{"id":"3f7c4a9c-a1c8-4891-8798-3679121e321e","arxiv_id":"2411.14217","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For mixed local-nonlocal singular quasilinear elliptic problems, the authors establish existence, uniqueness under a boundary-weight restriction, sharp Sobolev regularity, boundary behavior, and Hölder regularity with exponents depending on beta plus delta.","lead":"This paper proves existence, uniqueness, and Hölder regularity theorems for a family of mixed local and nonlocal elliptic equations containing a singular term that blows up near the boundary. The results matter because this operator family combines classical diffusion with long-range fractional effects, and the singularity models sources that grow as the boundary is approached.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The global Hölder regularity claimed in Theorem 2.31(iii)-(iv) is not justified: the two-case argument in §7.4 relies on local Hölder constants that blow up as R→0 when β+δ>1.","rationale":"The reader’s weakest assumption correctly identifies that uniqueness/comparison is proved only for β<2−1/p, while the abstract and Theorem 2.31 mention 'the unique solution' without this restriction. That is a real statement-level issue. However, the more load-bearing concern is in the proof of the central regularity claim itself. Theorem 2.31 is the paper’s main output, and its cases (iii)-(iv) assert global C^{0,ζ} regularity up to the boundary for β+δ>1. The two-case argument in §7.4 is a standard pattern, but it only works if the local Hölder estimate used in Case (I) has a constant that is uniform as the ball approaches the boundary. For the singular problem with β+δ>1, the source term f u^{−δ} behaves like d^{−β−δ}, and its L^n norm on a ball of radius R at distance R from the boundary is of order R^{1−β−δ}, which diverges as R→0. The local estimate in Theorem 2.19 depends on this norm, so the uniform control needed for the global Hölder conclusion is not supplied. Boundary growth u≤Cd^ζ alone is insufficient: one can construct smooth functions bounded between c d^ζ and C d^ζ that are locally Hölder on every compact subset but not globally C^{0,ζ} due to oscillations accumulating near the boundary. Thus the proof must use the equation at the boundary scale, which it currently does not. I am not claiming the theorem is false; the stated result is plausible and likely recoverable with a boundary barrier or Campanato argument. But as written, the central global Hölder assertion is not fully proven, which reinforces the need for a conditional verdict and a specific requested repair.","tokens_in":66529,"tokens_out":45935,"duration_ms":420815,"concrete_test":"Re-run the proof of Theorem 2.31(iii) with explicit constants: for the half-ball model −Δ_p U + (−Δ)_q^s U = d^{−β−δ} with 0≤U≤C d^ζ and ζ=(p−β)/(p−1+δ), attempt to derive the uniform Campanato estimate ⨍_{B_R^+}|U−(U)_{B_R^+}|^p ≤ C R^{pζ} for all R∈(0,ρ). If the only available bound from Theorem 2.19 contains ‖d^{−β−δ}‖_{L^n(B_R)} ≃ R^{1−β−δ}, the estimate fails when β+δ>1, confirming the gap. If a barrier argument independently yields this Campanato estimate, the concern is resolved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 2.31 in §7.4 establishes the up-to-boundary Hölder regularity for β+δ>1 by a two-case argument: if |x−y|<d(x)/64, apply local Hölder regularity in B_R(x) with R=d(x)/64; otherwise, use the boundary bound u≤Cd^ζ. The second case is fine, but the first case requires a uniform local Hölder seminorm on balls that approach the boundary. The only local estimate provided, Theorem 2.19(a)-(b), has a constant that depends on ‖f u^{−δ}‖_{L^n(B_R)} (see the constant in (4.30)). For β+δ>1, on a ball of radius R located at distance R from ∂Ω, the singular source satisfies f u^{−δ} ≃ d^{−β−δ}, so ‖f u^{−δ}‖_{L^n(B_R)} ≃ R^{1−β−δ} → ∞ as R→0. Thus the local Hölder constant can blow up, and the boundary growth u≤Cd^ζ alone does not rule out oscillations at scales much smaller than R: a smooth function comparable to d^ζ can have frequency growing near the boundary and be locally C^{0,η} on every compact subset while failing global C^{0,ζ}. In Case (I) the proof does not use the equation beyond citing this local estimate, so it does not control such oscillations. Consequently, the asserted global C^{0,ζ} regularity in Theorem 2.31(iii)-(iv) is not established by the argument as written; a separate boundary Campanato estimate for the singular weight d^{−β−δ} is missing.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the mixed local-nonlocal quasilinear singular problem -Δ_p u + (-Δ)_q^s u = f(x)u^{-δ} in Ω, u=0 outside Ω, where f behaves like dist(x,∂Ω)^{-β}. It develops a local Hölder and gradient Hölder theory for the regular operator, proves up-to-boundary C^{1,γ} regularity under distance-like assumptions, and then applies these tools to the singular problem. The main results claimed are existence for β∈[0,p), uniqueness for β<2-1/p, boundary behavior with explicit rates depending on β+δ, optimal Sobolev regularity, non-existence for β≥p, and global Hölder regularity of the solution in Theorem 2.31 with exponents depending on β+δ. The paper is long and detailed, with many estimates following the patterns of [24] and [1].","tokens_in":66849,"tokens_out":10174,"duration_ms":102860,"significance":"If Theorem 2.31 were fully established, the paper would deliver a substantial result: up-to-boundary Hölder and interior gradient Hölder regularity for a doubly singular mixed local-nonlocal problem. The local regularity theorems (Theorems 2.17-2.20), the barrier constructions for boundary behavior, the comparison principle in its valid range, and the non-existence result are useful contributions that are likely to be of independent interest. The proof of the global Hölder claim, however, contains a genuine gap in the transition from local Hölder estimates to the boundary, and the uniqueness statement in Theorem 2.31 is broader than what is proved. These issues concern the central advertised results and need to be repaired before the paper can be accepted.","major_comments":[{"comment":"Theorem 2.31 is stated for all β∈[0,p) and refers to 'the unique solution' of problem (2.13), but uniqueness is proved in Theorem 2.28 only under the additional assumption β<2-1/p. The restriction enters in Section 7.1, equation (7.39), where Hardy's inequality is used to show that the operator J_m is well-defined; this requires (1-β)p/(p-1)>-1, i.e. β<2-1/p. Without this condition, no comparison principle is established. Therefore the phrase 'the unique solution' in Theorem 2.31 is not justified for β∈[2-1/p,p). The theorem should either be restricted to the range where uniqueness is proved or the uniqueness statement should be reformulated as an open problem for the remaining range.","section":"Theorem 2.31 vs. Theorem 2.28"}],"minor_comments":[{"comment":"The theorem states 'u∈C^{0,η}(Ω) for all σ∈(0,1)', but the exponent should be η; the symbol σ is inconsistent with the notation η used in the same sentence.","section":"Theorem 2.31(ii)"},{"comment":"Line: 'we aim to we study Hölder regularity results' contains a typo; it should read 'we aim to study'.","section":"Section 1, Introduction"},{"comment":"The sentence 'set 64R=d(x)' is ambiguous; it should read 'set R=d(x)/64' to make clear that B_R(x) is a ball of radius R contained in Ω.","section":"Section 7.4, Case (I)"},{"comment":"The abstract says regularity is obtained 'albeit with different exponents depending on β+δ', but it does not mention that the uniqueness theorem is conditional on β<2-1/p; this limitation should be stated in the abstract or at least in the introduction.","section":"Abstract and Introduction"}],"recommendation":"major_revision","confidential_remarks":"The paper contains substantial useful material, and the local regularity and existence machinery appear sound. The main obstacle is the global Hölder claim in Theorem 2.31, which is not justified by the local estimates because the relevant constants blow up near the boundary. This is a repair problem rather than a fundamental invalidation of the entire paper, but it is central to the paper's advertised novelty. The uniqueness range issue in Theorem 2.31 should also be fixed. I recommend major revision; if the authors can supply the missing boundary Campanato estimate or suitably weaken/qualify Theorem 2.31, the paper could be acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague —\n\nThe headline: the paper’s main global-Hölder claim for the singular range β+δ>1 (Theorem 2.31(iii)–(iv)) is not actually proved. The two-case argument in §7.4 breaks down because the local Hölder constant from Theorem 2.19(a) depends on ‖f u^{−δ}‖_{L^n(B_R)}, and for balls at distance R from ∂Ω that norm behaves like R^{1−β−δζ} with ζ=(p−β)/(p−1+δ), which blows up as R→0 whenever β+δ>1. The boundary bound u≤Cd^ζ controls values but not oscillations at scales below R. A boundary Campanato estimate with the singular weight d^{−β−δζ} is missing. So the advertised up-to-boundary C^{0,ζ} regularity is unsubstantiated as written; the case β+δ=1 is borderline and may be okay.\n\nThat said, the paper is not a throwaway. The regular-problem results (Theorems 2.17–2.21) genuinely extend the De Filippis–Mingione and Antonini–Cozzi machinery to a nonhomogeneous (p,q) mixed operator with solutions in a weaker local class, and those look solid. The existence theory for the singular problem (Theorem 2.27), the boundary behavior in Definition 2.26, and the optimal Sobolev regularity are coherent and useful, and the comparison principle in Theorem 2.28 is a real achievement in the restricted range β<2−1/p. The authors are honest about the restrictions in the theorems; the problem is the abstract, which promises existence and uniqueness without them.\n\nOther soft spots: the proofs lean heavily on [24] and [1], which makes verification slow, but that is standard practice, not a flaw. The uniqueness limitation means Theorem 2.31’s phrase “the unique solution” only applies in the restricted parameter range; for larger β the constructed solution is not known to be unique.\n\nIf I were the editor, I’d send it to peer review rather than desk-reject: the topic is active, the regular results have independent value, and the gap in §7.4 is identifiable and potentially fixable. The referee should demand either a genuine boundary Campanato estimate or a clear explanation of why the local constants don’t blow up. The abstract also needs to match the theorem statements.\n\nFor a reading group: maybe — worth discussing the gap, but I wouldn’t present it as a finished global regularity result.","headline":"The global Hölder claim for β+δ>1 is not proved because the local regularity constants blow up near the boundary; the regular-problem results and existence theory are solid enough to merit peer review.","tokens_in":67379,"tokens_out":5873,"would_cite":true,"duration_ms":53401,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J75","35M10","35R11"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves up-to-boundary Hölder and gradient-Hölder regularity for the singular mixed local-nonlocal problem $-\\Delta_p u+(-\\Delta)_q^s u=f(x)u^{-\\delta}$ with $f$ blowing up like $d^{-\\beta}$.","keywords":["Mixed local-nonlocal equation","Hölder regularity","Singular nonlinearity","Distance function","Existence and uniqueness","Fractional p-Laplacian","Boundary behavior","Quasilinear elliptic equation"],"falsifier":"Take $\\Omega$ to be the unit ball, $p=q=2$, $s=1/2$, and $f=d^{-\\beta}$ with $\\beta+\\delta>1$; Theorem 2.31 predicts $u\\in C^{0,(2-\\beta)/(1+\\delta)}(\\Omega)$. Computing the boundary quotient $\\limsup_{x\\to\\partial\\Omega} u(x)/d(x)^{(2-\\beta)/(1+\\delta)}$ for the constructed solution—positive and finite if the predicted boundary behavior is exact, zero or infinite if it is not—would settle the boundary-regularity claim.","tokens_in":66320,"feed_emoji":"📏","tokens_out":15702,"duration_ms":142861,"temperature":0.7,"pith_summary":"This paper studies positive zero-outside solutions of $-\\Delta_p u+(-\\Delta)_q^s u=f(x)u^{-\\delta}$ in a bounded $C^2$ domain, where $1<q\\le p<\\infty$, $s\\in(0,1)$, and $f$ blows up near the boundary like $d(x)^{-\\beta}$, with $d(x)=\\mathrm{dist}(x,\\partial\\Omega)$. Since the local and nonlocal operators have different orders and homogeneity, and the singularity is double—the term $u^{-\\delta}$ and the weight $d^{-\\beta}$—the usual scaling, energy, and sub-supersolution arguments for single-operator problems do not apply directly. The paper establishes existence of a weak solution for all $\\beta\\in[0,p)$, a weak comparison principle and uniqueness when $\\beta<2-1/p$, optimal Sobolev regularity of the solution in terms of $\\beta+\\delta$, and nonexistence for $\\beta\\ge p$. Its main regularity theorem makes the boundary-Hölder exponent a function of $\\beta+\\delta$: $C^{1,\\sigma}(\\Omega)$ when $\\beta+\\delta<1$, $C^{0,\\eta}(\\Omega)$ for every $\\eta<1$ when $\\beta+\\delta=1$, and $C^{0,(p-\\beta)/(p-1+\\delta)}(\\Omega)$ when $\\beta+\\delta>1$, with local gradient-Hölder regularity in all these cases. The paper thereby gives the doubly singular problem a coherent boundary-regularity picture despite the non-homogeneous mixed operator.","feed_headline":"β + δ decides boundary Hölder regularity in mixed singular PDEs","feed_subtitle":"For a source blowing up like d^{-β}, solutions are C^{1,σ} when β+δ<1 and Hölder of explicit exponent otherwise.","key_machinery":"The central object is the distance function $d(x)=\\mathrm{dist}(x,\\partial\\Omega)$, together with barrier functions of the form $w(x)=\\Gamma(d(x)+\\varepsilon^{1/\\tau})^\\tau$, where $\\tau=(p-\\beta)/(p-1+\\delta)$. Because $\\partial\\Omega$ is $C^2$, $d$ is smooth in a boundary neighbourhood, and the paper computes the action of the $p$-Laplacian on these powers exactly, while showing through estimates for fractional powers of distance that the fractional $q$-Laplacian of the truncated barrier is controlled. These barriers produce the boundary behavior of the approximating solutions and support the weak comparison principle. In the interior, gradient Hölder regularity is obtained by comparison with solutions $h$ of a frozen homogeneous problem on each ball; the difference $|u-h|$ is controlled by energy estimates and a nonlocal tail functional that records the contribution of $u$ away from the ball. The singular weight parameter $\\beta$ enters through a weighted integrability condition, which is why the comparison principle is stated only for $\\beta<2-1/p$.","core_discovery":"The central claim, Theorem 2.31, is that for $\\beta\\in[0,p)$ and $\\delta>0$, the weak solution constructed by approximation—or the unique solution when the comparison principle applies—belongs to $C^{1,\\sigma}(\\Omega)$ for some $\\sigma\\in(0,1)$ if $\\beta+\\delta<1$; to $C^{0,\\eta}(\\Omega)$ for every $\\eta\\in(0,1)$ if $\\beta+\\delta=1$; and, if $\\beta+\\delta>1$, to $C^{0,(p-\\beta)/(p-1+\\delta)}(\\Omega)$ except in the case $\\beta=p-q's(p-1+\\delta)$, where it belongs to $C^{0,(p-\\beta_1)/(p-1+\\delta)}(\\Omega)$ for every $\\beta_1\\in(\\beta,p)$, with $q'=q/(q-1)$. In the last three cases the solution is also $C^{1,\\gamma}$ in the interior. These results rest on a systematic regularity theory for the nonsingular operator $\\mathcal{L}u=-\\operatorname{div}A(x,\\nabla u)+\\text{fractional nonlocal term}$: local Hölder regularity when the right-hand side is in $L^n_{\\mathrm{loc}}$, local gradient Hölder regularity when it is in $L^d_{\\mathrm{loc}}$ with $d>n$, and a boundary $C^{1,\\gamma}$ theorem under $0\\le f\\le Cd^{-\\sigma}$ with $\\sigma<1$ together with a one-sided bound $0\\le u\\le Cd^{\\epsilon}$. The singular solution is obtained as the increasing limit of solutions with $u^{-\\delta}$ replaced by $(u+\\varepsilon)^{-\\delta}$, and its boundary behavior is pinned down by distance barriers as $u\\asymp d^{(p-\\beta)/(p-1+\\delta)}$, with different powers when $\\beta+\\delta\\le1$.","pith_inferences":["The exponent $(p-\\beta)/(p-1+\\delta)$ is likely sharp: the proved lower and upper distance bounds have exactly this power, and boundary Hölder regularity cannot in general improve beyond the power governing the boundary behavior; a radial model would make this testable.","The restriction $\\beta<2-1/p$ in the uniqueness statement probably reflects the method rather than the phenomenon, since existence and boundary behavior are established for all $\\beta<p$; a comparison argument that avoids the weighted integrability step might extend uniqueness to the full range.","The interior regularity estimates are developed for a broad class of operators and for solutions only locally in $W^{1,p}$, so they could be reused for other singular, critical, or lower-order problems without repeating the distance-barrier construction."],"forward_implications":["When the weight is bounded at the boundary ($\\beta=0$) and $\\delta<1$, every weak solution is $C^{1,\\sigma}$ up to the boundary, not merely interior.","At $\\beta+\\delta=1$ the solution is Hölder continuous up to the boundary with every exponent below $1$, while its gradient is Hölder continuous in the interior; this almost-Lipschitz boundary regularity is obtained without requiring the solution to be in the energy space.","For $\\beta+\\delta>1$ the boundary Hölder exponent equals the power appearing in the boundary behavior $u\\asymp d^{(p-\\beta)/(p-1+\\delta)}$, so the Hölder and boundary-behavior statements are mutually consistent.","Existence is sharp in $\\beta$: no weak solution exists for $\\beta\\ge p$, and the Sobolev-regularity theorem says exactly when $u$, or a power $u^\\theta$, belongs to $W^{1,p}_0(\\Omega)$.","As a direct application, the singular perturbed problem $-\\Delta_p u+(-\\Delta)_q^s u=\\lambda u^{-\\delta}+b(x,u)$ with critical growth in $b$ has solutions in $C^{1,\\sigma}(\\Omega)$ when $\\delta<1$ and $\\beta=0$."],"supporting_citations":[{"why":"Supplies the local gradient-Hölder comparison machinery for mixed local-nonlocal operators, including energy estimates, tail functionals, and comparison with frozen homogeneous problems, which the paper adapts to local weak solutions.","marker":"[24]"},{"why":"Provides the global gradient regularity and boundary point estimate for mixed operators that the paper uses for boundary regularity and for the approximating solutions.","marker":"[1]"},{"why":"Introduces the monotone approximation of singular problems by replacing $u^{-\\delta}$ with $(u+\\epsilon)^{-\\delta}$, the method used here to construct the weak solution.","marker":"[14]"},{"why":"Gives the fractional-Laplacian estimates for powers of the distance function used in the boundary barriers.","marker":"[2]"},{"why":"Provides estimates for the strongly nonhomogeneous fractional $p,q$-Laplacian applied to distance powers, used in the boundary analysis.","marker":"[35]"},{"why":"Supplies the nonlocal singular-problem comparison argument that the paper adapts to prove uniqueness.","marker":"[19]"},{"why":"Supplies the finite-difference iteration used to obtain higher local Hölder regularity of solutions to mixed quasilinear equations.","marker":"[33]"}],"fun_headline_variants":["Mixed local-nonlocal PDEs: β+δ sets boundary smoothness","Explicit Hölder exponent for mixed singular PDEs","β+δ=1 marks a phase transition in boundary regularity","Distance singularity controls Hölder exponent","Boundary exponent (p−β)/(p−1+δ) for singular PDEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise for the uniqueness and comparison part is that the singular weight is integrable enough at the boundary for a weighted inequality of Hardy type to apply, and this holds only when $\\beta<2-1/p$; if that condition fails, the proof still gives existence and regularity but does not select a unique solution.","fun_headline_variants_meta":{"raw":{"variants":["Mixed local-nonlocal PDEs: β+δ sets boundary smoothness","Explicit Hölder exponent for mixed singular PDEs","β+δ=1 marks a phase transition in boundary regularity","Distance singularity controls Hölder exponent","Boundary exponent (p−β)/(p−1+δ) for singular PDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000973,"raw_usage":{"total_tokens":4311,"prompt_tokens":1298,"completion_tokens":3013,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":914,"completion_tokens_details":{"reasoning_tokens":2926}},"tokens_in":914,"tokens_out":3013,"duration_ms":21263,"temperature":1.0,"reasoning_tokens":2926,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:25:43.503448+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $\\Omega$ to be the unit ball, $p=q=2$, $s=1/2$, and $f=d^{-\\beta}$ with $\\beta+\\delta>1$; Theorem 2.31 predicts $u\\in C^{0,(2-\\beta)/(1+\\delta)}(\\Omega)$. Computing the boundary quotient $\\limsup_{x\\to\\partial\\Omega} u(x)/d(x)^{(2-\\beta)/(1+\\delta)}$ for the constructed solution—positive and finite if the predicted boundary behavior is exact, zero or infinite if it is not—would settle the boundary-regularity claim.","supporting_citations":[{"cited_title":"Gradient regularity in mixed local and nonlocal problems","cited_arxiv_id":null,"evidence_quote":"Supplies the local gradient-Hölder comparison machinery for mixed local-nonlocal operators, including energy estimates, tail functionals, and comparison with frozen homogeneous problems, which the paper adapts to local weak solutions."},{"cited_title":"Semilinear elliptic e quations with singular nonlinearities","cited_arxiv_id":null,"evidence_quote":"Introduces the monotone approximation of singular problems by replacing $u^{-\\delta}$ with $(u+\\epsilon)^{-\\delta}$, the method used here to construct the weak solution."},{"cited_title":"Regularity results for a class of nonlinear fractional Laplacian and singular problems","cited_arxiv_id":null,"evidence_quote":"Gives the fractional-Laplacian estimates for powers of the distance function used in the boundary barriers."},{"cited_title":"Interior and boundary regularity results for strongly nonhomogeneous p,q -fractional problems","cited_arxiv_id":null,"evidence_quote":"Provides estimates for the strongly nonhomogeneous fractional $p,q$-Laplacian applied to distance powers, used in the boundary analysis."},{"cited_title":"Higher Hölder regu larity for mixed local and nonlocal degenerate elliptic equations","cited_arxiv_id":null,"evidence_quote":"Supplies the finite-difference iteration used to obtain higher local Hölder regularity of solutions to mixed quasilinear equations."}],"review_version":1}