{"id":"b5d114a6-9eb6-43cb-b73e-a6dd3cd010af","arxiv_id":"2411.18505","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under p > sq, weak solutions of the mixed local-nonlocal problem with singular data are C^{1,α} up to the boundary in the weakly singular case and C^α in the strongly singular case.","lead":"This paper proves Hölder and gradient Hölder regularity up to the boundary for solutions of mixed local-nonlocal equations with singular right-hand sides, requiring only that the local exponent p exceed the product of the nonlocal exponent q and the fractional order s. These regularity results are foundational for existence, uniqueness, and multiplicity of solutions to singular nonlinear PDEs, and they close a parameter range left open in recent work.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The boundary barrier in Theorems 3.8, 2.3, and 2.5 requires α ∈ (qs/(q−1), 1), an interval that is empty unless q > 1/(1−s); the stated assumption p > sq does not prevent this failure.","rationale":"The reader identified the reliance on the preprint [2] and the barrier computation [35, Eqn (5.23)] as the weakest assumption. My reading agrees that these imports are load-bearing, but the more precise and more serious problem is that the quoted barrier computation is applied in a parameter regime where its hypothesis cannot be met. The proof of Theorem 3.8 needs α ∈ (qs/(q−1), 1); this interval is nonempty exactly when q > 1/(1−s). The paper states Theorems 2.3 and 2.5 under only p > sq, and p > sq does not imply q > 1/(1−s). For instance, q = 2, s = 1/2 satisfies p > sq for any p > 1, and the hypotheses of Theorem 2.5 can hold, yet the required α would have to satisfy α > 1 and α < 1 simultaneously. Therefore, the proof of the boundary regularity theorems has an internal gap in a large part of the claimed range. This is independent of whether [2] and [35] are correct. The result might be salvageable by adding the assumption q > 1/(1−s) or by constructing a different barrier, but as written the central claim 'minimal assumption p > sq' is not proven. This supports the reader's CONDITIONAL verdict, so I do not change the verdict, but my reason for conditionality is more concrete than the reader's. The concrete test I propose would demonstrate the empty interval and the failure of the L∞ bound for the fractional q-Laplacian of d^α, thereby confirming that the proof cannot be carried out for parameters satisfying the stated hypotheses.","tokens_in":35379,"tokens_out":21893,"duration_ms":184483,"concrete_test":"Restrict to a half-space and fix q = 2, s = 1/2, p = 3/2, γ = 1/2, δ = 1/4. All hypotheses of Theorem 2.5 are satisfied, including γ + (1/q − s)(1 − δ) < 1, yet the proof requires α ∈ (qs/(q−1), 1) = (1, 1), which is empty. To settle the concern directly, compute (-Δ)_2^{1/2}((x_N)_+)^α near x_N = 0 for any α < 1 using the integral representation; show that the integral diverges logarithmically, so the operator is not in L∞ near the boundary. This verifies that the barrier construction in Theorem 3.8 cannot be performed for parameters inside the stated range of Theorem 2.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem 3.8 and again in the proofs of Theorems 2.3 and 2.5, the upper bound u(x) ≤ C d(x)^α is obtained by testing the barrier ωρ = Γ d_e^α against the singular data. The proof needs two incompatible conditions on α: (i) α > qs/(q−1) so that, by [35, Eqn (5.23)], the fractional q-Laplacian of d^α lies in L∞ near the boundary; and (ii) α < 1 so that −Δ_p d^α has the positive sign needed for a supersolution and the exponent αp−α−p < −γ controls the singular term. Such an α exists only if qs/(q−1) < 1, that is, q > 1/(1−s). The paper never assumes this. For example, q = 2, s = 1/2 (with p > 1 and p > sq) gives qs/(q−1) = 1, so the interval (1,1) is empty; for q = 2, s = 0.8 the required α would be larger than 1 as well. Thus, for a substantial part of the claimed range p > sq, the barrier argument collapses. This is an internal gap, not merely a dependence on the unpublished preprint [2]: even granting [2] and [35] in full, the computation quoted from [35] cannot be applied because its hypothesis α > qs/(q−1) is incompatible with α < 1. The theorems may still be true, but they are not established as stated under the minimal assumption p > sq.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the mixed local-nonlocal quasilinear problem -Δ_p u + (-Δ)_q^s u = K_γ(x) u^{-δ} + F(x,u) with positive singular data and zero Dirichlet condition. It claims interior C^{1,θ}_{loc} regularity for locally bounded data, boundary C^{1,α} regularity when γ+δ<1, boundary C^α regularity when γ+δ>1, a strong comparison principle, and two applications (sublinear and subcritical perturbations). The proofs use Caccioppoli estimates, De Giorgi iteration, barrier functions, and a perturbative comparison with solutions of frozen-coefficient problems, following the strategy of De Filippis–Mingione and Antonini–Cozzi.","tokens_in":35762,"tokens_out":13360,"duration_ms":118580,"significance":"If the boundary regularity results hold in the advertised range sq<p<q, they would be a valuable step for singular mixed local-nonlocal problems, with consequences for comparison principles and existence theory. The paper develops useful technical machinery, including detailed interior estimates and a Campanato-style boundary gradient estimate. However, the central boundary barrier requires an additional condition on q,s that is not assumed, so the main advertised range is not established as stated. The strong comparison principle and local regularity parts appear more robust.","major_comments":[{"comment":"The proof constructs a barrier ω_ρ = (d_e)^α_+ and requires α ∈ (qs/(q−1), 1). Such an α exists only when qs/(q−1) < 1, i.e. q > 1/(1−s). The manuscript assumes only p > sq (Abstract and Section 1). For q=2, s=1/2 and p∈(1,2), one has p>sq but qs/(q−1)=1, so the admissible interval is empty. Thus even granting [35, Eqn (5.23)] and [2] in full, the two required inequalities α>qs/(q−1) and α<1 are incompatible. This is an internal gap, not merely a dependence on an unpublished preprint. Consequently Theorem 3.8(a), and with it Theorems 2.3 and 2.5 (and Theorem 2.10 via Theorem 2.5), are not proved under the stated minimal assumption p>sq. The statements must either add q>1/(1−s) or be supplied with a different boundary barrier.","section":"§3.3.1, Theorem 3.8 and Eq. (3.20)"},{"comment":"The lower bound C d(x) ≤ u is obtained by invoking Theorem 5.1, but Theorem 5.1 assumes γ ≤ min{1+s−1/q, 2−1/p}. This condition is not implied by the hypotheses of Theorem 2.5. For example, with s=0.2, q=2, δ=0.1 and γ=0.71, one has γ+δ=0.81<1 and γ+(1/q−s)(1−δ)=0.98<1, but 1+s−1/q=0.7<γ. The proof therefore needs either a comparison principle valid under the stated assumptions or an additional hypothesis guaranteeing the applicability of Theorem 5.1.","section":"§4.2, proof of Theorem 2.5, final paragraph"}],"minor_comments":[{"comment":"The displayed condition 'for every µ ∈ ( qs/(q−1), 1, )' is malformed and should be corrected; the intended interval is (qs/(q−1), 1).","section":"§3.3.1, Theorem 3.8 statement"},{"comment":"Lemma 4.2, Theorem 4.1 and Lemma 7.3 are only sketched or deferred to 'standard' arguments. Since these results justify the approximation u_ε and the existence steps in the applications, the authors should either give complete proofs or cite precise published statements.","section":"§4 and §7"},{"comment":"There are numerous typographical issues (e.g. 'T o', 'for for', inconsistent use of N and n in the fractional exponent, and duplicated definitions of d_e). A careful proofreading is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's advertised range 'minimal assumption p>sq' is overstated because the boundary barrier needs q>1/(1−s). This is a substantive but apparently fixable issue if the authors restrict the boundary theorems accordingly or find a genuinely different barrier. The paper also relies heavily on the unpublished preprint [2] and on [35]; the editor may wish to ensure those quoted results are verifiable. The local regularity and strong comparison principle portions are likely salvageable, so I do not recommend rejection at this stage."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is the first paper to claim boundary Hölder and gradient Hölder regularity for mixed local-nonlocal operators with singular data in the hard range sq < p < q, and it backs that with a genuine De Giorgi-based local boundedness result for p<q. Second, the main boundary barrier has an unstated compatibility condition: the proofs of Theorems 3.8 and 2.5 need an exponent α with qs/(q−1) < α < 1, which only exists when q > 1/(1−s). The paper assumes only p > sq throughout and never states q > 1/(1−s). For q=2, s=1/2 the interval is empty; for many other admissible parameters it is too. This is not merely a dependency on the unpublished Antonini–Cozzi preprint; even granting that preprint and the quoted [35, Eqn (5.23)] in full, the hypothesis of the quoted computation is incompatible with α < 1.\n\nWhat is genuinely good: the local boundedness theorem (Theorem 3.5) for p<q is proved by a careful Caccioppoli estimate adapted to the nonhomogeneous operator, and that part is new and looks sound. The interior C^{1,α} then follows from De Filippis–Mingione. The strong comparison principle (Theorem 2.8) is proved with a clean perturbative argument and is a useful tool. The applications sections are standard but complete enough.\n\nSoft spots: the gap above is load-bearing for the paper's main claims, so the theorems as stated are not established in the full range p>sq. A likely fix is to add the assumption q > 1/(1−s) to Theorems 2.3, 3.8, and 2.5, and to note that the strongly singular case (γ+δ>1) does not require it, since the proof there uses a different estimate from [37]. There is also heavy reliance on the unpublished preprint [2] for the Hopf lemma and global gradient estimates; that is a risk if the preprint changes. Several auxiliary lemmas (4.1, 4.2, 7.3) are only sketched, which is acceptable but should be fleshed out.\n\nWho should read this: people working on mixed local-nonlocal regularity and singular problems. It is worth a serious referee, but the referee should push on the range of validity of the boundary theorems.","headline":"Boundary regularity for singular data in the hard range sq<p<q is new and worth engaging, but the main barrier argument silently requires q > 1/(1−s), so the stated range is not proved.","tokens_in":36254,"tokens_out":5850,"would_cite":false,"duration_ms":49219,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J60","35J75","35B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under the minimal condition $p>sq$, solutions to the singular mixed local-nonlocal equation are $C^{1,\\alpha}$ up to the boundary when the singularity is mild, and $C^\\alpha$ when it is strong.","keywords":["singular elliptic equations","mixed local-nonlocal operators","fractional q-Laplacian","p-Laplacian","Hölder regularity","boundary regularity","strong comparison principle","singular nonlinearity"],"falsifier":"Take a half-ball and $u=d^\\alpha$ with $\\alpha=qs/(q-1)$; compute $(-\\Delta)_q^s u$ near the flat boundary. If it is bounded instead of diverging like a negative power of $d$, the barrier threshold in Theorem 3.8 is not necessary. Alternatively, numerically solve (1.1) in a disk with mild singularity and check whether the normal derivative at the boundary is continuous; a failure would disprove $C^{1,\\alpha}$ boundary regularity in that range.","tokens_in":35192,"feed_emoji":"📐","tokens_out":7998,"duration_ms":69658,"temperature":0.7,"pith_summary":"The paper proves regularity up to the boundary for weak solutions of the mixed local-nonlocal equation $-\\Delta_p u+(-\\Delta)_q^s u = K_\\gamma(x) u^{-\\delta}+F(x,u)$ in a bounded smooth domain, with $u=0$ outside the domain and a singular right-hand side. Under the minimal parameter assumption $p>sq$, it shows that for locally bounded data the solution is locally $C^{1,\\theta}$; for singular data, the solution is $C^{1,\\alpha}$ on the closed domain when $\\gamma+\\delta<1$ and $C^\\alpha$ on the closed domain when $\\gamma+\\delta>1$. It also proves a strong comparison principle for the singular problem and uses the regularity to obtain existence and uniqueness for sublinear and subcritical perturbations. This matters because earlier work on singular problems for this operator mostly stopped at Sobolev regularity, while boundary Hölder information is what enables comparison arguments, Hopf-type behavior, and multiplicity analysis.","feed_headline":"Singular mixed PDEs reach C^{1,α} regularity at the boundary","feed_subtitle":"Proof covers the hard range sq<p<q and yields boundary Hölder estimates plus a strong comparison principle.","key_machinery":"The argument is carried by barrier functions built from powers of the distance to the boundary, $d(x)^\\alpha$, with the exponent constrained by the fractional $q$-Laplacian: $(-\\Delta)_q^s d^\\alpha$ stays bounded in $L^\\infty$ near the boundary precisely when $\\alpha>qs/(q-1)$. The barriers give upper bounds $u\\le C d^\\alpha$ and, together with Hopf's lemma for the regular mixed operator, lower bounds $u\\ge C d$. Interior $C^{1,\\theta}_{\\rm loc}$ regularity is obtained from Caccioppoli estimates and De Giorgi iteration, and boundary gradient regularity by flattening the boundary and comparing $u$ with the solution of a constant-coefficient $p$-Laplacian Dirichlet problem on half-balls.","core_discovery":"On the paper's own terms, the central discovery is that the singular mixed operator $-\\Delta_p+(-\\Delta)_q^s$ behaves like its regular counterpart at the boundary: a weak solution with data of order $d(x)^{-\\gamma}$ still has a definite boundary trace of power type, and its gradient is Hölder continuous up to the boundary whenever the singularity is mild ($\\gamma+\\delta<1$). In the strongly singular case $\\gamma+\\delta>1$, the solution itself is Hölder continuous up to the boundary, though the gradient may blow up. The same framework yields a strong comparison principle for $C^{1,\\alpha}$ solutions with singular term, and existence and uniqueness results for perturbed problems. The authors state these are the first boundary regularity results of this kind for singular data in the range $sq<p<q$, where the nonlocal part is not dominated by the local gradient term.","pith_inferences":["The exponent threshold $\\alpha>qs/(q-1)$ is probably not an artifact of the proof: it is the same threshold that makes the fractional $q$-Laplacian of the distance power locally bounded, so the boundary Hölder exponent may be optimal; a test would be explicit radial solutions for $p=q$ in a ball.","The method likely extends to data with more general singular weights than $K_\\gamma(x)u^{-\\delta}$, as long as the barrier can dominate the weight; one can test by replacing $K_\\gamma$ with a weight that oscillates between two distance powers.","The strong comparison principle for the singular nonlinearity is a natural stepping stone to a Hopf-type lemma for the fractional part, which the authors note remains open for singular $(-\\Delta)_p^s$.","The regularity results should allow variational methods, such as Sobolev-versus-Hölder minimizer arguments, for mixed local-nonlocal singular functionals, by analogy with the local $p$-Laplacian theory."],"forward_implications":["If the central claim is correct, singular weak solutions of (1.1) have a definite boundary profile: comparable to the distance function from below and bounded by a power of the distance from above, with the power depending only on $q,s$ and the singularity parameters.","In the mildly singular range $\\gamma+\\delta<1$, the gradient is Hölder continuous up to the boundary, so boundary-value techniques such as Hopf-type arguments and Picone identities apply to the singular problem.","The strong comparison principle implies uniqueness of the solution for sublinear perturbations and gives strict ordering of solutions, which is the tool needed for subcritical superlinear existence by truncation.","Under the same regularity, the problem with $F(x,u)=u^l$ has a unique $C^{1,\\alpha}$ solution for sublinear $l$, and a $C^{1,\\alpha}$ solution for superlinear subcritical $l$ for small $\\lambda$."],"supporting_citations":[{"why":"It supplies the interior $C^{1,\\alpha}$ regularity theory and Caccioppoli estimates for the regular mixed operator that Theorem 2.2 invokes.","marker":"[20]"},{"why":"It supplies global gradient regularity, the Hopf lemma, and weak comparison for the regular mixed operator, used for lower bounds and boundary gradient estimates.","marker":"[2]"},{"why":"It supplies the computation that the fractional $q$-Laplacian of $d^\\alpha$ is bounded in $L^\\infty$ near the boundary when $\\alpha>qs/(q-1)$, which sets the barrier exponent.","marker":"[35]"},{"why":"It supplies Caccioppoli-type estimates and boundary regularity methods for fractional $(p,q)$ problems that are adapted to the mixed operator.","marker":"[37]"},{"why":"It supplies the De Giorgi iteration scheme used to prove local boundedness from the Caccioppoli inequality.","marker":"[14]"},{"why":"It supplies the Sobolev and Hölder regularity framework for singular nonhomogeneous quasilinear problems used as a template for the singular data argument.","marker":"[34]"},{"why":"It supplies the fractional strong comparison estimates used to prove the strong comparison principle for the singular mixed operator.","marker":"[45]"}],"fun_headline_variants":["Boundary C^{1,α} regularity for singular mixed PDEs","Singular data: gradient Hölder up to boundary","Strong comparison principle for singular mixed operators","Boundary Hölder estimates in hard range sq<p<q","Singular local-nonlocal PDEs: boundary Hölder regularity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the previously proved Hopf lemma and global gradient estimates for the same mixed operator without singularity remain valid for the singular weak solutions studied here; if that transfer fails, the boundary regularity argument collapses.","fun_headline_variants_meta":{"raw":{"variants":["Boundary C^{1,α} regularity for singular mixed PDEs","Singular data: gradient Hölder up to boundary","Strong comparison principle for singular mixed operators","Boundary Hölder estimates in hard range sq<p<q","Singular local-nonlocal PDEs: boundary Hölder regularity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001753,"raw_usage":{"total_tokens":6871,"prompt_tokens":841,"completion_tokens":6030,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":457,"completion_tokens_details":{"reasoning_tokens":5948}},"tokens_in":457,"tokens_out":6030,"duration_ms":34976,"temperature":1.0,"reasoning_tokens":5948,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:09:04.294267+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a half-ball and $u=d^\\alpha$ with $\\alpha=qs/(q-1)$; compute $(-\\Delta)_q^s u$ near the flat boundary. If it is bounded instead of diverging like a negative power of $d$, the barrier threshold in Theorem 3.8 is not necessary. Alternatively, numerically solve (1.1) in a disk with mild singularity and check whether the normal derivative at the boundary is continuous; a failure would disprove $C^{1,\\alpha}$ boundary regularity in that range.","supporting_citations":[{"cited_title":"Gradient regularity in mixed local and nonlocal problems","cited_arxiv_id":null,"evidence_quote":"It supplies the interior $C^{1,\\alpha}$ regularity theory and Caccioppoli estimates for the regular mixed operator that Theorem 2.2 invokes."},{"cited_title":"Global gradient regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type","cited_arxiv_id":"2308.06075","evidence_quote":"It supplies global gradient regularity, the Hopf lemma, and weak comparison for the regular mixed operator, used for lower bounds and boundary gradient estimates."},{"cited_title":"Sreenadh","cited_arxiv_id":null,"evidence_quote":"It supplies the computation that the fractional $q$-Laplacian of $d^\\alpha$ is bounded in $L^\\infty$ near the boundary when $\\alpha>qs/(q-1)$, which sets the barrier exponent."},{"cited_title":"Interior and boundary regularity results for strongly nonhomogeneous p, q-fractional problems","cited_arxiv_id":null,"evidence_quote":"It supplies Caccioppoli-type estimates and boundary regularity methods for fractional $(p,q)$ problems that are adapted to the mixed operator."},{"cited_title":"Hölder regula rity for weak solutions to nonlocal double phase prob- lems","cited_arxiv_id":null,"evidence_quote":"It supplies the De Giorgi iteration scheme used to prove local boundedness from the Caccioppoli inequality."},{"cited_title":"Sreenadh","cited_arxiv_id":null,"evidence_quote":"It supplies the Sobolev and Hölder regularity framework for singular nonhomogeneous quasilinear problems used as a template for the singular data argument."},{"cited_title":"Strong comparison principle for the fract ional p-laplacian and applications to starshaped rings","cited_arxiv_id":null,"evidence_quote":"It supplies the fractional strong comparison estimates used to prove the strong comparison principle for the singular mixed operator."}],"review_version":1}