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REVIEW 3 major objections 4 minor 68 references

Tangential touch between free and fixed boundaries for the fully nonlinear Alt-Phillips problem

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read For the fully nonlinear Alt-Phillips problem with γ in (1,2), every contact between the free boundary and the fixed boundary is tangential, and the proof is a uniqueness classification of global half-space solutions.

desk verdict Genuinely new tangency result for γ∈(1,2), but Lemma 4.1 is false as stated; the main theorem is likely salvageable with an extra hypothesis that the application already provides. read the letter →

arxiv 2509.02064 v1 pith:QHINLDGC submitted 2025-09-02 math.AP

Yamin Wang , Hui Yu This is my paper
classification math.AP MSC 35R3535J6035B40
keywords Alt-Phillipsproblemfreeboundaryfixedtangentialcontactfullynonlinearellipticequationsblow-upanalysisglobalclassificationdegeneratesemilinearequation
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

This paper proves that, for the fully nonlinear Alt-Phillips problem with parameter γ in (1,2), the free boundary—the boundary of the region where the solution is positive—meets the fixed boundary only tangentially at points where the Dirichlet data vanish. The result is new even for the Laplacian. The proof rescales a solution at a contact point and shows that every limit of the rescaled solutions is the same explicit one-dimensional profile, namely u(x)=((√2/β)(x_n)_+)^β with β=2/(2−γ). Because that profile's positive region stays strictly away from the fixed boundary in the limit, the free boundary approaching the contact must flatten onto the fixed boundary. The classification of the limiting profile is the core of the argument.

What carries the argument

The engine is the blow-up analysis tied to a uniqueness classification. With β=2/(2−γ), rescale u near a contact point by u_r(x)=u(rx)/r^β. Growth estimates give a uniform upper bound, nondegeneracy gives a lower bound, so rescaled solutions converge along subsequences; the fully nonlinear operator degenerates to the Laplacian in the limit, and the combination of monotonicity and improvement-of-monotonicity forces the limit to be β-homogeneous. The classification then shows that the only β-homogeneous half-space solution of Δu=γu^{γ-1} with zero boundary data is the explicit one-dimensional profile. This one profile has no free-boundary contact in the limit, and that absence of contact is wh

What would settle it

Compute, for n=2, γ=3/2 and F=Δ, the exact or numerical solution of (1.1) with 0 on the free boundary and inspect the rescaled sequence r^{-β}u(rx). If any subsequential limit is not ((√2/β)(x_n)_+)^β, or if rescaled free-boundary points accumulate along a positive angle to {x_n=0}, Theorem 1.1 fails. More directly, any nonzero global half-space solution of Δu=γu^{γ-1} with r^β growth and nondegeneracy different from the explicit profile falsifies Lemma 4.1.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: if F is a uniformly elliptic, convex, C^1 fully nonlinear operator with F(0)=0 and trace normalization, γ∈(1,2), and u solves the Alt-Phillips system (1.1) with 0 a free-boundary point on the fixed boundary, then the free boundary near 0 is contained in {x_n ≤ σ(|x|)|x|} for a universal modulus σ tending to 0. In other words, contact is tangential. The proof rests on Lemma 4.1, which classifies all global solutions of the limiting classical problem Δu=γu^{γ-1} in the half-space with zero boundary data and the natural growth and nondegeneracy bounds: the only such solution is the one-dimensional explicit profile. Blowing up any solution at a contact point yie

Load-bearing premise

The load-bearing premise is that every solution is differentiable up to the free boundary with both the value and the gradient vanishing there; if that boundary regularity were false for some γ∈(1,2) or some operator F, the classification of blow-up limits would not apply to actual solutions.

Editorial extensions

If this is right

  • Every free-boundary point on the fixed boundary is a tangential contact, for the Laplacian and for every operator in the class; no free boundary can enter the domain at a positive angle.
  • At a contact point, every blow-up of the solution is the same explicit power profile, so the leading-order shape of the solution near the contact is universal.
  • The flatness of the free boundary is controlled by a universal modulus σ(|x|) that depends only on dimension, γ, ellipticity, and the operator's modulus of continuity, not on the individual solution.
  • This completes the picture for the Laplacian in the range γ∈(1,2), where no boundary-touch result existed before, and matches the classical tangential-contact results for γ=0 and γ=1.
  • The classification of global half-space solutions implies rigidity: any half-space solution with the right growth and nondegeneracy is one-dimensional, so the free boundary cannot branch from the fixed boundary.

Reading between the lines

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

  • The same mechanism likely governs energy minimizers of the Alt-Phillips functional in this range, since the argument uses the PDE rather than minimality; this would unify interior and boundary regularity for minimizers.
  • Because the limiting profile is explicit, a full asymptotic expansion of u near the contact point could be obtained by linearizing around that profile; the overdetermined condition u=|∇u|=0 at the free boundary would fix the expansion coefficients.
  • The paper leaves γ∈(0,1) open, and the stated reasons—no convexity of global solutions and a continuum of homogeneous solutions—suggest that a direct extension of this classification will require new stability inputs rather than a rerun of the same proof.
  • A straightforward numerical check for the Laplacian at γ=3/2 in two dimensions could test the claimed universality: the rescaled free boundary should approach the fixed boundary with the slope bound σ(|x|)|x|, and all blow-ups should be the explicit profile.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the fully nonlinear Alt–Phillips problem in the upper half-ball with zero Dirichlet data on the fixed boundary, for 1 < γ < 2. It claims that the free boundary Γ(u) = ∂{u>0} ∩ {x_n>0} meets the fixed boundary tangentially wherever it touches points where the Dirichlet data vanish. The proof proceeds by rescaling at a contact point and proving a classification lemma (Lemma 4.1) for global solutions of the limiting classical problem Δu = γu^{γ-1} in the half-space: the asserted conclusion is that every such solution with the stated growth and nondegeneracy assumptions is the one-dimensional profile (√2 (x_n/β)_+)^β, β = 2/(2-γ). From this, the authors conclude that blow-up limits have empty free boundary, giving the tangential-touch statement. The argument uses a Weiss-type monotonicity formula, an improvement-of-monotonicity lemma, and boundary regularity estimates.

Significance. If the main theorem and the supporting classification were correct, the result would be a meaningful advance: the tangential-touch property is new for γ ∈ (1,2) even for the Laplacian, and the strategy of reducing boundary contact to a classification of global half-space solutions is attractive. The paper also offers a fully nonlinear extension of a line of results known previously only for γ = 0 and γ = 1. However, the central classification lemma is false as stated, and there are substantial unproved regularity assertions about the free boundary. The core idea is promising and likely repairable, but the manuscript in its current form cannot be accepted.

major comments (3)
  1. [Section 4, Lemma 4.1 and the paragraph after (4.21)] Lemma 4.1 is false as stated. For any c > 0, let g_c solve g_c'' = γ g_c^{γ-1}, g_c(0)=0, g_c'(0)=c. The first integral gives (g_c')^2/2 - g_c^γ = c^2/2, so g_c is positive and smooth on (0,∞), C^{1,α} up to 0, and u_c(x) = g_c(x_n) solves (4.1). For this u_c, Γ(u_c) = ∅, so hypothesis (ii) is vacuous, and (i) holds with a = min(c/2, (√2/β)^β/2) because g_c(t) ≥ ct for small t and g_c(t) ∼ (√2 t/β)^β for large t. Yet u_c is not the claimed profile. The proof's final step, 'Then (4.16) follows immediately', is invalid because the ODE g''=γg^{γ-1} with g(0)=0 has a one-parameter family of solutions g_c. The classification needs an additional hypothesis, e.g. |∇u(0)|=0, which is available for the blow-up limits used in the proof of Theorem 1.1 through Lemma 3.1. This false lemma is load-bearing for the advertised uniqueness claim and for the identification of the blow-up limit in the proof
  2. [Section 2.2, the paragraph beginning 'Let u be a continuous viscosity solution'] The assertion 'This implies an overdetermined condition on the free boundary, namely, u = |∇u| = 0 on Γ(u)' is not a consequence of Evans–Krylov regularity alone. Evans–Krylov gives classical solvability in {u>0}, not regularity up to the free boundary Γ(u). This overdetermined condition is used throughout: in Lemma 3.1, in condition (3.1) of Lemma 3.2, and in the compactness argument leading to (3.16). If this is a theorem proved in the authors' earlier work [WY], it must be cited explicitly at the point of use with the exact statement; otherwise it must be proved in this paper. As it stands, the regularity input is missing and is a load-bearing gap.
  3. [Section 3, proof of Lemma 3.2, around (3.10)–(3.11)] In the proof of the growth estimate, the authors write 'Due to the above and (3.10), D u_j(x_j)=0, D^2 u_j(x_j)=0'. The bound (3.10) is a C^{2,α} estimate in B^+_{3/4}, which does not by itself imply that the full Hessian vanishes at an interior free-boundary point x_j. The condition u_j=0 on one side of Γ(u_j) forces the tangential second derivatives to vanish, but not necessarily the normal second derivative. Therefore the Taylor estimate sup_{B_{2^{-k}}} u_j ≤ C2^{-k(2+α)} in (3.11) is not justified. This step is used to rescale the equation to obtain the limiting harmonic function in (3.12), so the gap affects the entire compactness argument. The missing input is an additional free-boundary regularity statement (vanishing of D^2 at free-boundary points) that is neither proved nor cited.
minor comments (4)
  1. [Section 2.3, Lemma 2.7] The notation 'eC0' is confusing. It appears to denote a large constant, but the exponential notation invites misreading. Please introduce a named constant, e.g. K, and state the choices of K and c0 explicitly.
  2. [Section 3, proof of Lemma 3.2] There are several typos in this proof: 'yn ≥ −ln' should be 'yn ≥ −l_j' in multiple places, and 'Arzalà-Ascoli' in Section 3 should be 'Arzelà–Ascoli'. These do not affect the mathematics but should be corrected.
  3. [Section 4, Lemma 4.3] In Case 2 of Step 1, the text reads 'there exists r1(p) such that ∂xn u(p)=0 in ...'; the argument of the function should be x, not p. Please fix.
  4. [Proof of Theorem 1.1] In the contradiction argument, the same symbol u is used for the fixed solution and for the sequence u_j. After passing to a subsequence, the rescaling of u_j at the origin should be explicitly distinguished; the invocation of (3.16) requires uniform compactness for the sequence, which should be stated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found: the profile and tangency are derived from the classification argument, not assumed; the same-author citation [WY] is contextual, and the main gaps are correctness issues rather than circular reductions.

full rationale

I walked the derivation chain. Theorem 1.1 is a parameter-free qualitative statement: no fitted constants enter, and the explicit profile (4.2) is obtained by proving beta-homogeneity (Lemma 4.2), monotonicity (Lemma 4.3), reduction to an ODE (Lemma 4.4), and then transferring monotonicity back to u in Lemma 4.1. At no point is the target tangency or the one-dimensional profile used as an input; they emerge from the proof. The only same-author citation, [WY], is used in the introduction for background (convexity of global solutions and interior free-boundary regularity for gamma in (1,2)) and is not invoked as the justification for the classification or the tangency. The Section 2.2 assertion that the overdetermined condition u=|∇u|=0 on Γ(u) follows from Evans-Krylov is not supported by the cited estimate, and this is a genuine missing-proof/correctness concern; however, it is an input assumption, not a conclusion secretly built into the definition. Similarly, the classification Lemma 4.1 appears to admit the one-parameter family u_c(x)=g_c(x_n) with (g_c')^2=c^2+2g_c^gamma and g_c(0)=0, for which Γ(u_c)=∅ so hypothesis (ii) is vacuous, making (4.22) unjustified without an extra hypothesis. This is a serious correctness gap in the proof of Lemma 4.1, but it is not circularity: the statement of Lemma 4.1 is not equivalent to its assumptions by construction, and no fitted parameter is renamed as a prediction. I therefore find no circular step and assign score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

Everything the proof rests on beyond standard published theorems is either a structural hypothesis on the operator F ((2.1)-(2.4)) or a regularity input for the degenerate problem. The free parameters list is empty: β, the profile constant, and the universal constants are derived or chosen existentially, not fitted to data. No new entities are postulated. The main weight sits on the asserted free-boundary regularity (u = |∇u| = 0 on Γ(u), Section 2.2), which is imported from the prior interior regularity theory of the same research program rather than re-proved here.

assumptions (6)
  • domain assumption F is uniformly elliptic with constant Λ, convex, C^1, F(0) = 0, |DF(M) - DF(0)| ≤ ω(|M|), and DF(0)(M) = trace(M) for all M (assumptions (2.1)-(2.4)).
    Defines the admissible operator class. The normalization (2.4) makes every blow-up limit solve the pure Laplacian equation in Lemma 2.1; without it the limit operator is anisotropic and the Section 4 classification would not directly apply.
  • domain assumption Solutions are C^1 up to the free boundary with u = |∇u| = 0 on Γ(u) (overdetermined condition, asserted in Section 2.2 and used in Lemma 3.1).
    Load-bearing: every growth, compactness, and blow-up step (Lemmas 3.2, 3.3, 3.4) uses it. Classical solvability inside the positive set does not imply gradient vanishing on the free boundary; this is a regularity theorem for the degenerate class, plausibly from [WY], cited elsewhere but not at this point of use.
  • standard math C^{1,α} and C^{2,α} regularity up to the flat fixed boundary for fully nonlinear equations F(D^2u) = f with bounded f, cited to [LZ] and [SS].
    Used in Lemma 3.2 (boundary C^{1,α} of rescalings, C^{2,α} bound (3.10)) and Lemma 3.3. Standard published results; not re-proved.
  • standard math Weiss monotonicity formula for the classical Alt-Phillips equation with the Laplacian (Lemma 2.5), 'essentially due to Weiss [W1]'.
    Gives β-homogeneity of blow-down limits in Lemma 4.2; the proof of homogeneity requires W(u,R) ≤ C along a sequence, asserted for the global solutions considered.
  • standard math Blow-up invariance lemma for β-homogeneous functions (Lemma 2.6), cited to [V, Lemma 10.9].
    Used in Lemma 4.3, Case 3, to reduce to the (n-1)-dimensional problem.
  • domain assumption Comparison principle for the fully nonlinear problem (Lemma 2.4), including the singular right-hand side γv^{γ-1}, stated without proof.
    Used in Lemma 3.4 (nondegeneracy) and Lemma 2.7 (barrier). The statement is plausible and standard for this class, but the paper does not prove it and the singular exponent v^{γ-1} needs care.

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Pith. "Pith review of Tangential touch between free and fixed boundaries for the fully nonlinear Alt-Phillips problem." pith.science (2026). https://pith.science/paper/QHINLDGC

@misc{pith2026250902064,
  author       = {Pith},
  title        = {Pith review of: Tangential touch between free and fixed boundaries for the fully nonlinear Alt-Phillips problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QHINLDGC}},
  note         = {Machine review of arXiv:2509.02064}
}
abstract

For the fully nonlinear Alt-Phillips problem with parameter $\gamma\in(1,2)$, we show that the free boundary intersects the fixed boundary tangentially where the Dirichlet data vanish. For this range of $\gamma$, this result is new even when the operator is the Laplacian.

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Works this paper leans on

68 extracted references · 66 canonical work pages

  1. [1]

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

    work page 1981

  2. [2]

    H. W. Alt, D. Phillips, A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math. , 368 (1986), 63-107

    work page 1986

  3. [3]

    Andersson, On the regularity of a free boundary near contact points with a fixed boundary, J

    J. Andersson, On the regularity of a free boundary near contact points with a fixed boundary, J. Differential Equations, 232 (2007), 285-302

    work page 2007

  4. [4]

    Andersson, N

    J. Andersson, N. Matevosyan, H. Mikayelyan, On the tangential touch between the free and the fixed boundaries for the two-phase obstacle-like problem, Ark. Mat., 44 (2006), 1-15

    work page 2006

  5. [5]

    Andersson, H

    J. Andersson, H. Mikayelyan, On the non-tangential touch between the free and the fixed boundaries for the two-phase obstacle-like problem, Math. Ann., 352 (2012), 357-372

    work page 2012

  6. [6]

    Andersson, H

    J. Andersson, H. Shahgholian, Global solutions of the obstacle problem in half-spaces, and their impact on local stability, Calc. Var. Partial Differential Equations, 23 (2005), 271-279

    work page 2005

  7. [7]

    D. E. Apushkinskaya, N. N. Uralotseva, On the behavior of the free boundary near the boundary of the domain, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 221 (1995); Kraev. Zadachi Mat. Fiz. i Smezh. Voprosy Teor. Funktsii. 26, 5–19, 253

    work page 1995

  8. [8]

    D. E. Apushkinskaya, N. N. Uralotseva, Boundary estimates for solutions of two-phase obstacle problems, J. Math. Sci. (N. Y.) 142 (2007), 1723–1732, Problems in mathematical analysis. No. 34

    work page 2007

Show all 68 references
  1. [9]

    Aris, The mathematical theory of diffusion and reaction in permeable catalysts, Oxford 1975

    R. Aris, The mathematical theory of diffusion and reaction in permeable catalysts, Oxford 1975

    1975

  2. [10]

    Alberti, L

    G. Alberti, L. Ambrosio, X. Cabr\'e. On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math. , 65 (2001), no. 1-3, 9-33

    2001

  3. [11]

    Audrito, J

    A. Audrito, J. Serra, Interface regularity for semilinear one-phase problems, Adv. Math. , 403 (2022), Paper No. 108380

    2022

  4. [12]

    Bonorino, Regularity of the free boundary for some elliptic and parabolic problems

    L. Bonorino, Regularity of the free boundary for some elliptic and parabolic problems. I, Comm. Partial Differential Equations , 26 (2001), 175-203

    2001

  5. [13]

    Bonorino, Regularity of the free boundary for some elliptic and parabolic problems

    L. Bonorino, Regularity of the free boundary for some elliptic and parabolic problems. II, Commun. Partial Differ. Equ., 26 (2001), 355-380

    2001

  6. [14]

    Bombieri, E

    E. Bombieri, E. De Giorgi, E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. , 7 (1969), 243-268

    1969

  7. [15]

    Caffarelli, The obstacle problem revisited, J

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

    1998

  8. [16]

    Cabr\'e, I

    X. Cabr\'e, I. Erneta, J. Felipe-Navarro, A Weierstrass extremal field theory for the fractional Laplacian, Adv. Calc. Var. , 17 (2024), no. 4, 1067-1093

    2024

  9. [17]

    Caffarelli, X

    L. Caffarelli, X. Cabr e , Fully nonlinear elliptic equations, AMS Colloquium Publications, 43, AMS Providence RI, 1995

    1995

  10. [18]

    Caffarelli, D

    L. Caffarelli, D. Jerison, C. Kenig, Global energy minimizers for free boundary problems and full regularity in three dimensions, Contempt. Math., 350 (2004), 83-97

    2004

  11. [19]

    Caffarelli, S

    L. Caffarelli, S. Salsa, A geometric approach to free boundary problems, Graduate Studies in Mathematics, 68. American Mathematical Society, Providence, RI, 2005

    2005

  12. [20]

    Chang-Lara, O

    H. Chang-Lara, O. Savin, Boundary regularity for the free boundary in the one-phase problem, Contemp. Math., 723 (2019), 149-165

    2019

  13. [21]

    Crandall, H

    M. Crandall, H. Ishii, P.-L. Lions. User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67

    1992

  14. [22]

    De Silva, Free boundary regularity for a problem with right hand side, Interfaces Free Bound., 13 (2011), 223-238

    D. De Silva, Free boundary regularity for a problem with right hand side, Interfaces Free Bound., 13 (2011), 223-238

    2011

  15. [23]

    De Silva, Existence and regularity of monotone solutions to free boundary problems, Amer

    D. De Silva, Existence and regularity of monotone solutions to free boundary problems, Amer. J. Math. 131 (2009), no. 2, 351-378

    2009

  16. [24]

    De Silva, F

    D. De Silva, F. Ferrari, S. Salsa, Two-phase problems with distributed source: regularity of the free boundary, Anal. PDE, 7 (2014), 267-310

    2014

  17. [25]

    De Silva, F

    D. De Silva, F. Ferrari, S. Salsa, Free boundary regularity for fully nonlinear non-homogeneous two-phase problems, J. Math. Pures Appl., 103 (2015), 658-694

    2015

  18. [26]

    De Silva, D

    D. De Silva, D. Jerison, A singular energy minimizing free boundary, J. Reine Angew. Math. , 635 (2009), 1-21

    2009

  19. [27]

    De Silva, D

    D. De Silva, D. Jerison, H. Shagholian, Inhomogeneous global minimizers to the one-phase free boundary problem, Comm. Partial Differential Equations , 47 (2022), 1193-1216

    2022

  20. [28]

    De Silva, O

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

    2021

  21. [29]

    De Silva, O

    D. De Silva, O. Savin, The Alt-Philips functional for negative powers, Bull. Lond. Math. Soc. , 55 (2023), no. 6, 2749-2777

    2023

  22. [30]

    De Silva, O

    D. De Silva, O. Savin, Uniform density estimates and -convergence for the Alt-Phillips functional of negative powers, Math. Eng. , 5 (2023), Paper No. 086

    2023

  23. [31]

    De Silva, O

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

    2023

  24. [32]

    De Silva, O

    D. De Silva, O. Savin, A note on higher boundary Harnack inequality, Discrete Contin. Dyn. Syst. , 35 (2015), no. 12, 6155-6163

    2015

  25. [33]

    Edelen, L

    N. Edelen, L. Spolaor, B. Velichkov, A strong maximum principle for minimizers of the one-phase Bernoulli problem, Indiana Univ. Math. J. , 73 (2024), 1061-1096

    2024

  26. [34]

    Engelstein, X

    M. Engelstein, X. Fern\'andez-Real, H. Yu, Graphical solutions to one-phase free boundary problems, J. Reine Angew. Math. , 804 (2023), 155-195

    2023

  27. [35]

    Engelstein, L

    M. Engelstein, L. Spolaor, B. Velichkov, Uniqueness of the blowup at isolated singularities for the Alt-Caffarelli functional, Duke Math. J. , 169 (2020), no. 8, 1541-1601

    2020

  28. [36]

    Fern\'andez-Real, F

    X. Fern\'andez-Real, F. Gr\"un, Continuity up to the boundary for minimizers of the one-phase Bernoulli problem, (2024), preprint: arXiv:2408.10019

    2024 arXiv

  29. [37]

    Fern\'andez-Real, X

    X. Fern\'andez-Real, X. Ros-Oton, On global solutions to semilinear elliptic equations related to the one-phase free boundary problem, Discrete Contin. Dyn. Syst. A , 39 (2019), no. 12, 6945-6959

    2019

  30. [38]

    Fern\'andez-Real, H

    X. Fern\'andez-Real, H. Yu, Generic properties in free boundary problems, (2023), preprint: arXiv: 2308.13209

    2023 arXiv

  31. [39]

    Ferreri, L

    L. Ferreri, L. Spolaor, B. Velichkov, On the boundary branching set of the one-phase problem, (2024), preprint, arXiv: 2407.15230

    2024 arXiv

  32. [40]

    Figalli, J

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

    2019

  33. [41]

    Gurtin, R

    M. Gurtin, R. MacCamy, On the diffusion of biological polulations, Math. Biosci. , 33 (1977), 35-49

    1977

  34. [42]

    Hong, The singular homogeneous solutions to one phase free boundary problem, Proc

    G. Hong, The singular homogeneous solutions to one phase free boundary problem, Proc. Amer. Math. Soc. , 143 (2015), 4009-4015

    2015

  35. [43]

    Indrei, Boundary regularity and nontransversal intersection for the fully nonlinear obstacle problem, Comm

    E. Indrei, Boundary regularity and nontransversal intersection for the fully nonlinear obstacle problem, Comm. Pure Appl. Math., 72 (2019), 1459-1473

    2019

  36. [44]

    Indrei, A

    E. Indrei, A. Minne, Nontransversal intersection of free and fixed boundaries for fully nonlinear elliptic operators in two dimensions, Anal. PDE, 9 (2016), 487-502

    2016

  37. [45]

    Hardt, L

    R. Hardt, L. Simon, Area minimizing hypersurfaces with isolated singularities, J. Reine Angew. Math. , 362 (1985), 102-129

    1985

  38. [46]

    Jerison, O

    D. Jerison, O. Savin, Some remarks on stability on cones for the one-phase free boundary problem, Geom. Funct. Anal. , 25 (2015), 1240-1257

    2015

  39. [47]

    Karakhanyan, T

    A. Karakhanyan, T. Sanz-Perela, Stable cones in the Alt-Phillips free boundary problem, (2024), preprint: arXiv:2403.13059

    2024 arXiv

  40. [48]

    Lee, Obstacle problems for the fully nonlinear elliptic operators, Thesis (PhD) , New York University, 1998

    K. Lee, Obstacle problems for the fully nonlinear elliptic operators, Thesis (PhD) , New York University, 1998

    1998

  41. [49]

    Y. Lian, K. Zhang, Boundary pointwise C^ 1, and C^ 1, regularity for fully nonlinear elliptic equations , J. Differential Equations, 269 (2020), 1172-1191

    2020

  42. [50]

    Matevosyan, Tangential touch between free and fixed boundaries in a problem from superconductivity, Comm

    N. Matevosyan, Tangential touch between free and fixed boundaries in a problem from superconductivity, Comm. Partial Differential Equations, 30 (2005), 1205-1216

    2005

  43. [51]

    Mooney, Y

    C. Mooney, Y. Yang, A proof by foliation that Lawson's cones are A_ -minimizing, Discrete Contin. Dyn. Syst. , 41 (2021), no. 11, 5291-5302

    2021

  44. [52]

    Matevosyan, P

    N. Matevosyan, P. A. Markowich, Behavior of the free boundary near contact points with the fixed boundary for nonlinear elliptic equation, Monatsh. Math. 142 (2004), 17-25

    2004

  45. [53]

    Petrosyan, H

    A. Petrosyan, H. Shahgholian, N. Uraltseva, Regularity of free boundaries in obstacle-type problems , Graduate Studies in Mathematics, 136, AMS Providence RI, 2012

    2012

  46. [54]

    Phillips, A minimization problem and the regularity of solutions in the presence of a free boundary, Indiana Univ

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

    1983

  47. [55]

    Restrepo, X

    D. Restrepo, X. Ros-Oton, C^ regularity in semilinear free boundary problems, (2024), preprint: arXiv:2407.20426

    2024 arXiv

  48. [56]

    Savin, H

    O. Savin, H. Yu, Regularity of the singular set in the fully nonlinear obstacle problem, J. Eur. Math. Soc. , 25 (2023), no. 2, 571-610

    2023

  49. [57]

    Savin, H

    O. Savin, H. Yu, Concentration of cones in the Alt-Phillips problem, (2025), preprint: arXiv:2503.03626

    2025 arXiv

  50. [58]

    Savin, H

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

    2025 arXiv

  51. [59]

    Savin, H

    O. Savin, H. Yu, Regularity of the singular set in the fully nonlinear obstacle problem, J. Eur. Math. Soc. , 25 (2023), 571-610

    2023

  52. [60]

    Shahgholian, N

    H. Shahgholian, N. Uraltseva, Regularity properties of a free boundary near contact points with the fixed boundary, Duke Math. J., 116 (2003), 1-34

    2003

  53. [61]

    Silvestre, B

    L. Silvestre, B. Sirakov, Boundary regularity for viscosity solutions of fully nonlinear elliptic equations, Comm. Partial Differential Equations, 39 (2014), 1694-1717

    2014

  54. [62]

    N. N. Uraltseva, C^1 regularity of the boundary of a noncoincident set in a problem with an obstacle, St. Petersburg Math. J., 8 (1997), 341-353

    1997

  55. [63]

    Velichkov, Regularity of the One-Phase Free Boundaries, Lecture Notes of the Unione Matematica Italiana, Springer Cham, 2023

    B. Velichkov, Regularity of the One-Phase Free Boundaries, Lecture Notes of the Unione Matematica Italiana, Springer Cham, 2023

    2023

  56. [64]

    Wang, Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order

    P. Wang, Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. I. Lipschitz free boundaries are C^1 , Comm. Pure Appl. Math. 53 (2000), 799-810

    2000

  57. [65]

    Wang, Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order

    P. Wang, Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. II. Flat free boundaries are Lipschitz, Comm. Partial Differential Equations 27 (2002), 1497-1514

    2002

  58. [66]

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

    1998

  59. [67]

    Weiss, Boundary monotonicity formulae and applications to free boundary problems I: The elliptic case, Electron

    G. Weiss, Boundary monotonicity formulae and applications to free boundary problems I: The elliptic case, Electron. J. Differ. Equ., 2004 (2004), 1-12

    2004

  60. [68]

    Y. Wu, H. Yu, On the fully nonlinear Alt-Phillips equation, Int. Math. Res. Not. , 2022, 8540-8570

    2022

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