1.3]; exhibiting an admissible sequence $F_n,g_n,u_n$ for which $\widetilde{F}_\infty$ violates one of those hypotheses, or producing a flat viscosity solution whose free boundary fails to be $C^{2,\alpha}$ under (H1)–(H3), would refute Theorem 1.5.

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Extended reading notes

Core claim

On its own terms, the paper establishes Theorem 1.5: there exist universal constants $\varepsilon_0>0$ and $\alpha_0\in(0,1)$ such that if $F\in E(\lambda,\Lambda)$, $f\in C^{0,\beta}(B_1)$, $g\in C^{1,\beta}(B_1,S^{d-1})$ satisfy (H1)–(H3), and a nonnegative continuous viscosity solution $u$ of (1.1) is trapped between $(g(0,\nu)(x\cdot\nu)-\varepsilon_0)_+$ and $(g(0,\nu)(x\cdot\nu)+\varepsilon_0)_+$, then $u\in C^{2,\alpha}(\Omega_u\cap B_{1/2})$ and the free boundary $\partial\Omega_u\cap B_{1/2}$ is a $C^{2,\alpha}$ manifold. The route is a quadratic improvement of flatness: at each scale the solution lies within error $r^{1+\alpha}$ of a quadratic polynomial solution of a linearized fully nonlinear oblique problem, and iteration over boundary points yields a second-order Taylor expansion with uniform rate. Corollary 1.6 then applies the hodograph transform to convert the $C^{2,\alpha}$ free boundary into a $C^{k+2,\beta}$, $C^\infty$, or analytic boundary according to the smoothness of $F$, $f$, and $g$. Theorem 1.4 supplies existence of a viscosity solution via Perron's method, with Lipschitz regularity and non-degeneracy of the constructed solution.

Load-bearing premise

The argument rests on an external $C^{2,\alpha}$ estimate for the linearized oblique problem (Proposition 2.7, citing [LZ18, Theorem 1.3]) that requires the operator to be concave or convex in the Hessian and the oblique direction to stay nondegenerate; the paper asserts these hypotheses pass to the limiting operator through (H2) and compactness but does not verify them in detail.

Editorial extensions

If this is right

Flat free boundaries are automatically $C^{2,\alpha}$ even when the operator is fully nonlinear, the right-hand side is nonzero, and the free boundary condition depends on the normal.
The quadratic improvement of flatness yields a uniform second-order Taylor expansion of the solution at every free boundary point, which is exactly the input needed for the hodograph transform.
If $F$, $f$, and $g$ are $C^{k,\beta}$, $C^{k,\beta}$, and $C^{k+1,\beta}$ respectively, the free boundary is $C^{k+2,\beta}$; for $C^\infty$ or analytic data the boundary is $C^\infty$ or analytic.
Perron's method gives a viscosity solution for any continuous nonnegative boundary datum, and the constructed solution is Lipschitz and non-degenerate in compact sets.
The Perron construction deliberately excludes degenerate or collapsed-boundary solutions such as $u(x)=x_d^2/2$ or $u(x)=c|x_d|$, clarifying that the existence result covers a special class of viscosity solutions.

Reading between the lines

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

A natural next target is the analogous two-phase problem with fully nonlinear operators and right-hand side; the same linearized oblique estimate would be the bottleneck, and the paper's partial Harnack argument looks transferable, but that is an extension, not a claim of the paper.
If the hypotheses of the external linearized estimate could be relaxed, the convexity/concavity assumption (H2) in Theorem 1.5 might be weakened; the present proof does not test this.
The dichotomy in the admissible family suggests a template for other non-homogeneous free boundary problems where the right-hand side prevents naive barrier constructions; one could test whether Perron's solutions vary continuously with the strict minorant and the boundary datum.

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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 strong paper that genuinely advances the one-phase free boundary theory for fully nonlinear operators with right-hand side and normal-dependent boundary condition. The existence theorem via a carefully chosen admissible family is a real novelty, and the C^{2,α} flatness strategy is the right approach. The technical work in Sections 3–4 is mostly careful, and the reliance on [LZ18] for the linearized problem is appropriate, though the hypotheses should be checked more explicitly.

The main problem is in Proposition 5.1 and the proof of Theorem 1.5. Proposition 5.1 states the second-order expansion with coefficient g(0,ν_{x0}). At a free boundary point x0, the free boundary condition gives |∇u(x0)| = g(x0,ν_{x0}), so the correct linear coefficient is g(x0,ν_{x0}). The proof rescales around x0 and applies Proposition 4.1 to the rescaled problem, whose boundary data is g_{x0,δ} with g_{x0,δ}(0,e_d) = g(x0,e_d). Proposition 4.1 therefore requires flatness relative to g(x0,e_d). But the proof only derives flatness relative to g(0,e_d) from the global assumption. The difference is a linear term that cannot be absorbed into the quadratic polynomial p0, and for x0 away from the origin it is not small. The paper gives no argument that |g(x0,e_d) − g(0,e_d)| is controlled by the flatness scale. Without that, the iteration cannot start at arbitrary free boundary points.

This is a real gap, but I think it is fixable. A compactness argument should show that the global flatness with small ε0 forces g(x0,e_d) to be close to g(0,e_d) uniformly on the free boundary; the heuristic is that a flat solution is close to a half-plane with slope g(0,e_d), so the normal derivative is close to that value, and the free boundary condition then forces g(x0,e_d) close to g(0,e_d). The current manuscript does not contain that argument.

The Perron existence proof and the higher-regularity bootstrapping look solid. The citation pattern is reasonable; the single self-citation is standard. No fitted parameters. This paper deserves a serious referee, but the referee should prominently flag the Proposition 5.1 issue and request a fix before the regularity theorem can be accepted as stated.

Referee Report

1 major / 5 minor

Summary. The paper studies the one-phase free boundary problem for fully nonlinear elliptic operators with a right-hand side and a free boundary condition |∇u| = g(x,ν) depending on the normal. It contains three main results: (i) existence of viscosity solutions by Perron's method (Theorem 1.4), together with local Lipschitz regularity and non-degeneracy; (ii) C^{2,α} regularity of flat free boundaries via a quadratic improvement of flatness (Theorem 1.5); and (iii) higher regularity of the free boundary by a hodograph transform (Corollary 1.6). The existence part follows a Perron-type construction with an admissible family of supersolutions and a strict minorant, and the regularity part uses a compactness/linearization argument leading to an oblique-boundary problem for the linearized operator. The hodograph step is standard once C^{2,α} regularity is available.

Significance. If correct, the paper would extend the De Silva-type improvement-of-flatness theory to fully nonlinear one-phase problems with nonzero right-hand side and normal-dependent boundary data, and would in particular give the first C^{2,α} flat-free-boundary theorem in that generality. The Perron existence theorem is a solid contribution in itself, and the paper is largely self-contained: no free parameters are fitted, and the central arguments rely on external regularity theorems rather than circular reasoning. However, the statement and proof of Proposition 5.1 contain a mismatch between the coefficient g(0,ν) and the local value g(x0,ν) that currently breaks the proof of Theorem 1.5. The gap is substantial but appears repairable by reworking the flatness iteration around the local coefficient g(x0,ν).

major comments (1)

[Section 5, Proposition 5.1] The first-order coefficient in the claimed Taylor expansion is the wrong one. At a point x0 ∈ ∂Ω_u ∩ B_{1/2} the free boundary condition gives |∇u(x0)| = g(x0, ν_{x0}), so any expansion u(x0+x) = a (x·ν_{x0}) + O(|x|^2) must have a = g(x0, ν_{x0}). Proposition 5.1 instead states, and the proof concludes, a = g(0, ν_{x0}); for g depending on x these differ by an O(1) amount, so the proposition is false as stated unless g is independent of x. The source of the error is visible in the iteration: for the rescaled datum g_n(x,ν) = g_{x0,ρ_n}(x,ν) = g(x0 + ρ_n x, ν), the coefficient in Proposition 4.1 is g_n(0,ν) = g(x0,ν), not g(0,ν). When the proof of Proposition 5.1 asserts the initial flatness (g(0,e_d)x_d − r0^{1+β})_+ ≤ u_{x0,δ} ≤ (g(0,e_d)x_d + r0^{1+β})_+ for u_{x0,δ}(x) = u(x0+δx)/δ, the linear term [g(x0,e_d) − g(0,e_d)]x_d is not absorbable into p0 ∈ P(e_d, F_{x0,δ}, ...), since the class P in (1.7) contains only homogeneous quadratic polynomials and the difference is not small uniformly for |x0| of order 1. Thus the first application of Proposition 4.1 is unjustified, and the iteration proving Theorem 1.5 collapses. The proof needs either a localization argument producing flatness at x0 relative to g(x0,e_d), or a reformulation of the flatness hypothesis, and the statement of Proposition 5.1 should use g(x0, ν_{x0}).

minor comments (5)

[Section 2.6 and Section 4.4] Please add a short verification that the limiting operator ~F∞ in (4.9) and the oblique vector τ in (4.10) satisfy the hypotheses of [LZ18, Theorem 1.3]. The verification is straightforward — concavity/convexity passes to the uniform limit of the difference quotients in (4.8), and τ·e_d = 1 is preserved — but it should be written out because Proposition 2.7 is a load-bearing external input.
[Section 4.5, Eq. (4.14)] The displayed formula for ν_n − e_d is garbled: it should first define the unit vector ν_n as the normalization of g_n(0,e_d)e_d + r^{1+α}ν and then expand the difference. The conclusion |ν_n − e_d| ≤ C r^{1+α} is correct, but the displayed computation should be fixed.
[Section 5, Lemma 5.2] The notation [g_{x0,δ}]_{C^{1,β}(B1)} should specify that the norm is taken only in the spatial variable x. Derivatives of g_{x0,δ} with respect to ν are not rescaled by δ and therefore cannot satisfy the stated smallness bound; the later use in the proof of Proposition 4.1 only needs boundedness of the ν-derivatives together with smallness of the x-derivatives.
[Section 1.2.2, Theorem 1.5] The conclusion '∂Ω_u ∩ B_1 is a (d−1)-dimensional manifold of class C^{2,α} in B_{1/2}' should read '∂Ω_u ∩ B_{1/2}'.
[Section 3.4, proof of Proposition 3.10] The boundary conditions for the auxiliary function v are stated inconsistently: 'v = 0 in B1 \ D_σ' and 'v = h on ∂B1' overlap on ∂B1 ∩ D_σ. The intended boundary split should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main results are derived from stated hypotheses plus external regularity theorems; the only self-citation is for standard compactness arguments and is not load-bearing.

full rationale

The derivation chain is self-contained and non-circular. The central regularity result (Theorem 1.5) follows from the quadratic improvement of flatness (Prop. 4.1), whose hypotheses (HF)-(Hp) are stated assumptions on F, f, g and a polynomial class P defined by (1.5), not by the conclusion. The contradiction proof produces the limiting linearized problem (4.7), then applies the external C^{2,α} estimate Prop. 2.7 = [LZ18, Thm 1.3]; this is a genuine external input with stated assumptions (concavity/convexity (H2), τ·e_d = 1, |τ| ≤ C), and the paper explicitly identifies α0 with the constant of that theorem. The iteration in Section 5 rescales the solution by (1.8) and verifies the hypotheses of Prop. 4.1 via Lemma 5.2; the size of the new polynomial is controlled by (4.1), giving a rate of convergence. No fitted parameter is renamed as a prediction: the flatness scale ε0 is chosen universal, and the expansion coefficient is the given boundary datum g(0,ν) evaluated at the origin, not a fitted value. The only self-citation is [Vel23] in Corollary 4.6 and Theorem 1.5 for 'standard arguments' (Hausdorff convergence of graphs and rate-of-convergence to C^{2,α} regularity); these routine compactness/regularity steps do not carry the load of the main estimate and are also cross-referenced to De Silva [DeS11]. Whether the coefficient in Prop. 5.1 should be g(x0,ν_{x0}) instead of g(0,ν_{x0}) is a question of correctness of the statement, not circularity: it does not make the conclusion equivalent to the hypothesis by construction.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central results rest on the stated structural hypotheses (uniform ellipticity, positive lower bound for g, concavity/convexity (H2), Hölder regularity) and on external regularity theorems ([LZ18], [SS14], [KT20], [ADN59], [Mor08]). No free parameters are fitted and no new entities are postulated.

assumptions (8)

domain assumption F is uniformly elliptic with constants 0<λ≤Λ and Lipschitz in ξ, with F(0,0,x)≡0 (condition (1.2)).
Defines the class E(λ,Λ); used in all sections.
domain assumption g satisfies inf g ≥ γ0 > 0 (H1).
Needed for non-degeneracy, barriers, and the half-plane approximating solutions; stated in Section 1.
domain assumption F(·,ξ,x) is concave or convex (H2).
Required to apply the C^{2,α0} estimate [LZ18] for the linearized oblique problem in Proposition 2.7; this is the key regularity input.
domain assumption F is C^{0,β} in x, f is C^{0,β}, g is C^{1,β} (H3 and Theorem 1.5 hypotheses).
Regularity hypotheses used in the improvement of flatness (Lemma 5.2 and Proposition 4.1).
standard math C^{2,α0} estimates for viscosity solutions of the linearized oblique problem hold as asserted in Proposition 2.7, citing [LZ18, Theorem 1.3].
This external theorem is the load-bearing input for the quadratic improvement of flatness (Proposition 4.1).
standard math The obstacle problem for fully nonlinear operators with gradient dependence has C^{1,γ}_loc viscosity solutions, cited from [KT20, Theorem 2.11, 2.12].
Used in Lemma 3.3 to replace admissible supersolutions by solutions of the interior equation.
standard math Boundary C^{1,α} regularity estimates for fully nonlinear equations on the cone-like domain Dσ apply, cited from [SS14, Theorem 1.1].
Used in Proposition 3.10 to construct the supersolution v with controlled gradient.
standard math The complementing condition and the Schauder theory of [ADN59, Mor08] apply to the hodograph-transformed problem (6.3).
Used in Corollary 1.6 to lift C^{2,α} regularity to C^{k+2,β} and analyticity.

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Pith. "Pith review of Existence and regularity in the fully nonlinear one-phase free boundary problem." pith.science (2026). https://pith.science/paper/LEUEGFW6

@misc{pith2026250112101,
  author       = {Pith},
  title        = {Pith review of: Existence and regularity in the fully nonlinear one-phase free boundary problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LEUEGFW6}},
  note         = {Machine review of arXiv:2501.12101}
}

abstract

We consider viscosity solution to one-phase free boundary problems for general fully nonlinear operators and free boundary condition depending on the normal vector. We show existence of viscosity solutions via the Perron's method and we prove $C^{2,\alpha}$ regularity of flat free boundaries via a quadratic improvement of flatness. Finally, we obtain the higher regularity of the free boundary via an hodograph transform.

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10 extracted references · 10 canonical work pages · cited by 2 Pith papers

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[ADN59] S. Agmon, A. Douglis, and L. Nirenberg. Estimates near the boundary for solutions of elliptic partial diﬀerential equations satisfying general boundar y conditions. I . Comm. Pure Appl. Math. 12 (1959), pp. 623–727. [AC81] H. W. Alt and L. A. Caﬀarelli. Existence and regularity for a minimum problem with free bou ndary. J. Reine Angew. Math. 325 (...

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Agmon, A

[ADN59] S. Agmon, A. Douglis, and L. Nirenberg. Estimates near the boundary for solutions of elliptic partial diﬀerential equations satisfying general boundar y conditions. I . Comm. Pure Appl. Math. 12 (1959), pp. 623–727. [AC81] H. W. Alt and L. A. Caﬀarelli. Existence and regularity for a minimum problem with free bou ndary. J. Reine Angew. Math. 325 (...

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Salsa, F

[STV19] S. Salsa, F. Tulone, and G. Verzini. Existence of viscosity solutions to two-phase problems for fully nonlinear equations with distributed sources . Math. Eng. 1:1 (2019), pp. 147–173. [SS14] L. Silvestre and B. Sirakov. Boundary regularity for viscosity solutions of fully nonli near elliptic equations. Comm. Partial Diﬀerential Equations 39:9 (20...

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New York University, Courant Institute of Mathema tical Sciences, New York; American Mathematical Society, Providence, RI, 1997, pp

Courant Lecture Notes in Math- ematics. New York University, Courant Institute of Mathema tical Sciences, New York; American Mathematical Society, Providence, RI, 1997, pp. x+144. [KN77] D. Kinderlehrer and L. Nirenberg. Regularity in free boundary problems . Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4:2 (1977), pp. 373–391. [KNS78] D. Kinderlehrer, L. Nir...

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Springer, Cham, 2023, pp

Lecture Notes of the Unione Matematica Italiana. Springer, Cham, 2023, pp. xiii+246. [Wan00] P. Wang. Regularity of free boundaries of two-phase problems for ful ly nonlinear elliptic equations of second order. I. Lipschitz free boundaries are C 1,α . Comm. Pure Appl. Math. 53:7 (2000), pp. 799–810. [Wan02] P. Wang. Regularity of free boundaries of two-ph...

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Argiolas and F

[AF09] R. Argiolas and F. Ferrari. Flat free boundaries regularity in two-phase problems for a class of fully nonlinear elliptic operators with variable coeﬃcien ts. Interfaces Free Bound. 11:2 (2009), pp. 177–199. [CC95] L. A. Caﬀarelli and X. Cabré. Fully nonlinear elliptic equations . Vol

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EMS Press, Berlin, 2022, pp

Zurich Lectures in Advanced Mathematics. EMS Press, Berlin, 2022, pp. viii+

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American Mathemati cal Society, Providence, RI, 1995, pp

American Mathemat- ical Society Colloquium Publications. American Mathemati cal Society, Providence, RI, 1995, pp. vi+104. [CCKS96] L. A. Caﬀarelli, M. G. Crandall, M. Kocan, and A. Swi ęch. On viscosity solutions of fully nonlinear equations with measurable ingredients . Comm. Pure Appl. Math. 49:4 (1996), pp. 365–397. [Caf87] L. Caﬀarelli. A Harnack ine...

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American Mathematical Society, Pr ovidence, RI, 2005, pp

Graduate Studies in Mathematics. American Mathematical Society, Pr ovidence, RI, 2005, pp. x+270. [CFS04] M. C. Cerutti, F. Ferrari, and S. Salsa. Two-phase problems for linear elliptic operators with variable coeﬃcients: Lipschitz free boundaries are C 1,γ . Arch. Ration. Mech. Anal. 171:3 (2004), pp. 329–348. [DRR V23] J. Da Silva, G. C. Rampasso, G. C....

work page 2004