{"id":"066b4f9b-9ba1-494d-aa4f-e506fe762d74","arxiv_id":"2509.01138","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A generalized small perturbation theorem yields interior C^{2,alpha} regularity for nonhomogeneous locally uniformly elliptic equations and measure-zero singular sets for sigma_k(D^2u)=f with positive Lipschitz f.","lead":"This paper proves that sufficiently small Holder perturbations of fully nonlinear elliptic equations still have smooth interior solutions, and draws the consequence that sigma-k Hessian equations with a positive Lipschitz right-hand side have singular sets of Lebesgue measure zero.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.10's proof is missing a measure-propagation step: a large set in B1 does not by itself lower-bound the L^ε norm on B1/4, and this Hölder estimate is what drives the compactness step in Lemma 3.1.","rationale":"The reader's verdict is CONDITIONAL, citing a sign issue when applying Corollary 2.11 and the unsupported Lemma 4.2 in the application. My stress-test focuses on the central claim (Theorem 1.7) and finds a more direct, load-bearing defect in Section 2.2. The proof of Theorem 2.10 asserts an L^ε lower bound on B1/4 from a measure lower bound in B1 without any argument. This is not a cosmetic issue: the Hölder estimate is the only tool used to obtain equicontinuity of the rescaled solutions v_k in Lemma 3.1, and without it the compactness limit v0 cannot be formed. The defect is in the proof rather than necessarily in the theorem—standard Krylov–Safonov machinery could repair it—so I do not recommend REJECT or UNVERDICTED; the appropriate posture remains CONDITIONAL, exactly as the reader concluded. I agree partially with the reader: they identified a symptom (non-negativity in Corollary 2.11) but not the deeper missing propagation step. The application-level concern about Lemma 4.2 is real but secondary to the central theorem. I would not change the reader's verdict.","tokens_in":21286,"tokens_out":30628,"duration_ms":367130,"concrete_test":"Verify the disputed inference in the proof of Theorem 2.10 by supplying the missing inequality directly from Lemma 2.8. Concretely, build a sequence of nonnegative v_j ∈ S^*_{ρ_j} with ||v_j||_{L∞(B1)} ≤ 1, f = 0, v_j = 0 on B_{1/4}, and |{v_j ≥ 1/2} ∩ B1| ≥ 1/2 (for example, smooth approximations of a bump supported in B1 \\ B_{3/4}). If such a sequence exists, the claimed lower bound on ||v_j||_{L^ε(B1/4)} is false and Theorem 2.10 needs repair. If not, identify the Pucci inequality that forces positive mass in B1/4 and insert the omitted covering step (cf. Lemma 2.6). This settles whether the compactness argument in Lemma 3.1 is valid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main theorem 1.7 relies on Lemma 3.1, whose compactness argument applies Corollary 2.11 (the translated Hölder estimate) to the rescaled solutions v_k. Corollary 2.11 is only a restatement of Theorem 2.10, so the validity of Theorem 2.10 is load-bearing. In the proof of Theorem 2.10, after normalizing v = (u(r_k·) − m_k)/ω_k with 0 ≤ v ≤ 1 and osc_{B1} v = 1, the author assumes |{v ≥ 1/2} ∩ B1| ≥ 1/2 and then asserts, without justification, that \nc0 ≤ ∥v_{r_k^2ρ/2}∥_{L^{ε0}(B1/4)} ≤ C(inf_{B1/4} v + ∥f̃∥_{L∞}).\nThe first inequality does not follow from a measure lower bound in B1: v could be ≥ 1/2 only in B1 \\ B1/4 and vanish in B1/4, in which case the L^ε norm on B1/4 is arbitrarily small. Lemma 2.8 (weak L^ε) gives an upper bound on superlevel sets, not a lower bound, so it cannot produce c0. What is needed is a covering/propagation argument—for instance using the already-proved Lemma 2.6—to convert large mass in B1 into positive mass in B1/4, or a splitting of cases according to whether the large set meets B1/4. No such argument is supplied. A second symptom of the same region is that Corollary 2.11 is applied to v_k without verifying its non-negativity hypothesis; this is patchable by a shift, but it shows the Hölder statement is being used beyond what is proved. Thus the proof of Theorem 1.7, as written, has a genuine gap in its main analytic ingredient.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a nonhomogeneous generalization of Savin's small perturbation theorem: if u is a viscosity solution to F(D^2u,Du,u,x)=f on B1, F is locally uniformly elliptic, F(0,0,0,x)=0, F has Lipschitz dependence on (p,z) and Hölder-small x-dependence, and both ||u||_L∞ and ||f||_C^{0,α} are sufficiently small, then u∈C^{2,α}(B_{1/2}) with a universal estimate (Theorem 1.7). The proof develops weak Harnack and Hölder estimates for locally uniformly elliptic equations via the method of sliding paraboloids (Section 2), then uses a compactness/improvement-of-flatness argument and Caffarelli iteration (Section 3). As an application, the paper claims a Lebesgue-a.e. twice differentiability result and a partial regularity result for σ_k(D^2u)=f with Lipschitz positive f (Proposition 1.10).","tokens_in":21728,"tokens_out":29653,"duration_ms":351868,"significance":"If the proof is completed, Theorem 1.7 is a meaningful extension of Savin's theorem to nonhomogeneous equations with a genuinely different technical route that avoids rescaling the ellipticity constants. The sliding-paraboloid weak Harnack machinery is of independent interest. Proposition 1.10 would be an attractive application connecting ε-regularity theory with Hessian equations. The paper is largely self-contained and contains no fitted parameters or circular reasoning. However, the current manuscript has several load-bearing gaps, in particular in the Hölder estimate of §2 and in the Hessian-measure lemma used for the application, so the advertised results are not yet fully established.","major_comments":[{"comment":"The induction step asserts c0 ≤ ||v_{r_k^2ρ/2}||_{L^{ε0}(B1/4)} after assuming |{v≥1/2}∩B1|≥1/2. This lower bound does not follow: the large set {v≥1/2} may be essentially disjoint from B1/4, in which case the L^ε norm on B1/4 is arbitrarily small. Corollary 2.8 is a distribution-function upper bound, so it cannot produce c0. A measure-propagation or covering argument (e.g. using Lemma 2.6) is needed to convert mass in B1 into mass in B1/4, or a separate treatment of the complementary case is needed. As written, the Hölder estimate — and hence the compactness step in Lemma 3.1 — is not proved.","section":"§2, Theorem 2.10, proof"},{"comment":"Corollary 2.11 is applied to the rescaled functions v_k = (u_k(r_k x)-P_k(r_k x))/r_k^{2+α}. Corollary 2.11 explicitly requires the function to be nonnegative in B1, but v_k may change sign because P_k is subtracted. This is not a vacuous hypothesis; the Hölder equicontinuity of {v_k} is the basis for passing to the limit v0. The gap is likely repairable by applying the corollary to a shifted, normalized version such as (v_k+2)/3, but the repair must be written.","section":"§3, Lemma 3.1, Step 1"},{"comment":"Lemma 4.2 asserts that every 2-convex function on B1 has a distributional Hessian that is a matrix of Radon measures, citing [CT05]. However, [CT05] is an Alexandrov-type theorem for k-convex functions with k>n/2; for k=2 in dimensions n≥5 this hypothesis is not met. The manuscript itself notes that a.e. twice differentiability may fail for k≤n/2 without an extra equation. Since the proof of Proposition 1.10(i) relies on Lemma 4.2 to obtain the Lebesgue-Radon-Nikodym decomposition of [D^2u], the application is not supported at present. The author must either prove the measure representation for solutions of σ_k(D^2u)=f or replace this step.","section":"§4, Lemma 4.2 and Proposition 1.10"}],"minor_comments":[{"comment":"Typo: 'area forluma' should be 'area formula'.","section":"§2, Lemma 2.3"},{"comment":"Typo: 'ellipricity' should be 'ellipticity'.","section":"§1, Remark 1.4"},{"comment":"The displayed range 'r 2ρ0/ρ ≤ r ≤ 1' is missing an inequality sign; it should read 'r ≥ 2ρ0/ρ'.","section":"§2, Theorem 2.10 and Corollary 2.11"},{"comment":"In the condition F(D2P0, DP(0), P(0), 0)=f(0), the arguments P(0) and DP(0) should be P0(0) and DP0(0).","section":"§3, Lemma 3.1 statement"},{"comment":"The bound on f^*_k is stated for the C^{0,α} norm, but the argument controls only the L∞ norm; the term r_k^2|v_k| is not shown to be Hölder small without a priori information. Since Corollary 2.11 needs only L∞ smallness, the estimate should be stated for ||·||_{L∞}.","section":"§3, Lemma 3.1, Step 1"},{"comment":"The formula for v(y) is missing parentheses; it should be v(y)=(u(r_k y)-m_k)/(M_k-m_k).","section":"§2, Theorem 2.10"},{"comment":"The proof divides by B := inf_{B1/4} u + ||f||_{L∞}/8. If B=0 the scaling v=u/B is undefined; a separate sentence treating this case (or taking a limiting argument) is needed.","section":"§2, Theorem 2.9"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the missing measure-propagation step in Theorem 2.10; without it, the Hölder estimate and hence the compactness argument for the main theorem are not established. I also recommend checking the use of [CT05] for Lemma 4.2, as the stated result appears to go beyond the cited theorem. The paper is clearly written and the approach is promising, so major revision rather than rejection seems appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper deserves a serious referee, but it is not ready as written. The main theorem—Savin's small perturbation result in the nonhomogeneous case with Hölder-small f and x-dependence—is the expected statement, and the proof via sliding paraboloids is coherent and genuinely different from the flawed Lian-Zhang attempt. The compactness argument in Lemma 3.1 is substantially worked out, and the paper is honest about the provenance of the theorem. No fitted parameters, no self-citation games.\n\nThe soft spots are real. The most serious is in Theorem 2.10, the Hölder estimate that drives everything. After rescaling, the author assumes |{v ≥ 1/2} ∩ B1| ≥ 1/2 and then asserts a lower bound on the L^ε norm of v on B1/4. That does not follow: the large set could sit entirely outside B1/4. What is needed is the standard expansion-of-positivity or covering argument that converts mass in B1 into mass in B1/4. The paper does not supply it. This is not a fatal conceptual flaw—this is exactly where the Krylov-Safonov machinery usually appears—but as written the main analytic ingredient is incomplete. The nearby application of Corollary 2.11 to v_k without checking nonnegativity is patchable by shifting, so I count that as minor.\n\nThe application to sigma_k is more doubtful. Lemma 4.2 is cited to Chaudhuri-Trudinger, but their Alexandrov-type theorem is for k > n/2; for 2-convex functions in n ≥ 5 the distributional Hessian need not be a matrix of Radon measures, and the paper does not derive that representation from the equation. If the lemma fails, the Radon-Nikodym step in Proposition 1.10(i) has no basis. The a.e. twice differentiability claim may still be true—Shankar-Yuan gives evidence in the sigma_2 case—but this proof does not establish it.\n\nWho this is for: PDE people working on fully nonlinear regularity, especially those interested in locally uniformly elliptic operators and Hessian equations. They will want to read the sliding-paraboloid weak Harnack section and check whether the propagation step can be filled.\n\nRecommendation: send to peer review. The main theorem is important and the proof strategy is worth referee time. Expect major revision, and the referee should be explicitly asked to verify the missing propagation step and to either prove Lemma 4.2 in the stated generality or restrict the application accordingly.","headline":"Serious, referee-worthy paper with a plausible main theorem and a genuinely different proof strategy, but the proof as written has a real gap in the Hölder estimate and the sigma_k application leans on an unsupported measure-theoretic lemma.","tokens_in":22211,"tokens_out":4316,"would_cite":false,"duration_ms":56191,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B65","35D40","35J60"],"pacs":[],"model":"deepseek-v4-flash","headline":"For locally uniformly elliptic fully nonlinear equations, uniformly flat viscosity solutions with Hölder-small forcing and coefficient oscillation are automatically C^{2,α} inside the ball.","keywords":["Savin small perturbation theorem","fully nonlinear elliptic equations","viscosity solutions","C^{2,α} regularity","sigma-k Hessian equation","partial regularity","weak Harnack inequality","sliding paraboloids"],"falsifier":"A 2-convex viscosity solution of σ_2(D^2u)=f in R^n with n≥5, with positive Lipschitz f, whose singular set has positive Lebesgue measure would falsify Proposition 1.10. Equally decisive for the main theorem: a sequence of flat solutions (u_j, f_j) satisfying Theorem 1.7's hypotheses with δ_j→0 but no uniform C^{2,α} bound on B_{1/2} would disprove Theorem 1.7.","tokens_in":21172,"feed_emoji":"📐","tokens_out":6692,"duration_ms":81477,"temperature":0.7,"pith_summary":"This paper tries to prove a robust regularity principle: if a viscosity solution of a fully nonlinear elliptic equation is uniformly tiny, and both the right-hand side and the equation's dependence on x are Hölder-small at the same scale, then the solution is automatically C^{2,α} in the interior. Savin's small perturbation theorem already did this for homogeneous problems without lower-order terms and with no x-dependence; the paper removes both restrictions under a Lipschitz structure condition in (p,z) and uniform continuity of the Hessian coefficient. The proof's engine is a compactness argument: badly behaved flat solutions are rescaled until they converge to a solution of the linearized constant-coefficient equation, which must be smooth. The advertised reward is a partial regularity theorem for σ_k(D^2u)=f: any k-convex viscosity solution with positive Lipschitz f is twice differentiable almost everywhere and C^{2,α} off a Lebesgue-null singular set.","feed_headline":"Flat solutions of fully nonlinear PDEs are automatically C^{2,α}","feed_subtitle":"Savin's small perturbation theorem now covers Hölder-small right-hand sides and x-dependence, with partial regularity for Hessian equations.","key_machinery":"The machinery is the method of sliding paraboloids: for a supersolution one slides concave paraboloids from below, estimates the measure of the contact set through Pucci operators and the area formula, and iterates by enlarging the paraboloid opening rather than rescaling the solution, which would change the ellipticity constants in locally uniformly elliptic problems. This yields the weak Harnack inequality and Hölder estimates. The second engine is Lemma 3.1 (improvement of flatness), which upgrades a quadratic approximation of a flat solution at scale r to a better quadratic approximation at scale ηr; repeated application gives the pointwise C^{2,α} estimate that is Theorem 1.7.","core_discovery":"Theorem 1.7 states that if F is elliptic and ρ-locally uniformly elliptic, vanishes at the origin, satisfies the structure inequality |F(M,p,z,x)−F(M,q,s,x)| ≤ b0|p−q| + c0|z−s|, and has DM F uniformly continuous, then any viscosity solution of F(D^2u,Du,u,x)=f with ||u||_{L∞} ≤ δ, ||f||_{C^{0,α}} ≤ δ, and |F(M,p,z,x)−F(M,p,z,x')| ≤ δ|x−x'|^α is automatically C^{2,α}(B_{1/2}) with a universal estimate. The constants depend only on n, α, ρ, λ, Λ, b0, c0, and the modulus of continuity ωF. The proof obtains weak Harnack and Hölder estimates for locally uniformly elliptic equations by sliding paraboloids, avoiding the rescaling that would destroy local uniform ellipticity, then uses a compactnes","pith_inferences":["If the uniform continuity of DM F were weakened to a Dini or VMO modulus, the same compactness-limiting scheme might yield C^{1,β} or C^{2,Dini} variants of the conclusion; the paper does not pursue this.","The σ_k application would likely extend to 1<k≤n/2 in any dimension once the Radon-measure representation of the Hessian is available for k-convex functions in that range; the paper's proof currently inherits that representation from a theorem stated for k>n/2.","One could test the sharp scaling of the thresholds by checking whether the δ in Theorem 1.7 must depend on α through a power law for model equations such as σ_k(D^2u)=f, where explicit examples might show the optimal relation.","Combining Theorem 1.7 with higher W^{3,ε}-type estimates and the blow-up analysis used for uniformly elliptic equations may upgrade the Lebesgue-null singular set in Proposition 1.10 to a Hausdorff-dimension estimate; the paper leaves that step implicit."],"forward_implications":["If Theorem 1.7 is correct, the classical small-perturbation regularity phenomenon extends to nonhomogeneous fully nonlinear equations with Hölder data, requiring no convexity or concavity of F.","The regularity principle implies partial regularity for σ_k Hessian equations: every k-convex viscosity solution of σ_k(D^2u)=f with positive Lipschitz f is twice differentiable a.e. and its singular set has Lebesgue measure zero.","The sliding-paraboloid weak Harnack and Hölder estimates in Section 2 apply to locally uniformly elliptic equations without rescaling, giving a reusable toolbox beyond the main theorem.","For uniformly elliptic F=F(M,x), Theorem 1.7 recovers and extends the earlier nonhomogeneous small-perturbation result, and for general locally uniformly elliptic F it supplies a proof different from a previous claimed one that the paper says contained a mistake."],"supporting_citations":[{"why":"Supplies the homogeneous small perturbation theorem that Theorem 1.7 generalizes.","marker":"[Sav07]"},{"why":"Proves the nonhomogeneous uniformly elliptic case that the present theorem extends to locally uniformly elliptic operators.","marker":"[dPT16]"},{"why":"Provides the viscosity/Pucci-class framework and the Jensen-envelope approximation used throughout the proof.","marker":"[CC95]"},{"why":"Introduced the sliding-paraboloid method used for the weak Harnack inequality.","marker":"[Cab97]"},{"why":"Credited with generalizing the sliding-paraboloid technique that the proof adapts.","marker":"[Sav03]"},{"why":"Supplies the measure-estimate and covering arguments modified in Lemma 2.5 and Corollary 2.7.","marker":"[LL17]"},{"why":"Source of the covering lemma used to pass from measure estimates at one scale to the weak Harnack inequality.","marker":"[IS16]"},{"why":"Establishes the partial regularity strategy, using small perturbation plus W^{2,ε}-type estimates, that motivates Proposition 1.10.","marker":"[ASS12]"},{"why":"Proves the σ_2 Hessian regularity result that Proposition 1.10 generalizes to σ_k equations with Lipschitz right-hand side.","marker":"[SY25]"},{"why":"Cited as the Alexandrov-type theorem that gives the Radon-measure Hessian representation on which the almost-everywhere twice differentiability part of Proposition 1.10 rests.","marker":"[CT05]"}],"fun_headline_variants":["Small perturbations yield C^2,α regularity for fully nonlinear PDEs","Savin's theorem extended to nonhomogeneous fully nonlinear equations","Automatic C^2,α regularity for small solutions of fully nonlinear PDEs","Partial regularity for sigma-k Hessian equations via Savin's theorem","New regularity result for fully nonlinear elliptic equations with Hölder data"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"For the advertised σ_k application, the load-bearing premise is that every 2-convex function has a distributional Hessian representable as a matrix of Radon measures; the paper cites this to a theorem stated for k>n/2 and does not derive it for k=2 in dimensions n≥5, so Proposition 1.10 collapses if that representation fails.","fun_headline_variants_meta":{"raw":{"variants":["Small perturbations yield C^2,α regularity for fully nonlinear PDEs","Savin's theorem extended to nonhomogeneous fully nonlinear equations","Automatic C^2,α regularity for small solutions of fully nonlinear PDEs","Partial regularity for sigma-k Hessian equations via Savin's theorem","New regularity result for fully nonlinear elliptic equations with Hölder data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000792,"raw_usage":{"total_tokens":3286,"prompt_tokens":664,"completion_tokens":2622,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":408,"completion_tokens_details":{"reasoning_tokens":2529}},"tokens_in":408,"tokens_out":2622,"duration_ms":20690,"temperature":1.0,"reasoning_tokens":2529,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T12:52:43.290965+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A 2-convex viscosity solution of σ_2(D^2u)=f in R^n with n≥5, with positive Lipschitz f, whose singular set has positive Lebesgue measure would falsify Proposition 1.10. Equally decisive for the main theorem: a sequence of flat solutions (u_j, f_j) satisfying Theorem 1.7's hypotheses with δ_j→0 but no uniform C^{2,α} bound on B_{1/2} would disprove Theorem 1.7.","supporting_citations":[],"review_version":1}