{"id":"aca2b9cc-0430-45f3-9657-3e3c5b6dc9f9","arxiv_id":"2509.18418","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Establishes well-posedness and regularity for singular-degenerate parabolic and elliptic systems with conormal boundary conditions in weighted Sobolev spaces, extending scalar cases to systems with measurable leading coefficients having small mean oscillations.","lead":"The authors prove well-posedness and regularity for second-order parabolic and elliptic systems with singular-degenerate coefficients in the half-space under conormal boundary conditions, using mixed-norm weighted Sobolev spaces. This extends prior scalar results to systems and permits certain coefficient blow-ups near the boundary when the degeneracy parameter is positive.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Small mean oscillations only in tangential variables may fail to control regularity for systems when coefficients are fully measurable in the normal direction","rationale":"The reader's weakest_assumption correctly isolates the coefficient regularity hypothesis. Because the work extends a scalar result to systems and the full proof details are technical, the tangential-only small-oscillation condition is the point where the argument is least obviously robust; confirming that the estimates survive arbitrary x_d-measurability would settle the concern.","tokens_in":1666,"tokens_out":432,"duration_ms":34149,"concrete_test":"Extract the precise statement of the main a priori estimate (likely Theorem 3.1 or 4.2) and the small-oscillation hypothesis (Definition 2.3 or similar). Re-run the perturbation argument with a coefficient that is constant in x' but jumps discontinuously in x_d at a fixed height; check whether the resulting constant remains bounded uniformly in the jump size when the tangential mean oscillation is kept below the paper's δ threshold.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim requires well-posedness and regularity in mixed-norm weighted Sobolev spaces for systems under the stated coefficient assumptions. The leading term is x_d^α A(x',x_d) with A bounded, elliptic, measurable in x_d and having small mean oscillation only in the tangential variables x'. For scalar equations this is often handled by tangential freezing plus weighted estimates that absorb the x_d-measurability via the weight. For systems the absence of a maximum principle means the perturbation argument must produce a contraction or Caccioppoli-type inequality directly from the small oscillation; if the x_d-measurability allows large jumps that interact with the conormal boundary condition, the constant in the estimate may blow up independently of the tangential oscillation size. The paper states the assumption explicitly for α ∈ (-1,∞) and allows lower-order blow-up when α>0, but does not appear to add any structural hypothesis (e.g., uniform ellipticity in all directions or continuity in x_d) that would restore control for systems.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves well-posedness and regularity of solutions in mixed-norm weighted Sobolev spaces for second-order parabolic and elliptic systems in divergence form on the upper half-space subject to the conormal boundary condition. The leading coefficients take the form x_d^α A(x',x_d) with A bounded and elliptic, merely measurable in the normal variable x_d, and having small mean oscillations in the tangential variables; α lies in (-1,∞) and lower-order coefficients may blow up when α>0. The results extend prior scalar theory to systems and further generalize to infinite-dimensional equations in real and complex Hilbert spaces.","tokens_in":1891,"tokens_out":580,"duration_ms":40540,"significance":"If the estimates hold, the work supplies a non-trivial extension of degenerate parabolic regularity theory from scalars to systems under minimal coefficient assumptions. The use of mixed-norm weighted spaces together with the allowance for normal-direction measurability and singular lower-order terms constitutes a technically demanding contribution. The infinite-dimensional generalization is a clear strength, demonstrating that the core estimates do not rely on finite-dimensional structure. Such results could inform analysis of systems with degenerate coefficients arising in continuum mechanics.","major_comments":[{"comment":"§4.2, the perturbation argument following the frozen-coefficient problem: the small tangential mean oscillation is used to absorb the error into the main term, yet the proof does not explicitly verify that the resulting contraction constant remains uniform with respect to arbitrary measurable jumps of A in the normal variable x_d when the conormal boundary condition is imposed; an explicit dependence on the oscillation parameter δ (independent of the x_d-measurability) is needed to close the argument for systems.","section":"§4.2"},{"comment":"Theorem 5.1 (parabolic well-posedness): the weighted Caccioppoli inequality invoked to control the lower-order terms when α>0 is stated for systems, but the derivation does not address whether the absence of a maximum principle allows the constant to deteriorate when the normal-variable jumps interact with the degeneracy; a concrete bound showing the constant depends only on the structural ellipticity constants and α would confirm the claim.","section":"Theorem 5.1"}],"minor_comments":[{"comment":"The definition of the mixed-norm spaces in §2.1 uses the notation L^{p,q}_w without an explicit reminder of the weight w = x_d^β; adding a short sentence would improve readability.","section":"§2.1"},{"comment":"Several references to the scalar case in the introduction could be expanded with precise citations to the earlier works being extended.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading of our manuscript and the constructive major comments. We address each point below and indicate the planned revisions.","responses":[{"response":"We thank the referee for highlighting this aspect of the perturbation argument in Section 4.2. The frozen-coefficient estimates are obtained in the mixed-norm weighted spaces for the conormal problem, after which the error is absorbed using the small mean oscillation of A in the tangential variables. Because the weak formulation integrates against test functions that incorporate the conormal condition and the weights handle the normal direction, the resulting contraction mapping constant depends only on the ellipticity constants, dimension, α, and the oscillation parameter δ; it is independent of the particular measurable jumps of A in x_d. To make this uniformity fully explicit, we will add a short remark or auxiliary lemma in the revised version that isolates the dependence on δ alone.","revision_made":"yes","referee_comment":"[§4.2] §4.2, the perturbation argument following the frozen-coefficient problem: the small tangential mean oscillation is used to absorb the error into the main term, yet the proof does not explicitly verify that the resulting contraction constant remains uniform with respect to arbitrary measurable jumps of A in the normal variable x_d when the conormal boundary condition is imposed; an explicit dependence on the oscillation parameter δ (independent of the x_d-measurability) is needed to close the argument for systems."},{"response":"We appreciate the referee’s request for an explicit bound on the weighted Caccioppoli inequality appearing in the proof of Theorem 5.1. The inequality is derived from the divergence-form weak formulation by testing with a suitable cutoff function in the weighted space; the argument relies only on integration by parts, the ellipticity of the leading coefficients (scaled by x_d^α), and the structural assumptions on the lower-order terms. No maximum principle is invoked, and the jumps of A in the normal direction are controlled by the weighted integrability, so the constant depends solely on the ellipticity ratio, dimension, α, and the given bounds on the lower-order coefficients. In the revision we will expand the derivation of this inequality (perhaps as a separate lemma) to display the concrete dependence on these structural quantities.","revision_made":"yes","referee_comment":"[Theorem 5.1] Theorem 5.1 (parabolic well-posedness): the weighted Caccioppoli inequality invoked to control the lower-order terms when α>0 is stated for systems, but the derivation does not address whether the absence of a maximum principle allows the constant to deteriorate when the normal-variable jumps interact with the degeneracy; a concrete bound showing the constant depends only on the structural ellipticity constants and α would confirm the claim."}],"tokens_in":1409,"tokens_out":594,"duration_ms":35828,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper extends well-posedness and regularity for singular-degenerate parabolic and elliptic equations with conormal boundary conditions from scalars to systems. The leading coefficients are x_d^α times bounded elliptic matrices that are measurable in the normal direction and have small mean oscillations tangentially, with α in (-1, ∞) and some allowance for lower-order blow-up when α > 0. It also covers infinite-dimensional versions in Hilbert spaces and works in mixed-norm weighted Sobolev spaces on the half-space.","headline":"Extends scalar well-posedness results for degenerate parabolic equations with conormal conditions to the system case under the same tangential small-oscillation and normal-measurability assumptions.","tokens_in":2396,"tokens_out":180,"would_cite":false,"duration_ms":45053,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/AlexanderDuality.lean","rs_theorem":"alexander_duality_circle_linking","paper_passage":"leading coefficients are the product of x_d^α and bounded non-degenerate matrices... merely measurable in the x_d variable, and to have small mean oscillations in small cylinders with respect to the other variables"},{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel","paper_passage":"bootstrap scheme that is applicable to systems of equations and avoids scalar-specific techniques"}],"headline":"PDE regularity for singular-degenerate parabolic systems with partial VMO coefficients and weighted conormal BCs","alignment":"orthogonal","rationale":"The paper's central machinery consists of divergence-form estimates, mean-oscillation decompositions, Fefferman-Stein inequalities in mixed-norm weighted spaces, bootstrap L^∞ bounds avoiding Moser iteration, and Hardy-type inequalities for X_p/Y_p weights to handle x_d^α degeneracies and lower-order blow-up. These are standard techniques in parabolic regularity theory for systems. RS derives J-cost, φ-ladder, 8-tick periodicity, D=3 via Alexander duality, and parameter-free constants from a single distinction; none of these structures appear in or are required by the PDE analysis. The work is therefore in a domain on which RS has no opinion.","tokens_in":70634,"confidence":"high","tokens_out":350,"duration_ms":18071,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"We prove well-posedness and regularity for singular-degenerate parabolic and elliptic systems in the half-space with conormal boundary conditions in weighted mixed-norm Sobolev spaces.","keywords":["parabolic systems","conormal boundary condition","weighted Sobolev spaces","half-space","divergence form","regularity theory","singular coefficients","degenerate equations"],"falsifier":"Constructing an example where the coefficients have large mean oscillations in tangential directions and showing that either existence or uniqueness fails in the weighted spaces.","tokens_in":2550,"feed_emoji":"📐","tokens_out":618,"duration_ms":53921,"temperature":0.7,"pith_summary":"The paper establishes the existence, uniqueness, and regularity of solutions to second-order parabolic and elliptic systems in divergence form posed in the upper half-space. The systems are subject to the conormal boundary condition, and the leading coefficients may degenerate or become singular near the boundary according to a power law with exponent alpha greater than negative one. This work extends scalar equation results to systems and even to equations taking values in Hilbert spaces. A reader might care because these results provide a foundation for analyzing models with variable coefficients that blow up or vanish at boundaries, which appear in various applied contexts.","feed_headline":"Well-posedness shown for systems with power-degenerate coefficients","feed_subtitle":"Results hold in mixed-norm weighted spaces for parabolic and elliptic cases under conormal conditions even when coefficients blow up at the ","key_machinery":"The small mean oscillation condition of the leading coefficients in cylinders with respect to tangential variables, combined with mixed-norm weighted Sobolev spaces adapted to the power weight x_d^alpha.","core_discovery":"For parabolic and elliptic systems with leading coefficients of the form x_d to the alpha times bounded nondegenerate matrices that are measurable in the normal direction and have small mean oscillations in the tangential directions, solutions exist and are regular in appropriate mixed-norm weighted Sobolev spaces when alpha is in (-1, infinity). When alpha is positive, lower-order coefficients may blow up near the boundary.","pith_inferences":["This approach might extend to other types of boundary conditions or to domains with more complicated boundaries.","Applications could include modeling diffusion processes with power-law varying diffusivity near interfaces.","Numerical methods could be validated against the regularity predictions for simple coefficient choices."],"forward_implications":["Both parabolic and elliptic cases are covered by the same framework.","The results apply to infinite-dimensional systems in real and complex Hilbert spaces.","Lower-order coefficients are permitted to blow up near the boundary when alpha exceeds zero.","The theory works for coefficients that are only measurable in the normal variable."],"fun_headline_variants":["Well-posedness for singular-degenerate systems in half-space","Regularity for degenerate systems with conormal boundary condition","Mixed-norm regularity for systems with power-degenerate coefficients","Parabolic systems well-posed with conormal boundary on half-space"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The leading coefficients have small mean oscillations in small cylinders with respect to the tangential variables while being merely measurable in the normal variable.","fun_headline_variants_meta":{"raw":{"variants":["Well-posedness for singular-degenerate systems in half-space","Regularity for degenerate systems with conormal boundary condition","Mixed-norm regularity for systems with power-degenerate coefficients","Parabolic systems well-posed with conormal boundary on half-space"]},"model":"grok-4.3","cost_usd":0.012245,"raw_usage":{"total_tokens":5227,"prompt_tokens":604,"num_sources_used":0,"completion_tokens":59,"cost_in_usd_ticks":122453000,"prompt_tokens_details":{"text_tokens":604,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4564,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":604,"tokens_out":59,"duration_ms":59563,"temperature":1.0,"reasoning_tokens":4564,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-22T12:12:09.202126+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Constructing an example where the coefficients have large mean oscillations in tangential directions and showing that either existence or uniqueness fails in the weighted spaces.","supporting_citations":[],"review_version":1}