{"id":"f712d621-276d-4d36-ad28-d691e2396719","arxiv_id":"2501.19033","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Weak solutions to weighted elliptic equations with a distance-to-a-manifold weight are shown to be C^{0,α} or C^{1,α} up to the characteristic manifold under a homogeneous conormal condition.","lead":"This paper proves Hölder and Schauder regularity estimates for solutions of elliptic equations whose coefficients degenerate or blow up on a lower dimensional set, such as a point or a line inside the domain. It provides explicit smoothness exponents and a general perforation method that transfers uniform estimates from approximations to the original singular problem.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the A3-regularity issue flagged by the reader is bypassed by the separate a priori estimate, and the central blow-up/Liouville argument is internally coherent.","rationale":"The reader and I agree that a conditional verdict is appropriate, but not because of the A3-regularity issue. That issue is explicitly addressed by the paper's structure: the perforation-based Theorem 1.3 is applied only after mollification in the final step, and the removal of the extra regularity uses the separate Proposition 7.1, whose contradiction proof works directly in the unperforated problem with C0,α coefficients. The actual soft spots are the deferred standard lemmas: Proposition 4.6 supplies the ε-uniform L∞ control needed to start the compactness argument, and Proposition 5.1 supplies the polynomial-in-x growth used to reduce the Liouville theorem to a one-dimensional-in-y spectral problem. Both are standard in spirit, but neither is proved in the paper, so the appropriate verification is to re-derive them in the perforated/weighted setting. If either fails, the blow-up limit could fail to be an entire solution with the claimed growth, breaking the contradiction. I do not see such a failure, and the surrounding argument is detailed and coherent, so the reader's conditional verdict should stand unchanged.","tokens_in":73481,"tokens_out":34675,"duration_ms":341065,"concrete_test":"Re-derive Proposition 5.1 for the cylindrical perforated domain R^{d-n}×(R^n\\Σ^A_ε) with constant A: verify that the x-difference quotients preserve the conormal boundary condition and that the iterative growth argument yields a polynomial of degree at most ⌊γ⌋ in x. If the polynomial-growth claim fails in the presence of the hole, the Liouville theorem in Case 3 collapses; if it succeeds, the central chain Theorem 1.3 → Theorem 1.2 is sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the paper as establishing a conditional Schauder theory: mollify the coefficients, apply the perforated-domain stable estimates, and then remove the extra C1 regularity by a separate a priori estimate (Proposition 7.1). The reader's weakest assumption—that the perforation geometry requires A3 to be at least C1—is therefore not load-bearing for Theorems 1.1 and 1.2, because the final reduction applies Proposition 7.1 to the mollified solutions and never needs C1 control of the original A3. The genuinely soft points are the deferred proofs: Proposition 4.6 (ε-stable L∞ bounds) is quoted from [29], and Proposition 5.1 (polynomial growth in x forces polynomial in x) is quoted as standard. Both are used essentially in the blow-up compactness and in the Liouville theorem. I found no internal inconsistency and no concrete regime where these standard facts fail; they are routine in this setting, but an independent check is the right way to confirm the conditional verdict.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a local regularity theory for degenerate/singular elliptic equations in divergence form whose weight is a power of the distance to a lower-dimensional set Sigma_0 of codimension n>=2. The main results are C^{0,alpha} (Theorem 1.1) and C^{1,alpha} (Theorem 1.2) estimates up to Sigma_0 under explicit assumptions on the data and a critical exponent alpha* defined through the ellipticity constants restricted to Sigma_0. The proof strategy is a regularization-approximation scheme: coefficients are mollified, the domain is perforated around Sigma_0 by anisotropic holes adapted to the matrix A, and epsilon-uniform regularity estimates (Theorem 1.3) are obtained by contradiction, blow-up, and a new Liouville theorem in perforated space (Theorem 1.4). A separate a priori estimate (Proposition 7.1) removes the extra C^1 or C^{1,alpha} regularity of the coefficients needed for the perforation geometry, yielding the final theorems under the optimal assumptions on A. The paper also extends the results to curved characteristic manifolds (Corollaries 8.4 and 8.5).","tokens_in":73635,"tokens_out":13065,"duration_ms":123428,"significance":"If correct, this is the first Schauder theory for elliptic equations degenerating on characteristic manifolds of codimension n>=2, and it gives a quantitative conormal boundary condition on Sigma_0 that is new even for the Laplacian in the appropriate parameter range. The epsilon-stable estimates in perforated domains, with holes whose shape follows the anisotropy of A, are of independent interest, and the connection to spectral stability for Neumann-perforated domains is a valuable byproduct. The paper is carefully written and contains a substantial amount of original technical work, including weighted functional inequalities, geometry of the perforated domains, and a Liouville theorem for entire solutions in perforated spaces. The main weakness is not an internal inconsistency but the deferral of a few key estimates to companion papers or standard references without full proofs.","major_comments":[{"comment":"Proposition 4.6 (epsilon-stable L-infinity bounds) is stated without proof, with the text saying the proof is carried out in [29, Sec. 2.4]. This estimate is used in an essential way in Step 1 of the proofs of Theorem 1.3(i) and (ii), and it feeds into (6.22), which is then used throughout the C^{1,alpha} proof and in Proposition 7.1. Because Theorem 1.3 is the engine of the paper, this omission leaves a load-bearing gap. The authors should provide a self-contained proof or state precisely which statements of [29] they use and confirm that [29] does not itself rely on results of the present paper.","section":"Sec. 4.2, Prop. 4.6"},{"comment":"Proposition 5.1, which asserts that an entire solution to (1.15) with growth |u(z)| <= c(1+|z|^gamma) is a polynomial of degree at most floor(gamma) in the x-variables, is used essentially in the proof of Theorem 1.4, both in the gamma<1 and gamma<2 cases. Its proof is omitted with a reference to a standard difference-quotients technique and to [52, Corollary 4.2, Lemma 4.3]. Since Theorem 1.4 is one of the paper's new Liouville theorems and is invoked in every blow-up contradiction argument, this proof should be included, or the exact statement from [52] should be quoted and the hypotheses verified for the present weighted, perforated setting.","section":"Sec. 5, Prop. 5.1"},{"comment":"In Step 5, when assuming r_k -> \\bar{r} > 0, the paper claims that \"Omega_infty = B_{1/\\bar{r}}(ς) \\setminus \\Sigma_0 for some ς\". This is not correct in general: if \\bar{\\varepsilon} = lim \\varepsilon_k/r_k > 0, the limiting hole is the cylinder \\Sigma^{\\bar{A}}_{\\bar{\\varepsilon}} = { \\bar{A}_3^{-1} y \\cdot y \\leq \\bar{\\varepsilon}^2 }, not the set \\Sigma_0. The subsequent contradiction that \\bar{v} is linear follows from the estimates already established and does not require the exact domain characterization, so the claim can be corrected or removed; as written it is a flaw in the proof of r_k -> 0.","section":"Sec. 6.2, Step 5 of the proof of Thm. 1.3(ii)"},{"comment":"Theorem 1.1 and Proposition 7.1(i) are stated with the proof omitted, with the text saying \"the proof for the other case follows the same argument and is easier to establish.\" Since Theorem 1.1 is one of the two main theorems of the paper, a proof or a detailed reduction to the estimates already established should be included; otherwise the Holder regularity result rests on an unproved a priori estimate that is not available in the existing literature in this form.","section":"Sec. 7, Prop. 7.1(i) and Thm. 1.1"}],"minor_comments":[{"comment":"In the definition of G_epsilon, the expression \"epsilon log epsilon\" for the case a+n=2 should be \"epsilon |log epsilon|\" (or the absolute value should be noted), since the constant must be positive.","section":"Sec. 3.3, Lemma 3.6"},{"comment":"The notation C^\\infty_c(B_R \\setminus \\Sigma^A_\\varepsilon) is defined twice with different support conditions; please use distinct symbols (e.g., different subscripts) to avoid ambiguity.","section":"Sec. 3.1.1"},{"comment":"In Step 1, the sentence \"By continuity, we can extend v in the whole B_{3/4}\" should read \"extend u\" rather than \"extend v\", as the letter v is not otherwise introduced there.","section":"Sec. 7.2, proof of Thm. 1.2"},{"comment":"In the change of variables leading to (6.15), the Jacobian determinant factor is not written explicitly; adding it would improve clarity and make the estimate easier to follow.","section":"Sec. 6.2, Eq. (6.15)"},{"comment":"Inequality (3.3) in Proposition 3.5 is missing the volume element dz in the integrand on the left-hand side.","section":"Sec. 2.2, Prop. 2.3"},{"comment":"The lower bound for mu_1 in the case n=2 is cited to [47, Lemma 1]; please confirm that the formula used in (5.4) exactly matches the statement of that lemma, since a small mismatch in the exponent |a|/4 would propagate into the definition of alpha*.","section":"Sec. 5, Lemma 5.2"}],"recommendation":"major_revision","confidential_remarks":"The paper's main results appear credible and the proof strategy is coherent, but the reliance on [29] and [52] for key estimates (Propositions 4.6 and 5.1) and the omission of proofs for Theorem 1.1 and Proposition 7.1(i) are the main concerns. The companion paper [29] is by one of the authors and is listed as a preprint; the editors may wish to verify that it is available and that there is no circular dependence. The Step 5 domain-characterization issue in Theorem 1.3(ii) is a local error that the authors can fix without changing the proof's overall structure."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a genuine advance, not a routine codimension bump. The codimension-one theory was settled by Terracini–Tortone–Vita and others, but going to codimension n ≥ 2 changes the capacity ranges, forces the anisotropic perforation Σ^A_ε shaped by the block A_3, and produces an explicit exponent α* that depends on the restricted ellipticity ratio λ*/Λ*. Theorem 1.3, the ε-stable estimates in perforated domains with the quantitative conormal bound (1.14), is new even for the Laplacian, and the Liouville theorem in perforated space (Theorem 1.4) is a standalone result. If the main theorems hold, this is the first Schauder theory up to a characteristic manifold of codimension at least two.\n\nI read the core proofs and they hold together. The blow-up scheme with its three regimes—interior, boundary of the hole away from Σ0, and both at once—is coherent, and the geometry lemmas in Appendix A supply exactly what the domain-characterization step needs. The reader's worry about Step 5 in the C^{1,α} proof (what if ε_k/r_k has a positive limit) dissolves: in that contradiction argument ε_k → 0 by construction, so if r_k fails to go to zero the limit domain is a bounded punctured ball and v̄ is forced to be linear, which contradicts the non-constancy of ∇v̄. Likewise, the A_3 ∈ C^1 requirement for the perforation geometry is real but not load-bearing for Theorems 1.1–1.2, because the mollified a priori estimates of Proposition 7.1 remove that extra regularity afterwards. The two-step regularization is coherent.\n\nThe actual soft spots are two deferred results. Proposition 4.6 (ε-stable L∞ bounds) is quoted from the companion preprint [29], and Proposition 5.1 (polynomial growth in x forces polynomial in x) is dismissed as standard without proof. Both are used essentially—in the blow-up compactness and in the Liouville reduction—and neither is available in published form in the perforated setting. I expect both are true; this is Moser iteration and difference quotients. But a referee should insist on proofs or published references before the paper is accepted.\n\nAudience: people working in degenerate elliptic regularity, extension problems for fractional Laplacians, or thin-obstacle free boundaries. It deserves a serious referee. My recommendation: send it to review and require the two deferred arguments to be written out or replaced by published references.","headline":"Genuine extension of Schauder theory to codimension n≥2 with a new anisotropic perforation scheme; the main structure holds up, but two load-bearing estimates are deferred to a companion preprint or to 'standard' status.","tokens_in":74198,"tokens_out":5295,"would_cite":true,"duration_ms":49155,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B65","35J70","35J75","35B40","35B44","35B45","35B53"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves explicit Hölder and Schauder estimates up to a characteristic manifold of codimension at least two, via perforation, blow-up, and a new Liouville theorem.","keywords":["weighted elliptic equations","degenerate ellipticity","Schauder regularity estimates","lower dimensional boundaries","Liouville theorems","perforated domains","codimension at least two"],"falsifier":"Compute the model profile $u_i(y)=y_i|y|^{\\gamma_1^+-1}$ for the isotropic case $A=I$ with $a<0$ and $n=2$, where $\\gamma_1^+=\\alpha_*$ as defined in (1.7). By construction this profile should be a weak solution of the homogeneous conormal problem with growth exactly at the threshold; its second derivatives behave like $|y|^{\\gamma_1^+-2}$, so if the exponent $\\alpha_*$ is correct the $C^{1,\\alpha}$ regularity in Theorem 1.2 cannot hold for $\\alpha>\\gamma_1^+-1$. Checking that the computed blow-up rate matches this prediction confirms the threshold, while any discrepancy in the rate, or any failure of the profile to satisfy the conormal condition, would refute the claimed exponent.","tokens_in":73257,"feed_emoji":"📐","tokens_out":13920,"duration_ms":132178,"temperature":0.7,"pith_summary":"Elliptic equations whose coefficients vanish or blow up like $|y|^a$ on a $(d-n)$-dimensional plane lose uniform ellipticity exactly on that set. This paper proves that, as long as $a+n>0$, weak solutions satisfying a natural homogeneous conormal condition are Hölder continuous up to the singular set, and in a parameter range singled out by an explicit exponent $\\alpha_*$ they are $C^{1,\\alpha}$ up to it. The proof replaces the singular set by a small anisotropic hole, solves a Neumann problem on the perforated domain, and shows the estimates are stable as the hole shrinks to zero. If the argument is correct, it gives the first Schauder theory for degeneracies on manifolds of codimension $n\\ge 2$, plus $\\varepsilon$-uniform stable estimates that are new even for the Laplacian. The theory extends to equations degenerating on smooth curved thin manifolds.","feed_headline":"Codimension-2 degeneracies get Hölder and Schauder regularity","feed_subtitle":"New estimates control degenerate-elliptic solutions up to the singular manifold, stable even for the Laplacian.","key_machinery":"The load-bearing object is the anisotropic perforation $\\Sigma^A_\\varepsilon = \\{A_3^{-1}(x,y)y\\cdot y \\le \\varepsilon^2\\}$, an $\\varepsilon$-neighborhood of $\\Sigma_0$ shaped by the lower-right block $A_3$ of the coefficient matrix; on its boundary the approximating solution is given the conormal Neumann condition $(A\\nabla u_\\varepsilon+F)\\cdot \\nu=0$. The estimates are proved by contradiction: if the $\\varepsilon$-uniform bound failed, a fine blow-up around points near $\\partial\\Sigma^A_\\varepsilon$ would produce an entire limiting profile defined either on $\\mathbb{R}^d$, a half-space, or $\\mathbb{R}^d$ minus an unbounded cylinder, and the new Liouville theorem rules out every nonconstant or non-linear profile of the allowed growth. The quantitative engine inside the Liouville theorem is the spectral gap of a weighted spherical operator: its first nontrivial eigenvalue $\\mu_1$, bounded below via Lemma 5.2, enters the explicit threshold $\\alpha_*$ in (1.7), and the condition $\\alpha_*>1$ is exactly what forces the limiting gradients to be constant.","core_discovery":"For the equation $-\\mathrm{div}(|y|^a A(x,y)\\nabla u)=|y|^a f+\\mathrm{div}(|y|^aF)$ in $B_1$, with $2\\le n\\le d$ and $a\\in\\mathbb{R}$, Theorem 1.1 says that whenever $a+n>0$ every weak solution in $H^{1,a}$ that solves across $\\Sigma_0=\\{y=0\\}$ belongs to $C^{0,\\alpha}_{\\mathrm{loc}}(B_1)$ for the explicit range (1.8). Theorem 1.2 is the central claim: if the threshold $\\alpha_*$ defined in (1.7), built from $a$, $n$, and the restricted ellipticity ratio $\\lambda_*/\\Lambda_*$, exceeds 1, then for $\\alpha$ satisfying (1.9), every such solution is $C^{1,\\alpha}_{\\mathrm{loc}}(B_1)$, satisfies the pointwise conormal condition $(A\\nabla u+F)\\cdot e_{y_i}=0$ on $\\Sigma_0$, and obeys estimate (1.11) with constants independent of $u$. The supporting Theorem 1.3 shows that the approximating Neumann problems on the perforated domains $B_1\\setminus\\Sigma^A_\\varepsilon$ admit $\\varepsilon$-uniform $C^{0,\\alpha}$ and $C^{1,\\alpha}$ bounds together with the quantitative boundary estimate (1.14), while Theorem 1.4 classifies entire solutions on the perforated space: sublinear growth forces constants and subquadratic growth forces linear functions. Corollaries 8.4 and 8.5 carry the flat result to equations whose weight degenerates on $C^{1,\\alpha}$ curved manifolds.","pith_inferences":["The explicit exponent $\\alpha_*$ suggests an optimal-Hölder-exponent picture for this degeneracy analogous to known optimal exponents for bounded measurable coefficients; the paper notes it remains open whether the reduction of $\\alpha_*$ by $\\lambda_*/\\Lambda_*$ is intrinsic or an artifact of the method.","Because Theorem 1.3(i) is stable in $\\varepsilon$, it should yield quantitative rates for Neumann eigenvalue problems in domains with small holes, converting the known qualitative convergence into explicit control of eigenfunctions near the holes.","The failure of $C^{2,\\alpha}$ stability under perforation documented in Remark 6.2 indicates that higher regularity, if true, needs a different regularization than hole removal; one testable route is to iterate only tangential derivatives and impose axial symmetry in $y$, under which the operator commutes enough to bootstrap.","The pointwise conormal condition (1.10) gives a higher-codimension analogue of the boundary condition appearing in fractional-Laplacian extension problems, so the estimates may serve as a regularity tool for very thin obstacle problems with obstacles of codimension at least two."],"forward_implications":["For every $a+n>0$, weak solutions of the homogeneous conormal problem are Hölder continuous up to the characteristic manifold, with an explicit exponent range depending on the ellipticity ratio restricted to $\\Sigma_0$.","When $\\alpha_*>1$, weak solutions are $C^{1,\\alpha}$ up to $\\Sigma_0$ and satisfy the strong conormal condition $(A\\nabla u+F)\\cdot e_{y_i}=0$ pointwise, not just in the weighted sense of (1.3).","The $\\varepsilon$-uniform stable estimates in perforated domains are new even for the Laplacian with Neumann holes and give stable $\\alpha$-Hölder bounds for eigenfunctions in Neumann-perforated domains.","Sublinear entire solutions of the degenerate problem on the perforated space are constant and subquadratic ones are linear, which is the rigidity fact powering the blow-up argument.","The flat results transfer to equations degenerating on smooth curved thin manifolds of codimension $n\\ge 2$, with the conormal condition normal to the manifold."],"supporting_citations":[{"why":"supplies the classical weighted De Giorgi-Nash-Moser theory and local Hölder continuity for 2-admissible weights that the paper's $C^{0,\\alpha}$ result refines.","marker":"[26]"},{"why":"introduces the regularization by domain perforation and blow-up strategy for codimension 1 that the present approximation scheme adapts to codimension $n$.","marker":"[50]"},{"why":"provides the half-space Liouville theorem with homogeneous conormal condition used to rule out the flat blow-up limits.","marker":"[52]"},{"why":"supplies the sharp lower bound on the first nontrivial spherical eigenvalue for $n=2$ and the optimal Hölder-exponent model that $\\alpha_*$ mirrors.","marker":"[47]"},{"why":"supplies the 2-admissible weight framework used for the Poincaré-Wirtinger and Sobolev inequalities that ground the functional setting.","marker":"[34]"},{"why":"provides the Caffarelli-Kohn-Nirenberg inequality used in the $\\varepsilon$-stable Sobolev embeddings for the perforated domains.","marker":"[5]"}],"fun_headline_variants":["Hölder and Schauder regularity for elliptic equations degenerating on submanifolds","Degenerate ellipticity on low-dimensional sets gains C^{1,α} and Hölder bounds","Schauder estimates up to a singular manifold for degenerate elliptic PDEs","Regularity across degeneracy loci for elliptic equations with weighted coefficients","Explicit Hölder and C^{1,α} bounds for degenerate elliptic equations on thin sets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The stability of the estimates in the perforated domains depends on the boundary of the anisotropic hole having a well-controlled normal, which requires the lower-right coefficient block $A_3$ to be at least $C^1$ ($C^{1,\\alpha}$ in the Schauder case); if $A_3$ were only continuous at that stage, the $\\varepsilon$-uniform trace and Sobolev inequalities used in the blow-up contradiction would fail, and the paper later removes this extra assumption only by a separate mollification argument.","fun_headline_variants_meta":{"raw":{"variants":["Hölder and Schauder regularity for elliptic equations degenerating on submanifolds","Degenerate ellipticity on low-dimensional sets gains C^{1,α} and Hölder bounds","Schauder estimates up to a singular manifold for degenerate elliptic PDEs","Regularity across degeneracy loci for elliptic equations with weighted coefficients","Explicit Hölder and C^{1,α} bounds for degenerate elliptic equations on thin sets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001272,"raw_usage":{"total_tokens":5330,"prompt_tokens":1197,"completion_tokens":4133,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":813,"completion_tokens_details":{"reasoning_tokens":4025}},"tokens_in":813,"tokens_out":4133,"duration_ms":27727,"temperature":1.0,"reasoning_tokens":4025,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T21:33:37.632649+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the model profile $u_i(y)=y_i|y|^{\\gamma_1^+-1}$ for the isotropic case $A=I$ with $a<0$ and $n=2$, where $\\gamma_1^+=\\alpha_*$ as defined in (1.7). By construction this profile should be a weak solution of the homogeneous conormal problem with growth exactly at the threshold; its second derivatives behave like $|y|^{\\gamma_1^+-2}$, so if the exponent $\\alpha_*$ is correct the $C^{1,\\alpha}$ regularity in Theorem 1.2 cannot hold for $\\alpha>\\gamma_1^+-1$. Checking that the computed blow-up rate matches this prediction confirms the threshold, while any discrepancy in the rate, or any failure of the profile to satisfy the conormal condition, would refute the claimed exponent.","supporting_citations":[{"cited_title":"Fabes, C","cited_arxiv_id":null,"evidence_quote":"supplies the classical weighted De Giorgi-Nash-Moser theory and local Hölder continuity for 2-admissible weights that the paper's $C^{0,\\alpha}$ result refines."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces the regularization by domain perforation and blow-up strategy for codimension 1 that the present approximation scheme adapts to codimension $n$."},{"cited_title":"Terracini, G","cited_arxiv_id":null,"evidence_quote":"provides the half-space Liouville theorem with homogeneous conormal condition used to rule out the flat blow-up limits."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the sharp lower bound on the first nontrivial spherical eigenvalue for $n=2$ and the optimal Hölder-exponent model that $\\alpha_*$ mirrors."},{"cited_title":"Heinonen, T","cited_arxiv_id":null,"evidence_quote":"supplies the 2-admissible weight framework used for the Poincaré-Wirtinger and Sobolev inequalities that ground the functional setting."},{"cited_title":"Caffarelli, R","cited_arxiv_id":null,"evidence_quote":"provides the Caffarelli-Kohn-Nirenberg inequality used in the $\\varepsilon$-stable Sobolev embeddings for the perforated domains."}],"review_version":1}