{"id":"ed428661-3ca2-4d9e-b8b4-7629def1db4d","arxiv_id":"2411.15311","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Viscosity solutions to uniformly elliptic fully nonlinear equations gain W^{γ,p} regularity with γ>1 up to the boundary, without convexity or concavity assumptions.","lead":"This paper proves global fractional Sobolev regularity, with more than one derivative, for solutions of fully nonlinear elliptic equations, all the way up to the boundary. If correct, it extends a known interior result to boundary value problems and to equations depending on the solution and its gradient.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.7's reduction to a viscosity equation for w=(v-h)/2 is unjustified: Proposition 3.1 supplies only an interior C^{1,α0} limit solution of a different operator, not a C^2 (or boundary-regular) solution of the same \\tilde F; the boundary decay chain collapses.","rationale":"Both the reader and I agree the proof is conditional, but I locate the failure more precisely. The power decay of |A_t| at the boundary (Lemma 3.6, strengthened in Lemma 3.8) is the engine of Proposition 3.2, which yields θ∈L^p. That engine is powered by Lemma 3.7, whose proof requires comparing v with a C^{1,α0} function h that is close and has no A_N near the boundary. The approximation h produced by Proposition 3.1 fails on three counts: (i) it solves only a constant-coefficient limit F∞, not the same operator \\tilde F; (ii) it is only C^{1,α0} in the interior, so it is not an admissible test function (C^2) for the viscosity equation used to claim w solves a PDE; (iii) it is not shown to have C^{1,α0} regularity up to the flat boundary or zero trace, so AN(h,·) need not be empty on cubes touching {x_d=0}. Without a fix, the conclusion of Theorem 2.1 is unsupported. A separate typo in the proof of Theorem 2.1 (−2v(y) instead of −2v(x0)) is easily corrected and does not change this assessment. The p∈(d−ε0,∞) improvement in Theorem 2.2 is also asserted rather than proved; Proposition 3.2 uses the maximal function on L^{p/d}, which requires p>d, leaving the pd−ε0. The proof uses C^{1,α}-cones, a Calderón–Zygmund decomposition at the boundary, an approximation lemma, and a reduction to a fractional Laplacian via a singular-integral bound.","tokens_in":23154,"tokens_out":8666,"duration_ms":82263,"significance":"If the main results were fully established, they would constitute a notable extension of fractional Sobolev regularity to the boundary for fully nonlinear elliptic equations without convexity assumptions, complementing the interior results of [11] and improving on the W^{1,p} and W^{2,p} boundary estimates of Winter [17] under very mild structural hypotheses. The paper has a clear and attractive strategy: it reduces the boundary problem to a fractional-Laplacian representation and then uses known regularity for the fractional Laplacian. The paper also carefully states the C^{1,α}-cone machinery and the Calderón–Zygmund decomposition at the boundary. However, several load-bearing steps in the written proof are incomplete or unjustified, most notably the boundary approximation lemma (Proposition 3.1), the viscosity-solution reduction in Lemma 3.7, and the explicit omission of the mollification argument in Proposition 4.1. These gaps currently prevent the main claims from being considered proven.","major_comments":[{"comment":"The proof of the approximation lemma is not established up to the boundary. The contradiction argument produces a limit u∞ that solves the constant-coefficient equation F∞(D^2u∞)=0 in the open half-ball B^+_{12√d}, and the paper then asserts u∞∈C^{1,α0}(B^+_{12√d}) merely from the fact that F∞ is uniformly elliptic. No boundary C^{1,α0} estimate for the flat-boundary problem is proved or cited, and the zero trace u=0 on {x_d=0} may be lost under locally uniform convergence in the open half-ball. This matters because Lemma 3.7 later uses h from Proposition 3.1 to control the sets A_N(h,B^+_{12√d})∩((Q^{d-1}_1×(0,1))+x0), which requires regularity and trace information up to the flat boundary.","section":"Section 3, Proposition 3.1"},{"comment":"The definition w:=(v−h)/2 and the assertion that w is a viscosity solution of G(M,x)=1/2 \\tilde F(2M+D^2h,x)=\\tilde f are not justified. The function h obtained from Proposition 3.1 is only C^{1,α0}_{loc} in the open half-ball and solves the limit equation F∞(D^2h)=0, not the equation \\tilde F(D^2h)=0 for the same operator for which v is a solution. The viscosity test-function argument requires h to be an admissible perturbation, e.g., h∈C^2 or a classical solution of the same operator; neither condition is supplied. Since Lemma 3.7 feeds into Lemma 3.8 and Proposition 3.2, the boundary decay mechanism for the cone sets is not proven as written.","section":"Section 3, Lemma 3.7"},{"comment":"There is a contradiction between the stated range of σ and the proof. The theorem states that σ∈(0,1+ε), while the proof works with '1<σ<1+α' and Corollary 2.1 concludes u∈W^{γ,p} for γ<σ. If σ can be smaller than 1, the claimed conclusion that solutions have differentiability of order strictly greater than one does not follow. The intended statement is presumably σ∈(1,1+ε), but as written the theorem does not imply the advertised regularity.","section":"Theorem 2.1 and its proof"},{"comment":"The singular integral is defined with the wrong middle term: I_{σ/2}(v)(x0)=∫[v(x0+y)+v(x0−y)−2v(y)]/|y|^{d+σ}dy, and the same erroneous expression −2\\tilde u(y) is used in the estimate that follows. The cone-touch bound gives control of |u(x0+y)+u(x0−y)−2u(x0)|, not of the printed numerator. As written, the L^p bound on the fractional Laplacian of \\tilde u is not justified. Even if this is a typo, it appears in the central step and must be corrected and re-verified.","section":"Proof of Theorem 2.1, singular integral"},{"comment":"The passage from the gradient-dependent operator F(D^2u,Du,u,x) to the x-dependent operator \\tilde F(D^2u,x)=F(D^2u,0,0,x) relies on an approximation argument that the paper explicitly omits ('Although we omit the detailed argument here'). This step is load-bearing for Theorem 2.2: one must show that the mollified operators satisfy Assumption A4 with uniform constants, that the corresponding solutions u_j exist and solve the approximated equations, and that u_j→u weakly in W^{γ,p}; none of this is demonstrated. Without these details, the reduction to Theorem 2.1 is incomplete.","section":"Section 4, Proposition 4.1"}],"minor_comments":[{"comment":"The notation '≤β_0^d' in A4 is confusing: β_0 is a small constant and d is the dimension; it is likely intended to be a power of the constant, but the dimension in the exponent is not explained.","section":"Definition 2.7 and Assumption A4"},{"comment":"In the covering argument, the term |A(u,Ω)∩Q_ε(x_i)| is missing the subscript t; it should read |A_t(u,Ω)∩Q_ε(x_i)|.","section":"Lemma 3.6, proof, second case"},{"comment":"The symbol φ is overloaded: it denotes the boundary data in the theorem statement but also a function φ∈W^{2,p}(B^+_1) in the flattening argument. The definition of \\tilde F also mixes v, φ and ϑ, making the pulled-back equation difficult to follow.","section":"Proof of Theorem 2.2"},{"comment":"Remark 2.2 refers to Proposition 5.1 as establishing that F-harmonic functions belong to W^{2,p}(B^+_1), but Proposition 5.1 only proves density of W^{2,p} solutions; the actual W^{2,p} estimate is Theorem 5.1, which requires additional assumptions A6 and A7 that are not present in the main theorems. This discrepancy should be clarified.","section":"Remark 2.2 and Appendix, Proposition 5.1"},{"comment":"There are numerous typographical issues, including missing spaces and missing superscripts (for example 'u ∈C(B6r√d)' in Lemma 3.1, and 'B1/2' in several places). These do not affect the mathematics but should be corrected in a revision.","section":"Various places"}],"recommendation":"major_revision","confidential_remarks":"The paper's central idea is promising and the intended results are natural extensions of [11], but the written proof has several substantial gaps that are not purely cosmetic. In particular, the boundary approximation lemma (Proposition 3.1) and the viscosity reduction in Lemma 3.7 are the heart of the boundary estimate and are currently unjustified, while Proposition 4.1 explicitly omits an essential mollification argument. The inconsistency in the range of σ in Theorem 2.1 also needs attention. I recommend major revision rather than rejection because the approach appears salvageable, but the authors will need to supply missing boundary regularity results and complete the omitted approximations. Given that one of the authors (Santos) is also an author of [11], the authors should take extra care to state precisely which parts are new and which are direct consequences of [11]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou asked about arXiv:2411.15311. Here's my read.\n\nThe paper aims to prove global W^{γ,p} regularity with γ>1 up to the boundary for uniformly elliptic fully nonlinear equations without convexity, including gradient dependence and variable coefficients. That would be a genuine extension of the interior result in [11] and would improve on the boundary results in [17] and [13]. The theorem statements are new and worth taking seriously.\n\nWhat is good: the cone method from [11] is adapted to the boundary in a natural way, and the authors correctly identify that the main work is to get a distributional estimate for the aperture function θ near the boundary. The writing is clear, the assumptions are standard (A1-A5), and the interior appendix extends [11] to gradient-dependent operators in a plausible way. If the proof were solid, this would be a nice paper.\n\nWhere it falls down: the boundary decay chain rests on Lemma 3.7, and I don't think the proof works as written. The step \"w = (v-h)/2 is a viscosity solution of G\" is not justified. The approximating h from Proposition 3.1 is only C^{1,α0}_{loc}, not C^2, and it solves a constant-coefficient limit equation F∞(D^2h)=0, not the original operator \\tilde F(D^2h,x)=0. Without a C^2 solution of the same operator, the viscosity solution argument cannot be made. This is not a minor technicality: it is the mechanism that transfers the decay estimate for A_t(w) to the original u, and without it the proof of θ∈L^p near the boundary collapses. The stress-test note on this point is accurate.\n\nThere are smaller issues too. The singular integral in Theorem 2.1's proof has the wrong middle term (-2u(y) instead of -2u(x0)); that looks like a typo, but as written the L^p estimate by θ doesn't follow. The range of σ in the statement is loose but not contradictory. The approximation in Proposition 4.1 and the appendix are sketched with references to [17] and [12], and the appendix adds assumptions A6-A7 that are not in the main theorems; the authors should clarify whether the main result depends on them.\n\nIf I were refereeing, I would ask the authors to fix Lemma 3.7 or supply a different argument for the boundary decay. The result is plausible and likely fixable, but the current manuscript doesn't carry the proof.\n\nWho should look at it: researchers in fully nonlinear elliptic regularity, especially those working with the C^{1,α}-cone method. It deserves a serious referee, not a desk reject, because the claimed theorem is important and the flaws are specific and potentially repairable.\n\nI wouldn't cite it in my own work until the proof is repaired.\n\nBest,\n[You]","headline":"The global boundary regularity result is attractive, but the proof has a central gap in Lemma 3.7 that breaks the boundary decay chain, so it needs referee work before it can be trusted.","tokens_in":23680,"tokens_out":12882,"would_cite":false,"duration_ms":117828,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B65","35J60","35J15","35J25"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that bounded viscosity solutions of uniformly elliptic fully nonlinear equations belong to W^{γ,p}(Ω) up to the boundary for every γ<1+ε, with quantitative estimates, assuming only uniform ellipticity and a small…","keywords":["fractional Sobolev regularity","fully nonlinear elliptic equations","viscosity solutions","C^{1,α}-cones","fractional Laplacian","boundary regularity","W^{γ,p} estimates","uniform ellipticity"],"falsifier":"If a uniformly elliptic operator F satisfying A1–A4 admits a bounded viscosity solution on the half-ball with zero boundary data that fails to lie in $W^{{1+δ,p}}$ for some δ>0, or if the constant-coefficient limit equation in Proposition 3.1 has a solution that is not $C^{{1,α0}}$ up to the flat boundary, the boundary regularity claim collapses.","tokens_in":22527,"feed_emoji":"📐","tokens_out":7284,"duration_ms":58754,"temperature":0.7,"pith_summary":"The paper establishes global fractional Sobolev regularity for viscosity solutions of fully nonlinear elliptic equations of the form F($D^{2}$u,Du,u,x)=f, under only uniform ellipticity and a small oscillation condition on F in the x-variable. The main results show that solutions belong to $W^{{γ,p}}$ up to the boundary for every γ<1+ε, with ε∈(0,α0), where α0 is the interior $C^{{1,α0}}$ Hölder exponent of F-harmonic functions. The proof works by showing that a solution extended by zero near a flat boundary is a weak solution of a fractional Laplacian equation of order σ/2 with σ<1+ε, so that the fractional regularity follows from known theory for the fractional Laplacian. This matters because it yields differentiability of order strictly greater than one at the boundary without convexity or concavity assumptions on the operator, a regime where classical $C^{{2,α}}$ or $W^{{2,p}}$ boundary estimates are not available.","feed_headline":"Fractional differentiability up to the boundary for nonlinear PDEs","feed_subtitle":"Viscosity solutions get fractional differentiability past first order at the boundary, without convexity.","key_machinery":"The proof is carried by $C^{{1,α}}$-cones, functions of the form ψ(x)=ℓ(x)±(M/2)|x−x0|^{1+α} with ℓ affine. For each scale, the sets G_M(u,Ω) of points that can be touched from above and below by such cones, and their complements A_M, are studied through a Calderón–Zygmund decomposition; the aperture function θ(x)=inf{M: x∈G_M} measures how much cone opening is needed at x. The main mechanism is the estimate $θ^{{1+α}}$∈L^p, obtained from the decay of |A_{M^k}| with k, which follows from the approximation lemma (Proposition 3.1). Once θ is integrable, the second-difference quotient $Δ_h^{{1+α}}$ũ is bounded by θ, and the singular integral I_{σ/2} representing (−Δ)^{σ/2} is controlled in L^p, producing the fractional Laplacian equation.","core_discovery":"On the paper's own terms, the central discovery is that viscosity solutions to uniformly elliptic fully nonlinear equations inherit fractional-diffusion structure up to the boundary: if u solves F($D^{2}$u,x)=f in the upper half-ball with zero Dirichlet data, then the zero-extension ũ is a weak solution of (−Δ)^{σ/2}ũ=g for some g∈L^p and every σ<1+ε. This yields u∈$W^{{γ,p}}$(B^+_{1/2}) for all γ<σ, with the estimate ‖u‖_{$W^{{γ,p}}$} ≤ C(‖u‖_{L^∞}+‖f‖_{L^p}). The boundary statement is then transported to general $C^{{1,1}}$ domains and non-zero $W^{{2,p}}$ Dirichlet data by flattening the boundary and subtracting the boundary values, giving the global estimate of Theorem 2.2. The regularity is quantitative and holds in the same range p>d−ε0 that is known for interior $W^{{1+ε,p}}$ estimates.","pith_inferences":["A natural test of the method is whether the exponent ε0 in Theorem 2.2 can be improved to the Escauriaza exponent for the boundary problem, since the paper uses the interior value; failure for p between d−ε0 and the boundary-optimal range would indicate the boundary argument is not sharp.","The same C^{1,α}-cone setup could be adapted to Neumann or oblique boundary conditions, where the flat-boundary fractional Laplacian structure would take a different form because the extension beyond the boundary is not zero.","The paper leaves implicit that the approximation lemma's boundary C^{1,α0} assumption might be supplied by existing boundary regularity theory under additional structure; if that structure is necessary, the result would hold only for a subclass of uniformly elliptic operators."],"forward_implications":["Viscosity solutions of uniformly elliptic fully nonlinear equations possess fractional differentiability of order greater than one up to the boundary, without convexity or concavity assumptions.","The fractional Sobolev norm is controlled by the L^∞ norm of the solution, the L^p norm of the source, and the W^{2,p} norm of the boundary data, with a constant depending only on universal parameters.","The result extends to gradient-dependent operators satisfying the structural condition A5, in the range p>d−ε0.","Since W^{1+ε,p} embeds into C^{1,α} for suitable p, the order 1+ε is essentially optimal, consistent with known counterexamples to C^{1,β} regularity.","The boundary regularity is achieved by reducing the original problem to a fractional Laplacian equation, so known fractional Calderón–Zygmund theory applies directly."],"supporting_citations":[{"why":"Supplies the C^{1,α}-cone method and the interior fractional-Laplacian result that this paper extends to the boundary.","marker":"[11]"},{"why":"Provides the boundary W^{1,p} and W^{2,p} estimates and the flattening/reduction strategy used to prove Theorem 2.2.","marker":"[17]"},{"why":"Introduces the oscillation condition and approximation framework used in Lemma 3.1 and the approximation lemmas.","marker":"[2]"},{"why":"Establishes the exponent ε0 allowing p>d−ε0, which Theorem 2.2 relies on.","marker":"[5]"},{"why":"Supplies the L^p-viscosity solution theory with measurable ingredients and the a.e. pointwise differentiability used to handle gradient-dependent operators.","marker":"[3]"},{"why":"Gives the global fractional Calderón–Zygmund regularity for weak solutions of the fractional Laplacian, used in Corollary 2.1.","marker":"[1]"},{"why":"Provides boundary C^{1,α} and C^{2,α} regularity under structural conditions, the baseline the paper compares against.","marker":"[13]"},{"why":"Supplies boundary regularity theory for homogeneous fully nonlinear equations, one source for the limiting behavior in the approximation lemma.","marker":"[7]"}],"fun_headline_variants":["Boundary fractional regularity for fully nonlinear elliptic PDEs","Viscosity solutions gain extra fractional smoothness at boundary","No convexity: fractional differentiability up to boundary","Fractional Sobolev regularity at boundary without convexity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The approximation step in Proposition 3.1 assumes that the limit of solutions with vanishing right-hand sides is $C^{{1,α0}}$ on the closed half-ball, which requires boundary $C^{{1,α0}}$ regularity for constant-coefficient fully nonlinear equations with zero data on the flat boundary.","fun_headline_variants_meta":{"raw":{"variants":["Boundary fractional regularity for fully nonlinear elliptic PDEs","Viscosity solutions gain extra fractional smoothness at boundary","No convexity: fractional differentiability up to boundary","Fractional Sobolev regularity at boundary without convexity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000659,"raw_usage":{"total_tokens":2961,"prompt_tokens":841,"completion_tokens":2120,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":457,"completion_tokens_details":{"reasoning_tokens":2055}},"tokens_in":457,"tokens_out":2120,"duration_ms":16625,"temperature":1.0,"reasoning_tokens":2055,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:27:19.206700+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"If a uniformly elliptic operator F satisfying A1–A4 admits a bounded viscosity solution on the half-ball with zero boundary data that fails to lie in $W^{{1+δ,p}}$ for some δ>0, or if the constant-coefficient limit equation in Proposition 3.1 has a solution that is not $C^{{1,α0}}$ up to the flat boundary, the boundary regularity claim collapses.","supporting_citations":[{"cited_title":"8, 1539– 1558","cited_arxiv_id":null,"evidence_quote":"Supplies the C^{1,α}-cone method and the interior fractional-Laplacian result that this paper extends to the boundary."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the boundary W^{1,p} and W^{2,p} estimates and the flattening/reduction strategy used to prove Theorem 2.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the oscillation condition and approximation framework used in Lemma 3.1 and the approximation lemmas."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the exponent ε0 allowing p>d−ε0, which Theorem 2.2 relies on."},{"cited_title":"4, 365–398","cited_arxiv_id":null,"evidence_quote":"Supplies the L^p-viscosity solution theory with measurable ingredients and the a.e. pointwise differentiability used to handle gradient-dependent operators."},{"cited_title":"Partial Differential Equations 39 (2014), no","cited_arxiv_id":null,"evidence_quote":"Provides boundary C^{1,α} and C^{2,α} regularity under structural conditions, the baseline the paper compares against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies boundary regularity theory for homogeneous fully nonlinear equations, one source for the limiting behavior in the approximation lemma."}],"review_version":1}