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Interior and Boundary Regularity of Mixed Local Nonlocal Problem with Singular Data and Its Applications

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2411.18505 v2 pith:4LLHLA7I submitted 2024-11-27 math.AP

R. Dhanya , Jacques Giacomoni , Ritabrata Jana This is my paper · ORCID
classification math.AP MSC 35J6035J7535B65
keywords singularellipticequationsmixedlocal-nonlocaloperatorsfractionalq-Laplacianp-LaplacianHölderregularityboundarystrongcomparisonprinciplenonlinearity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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 $\lambda$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

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.

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 (2)
  1. [§3.3.1, Theorem 3.8 and Eq. (3.20)] 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.
  2. [§4.2, proof of Theorem 2.5, final paragraph] 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.
minor comments (3)
  1. [§3.3.1, Theorem 3.8 statement] The displayed condition 'for every µ ∈ ( qs/(q−1), 1, )' is malformed and should be corrected; the intended interval is (qs/(q−1), 1).
  2. [§4 and §7] 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.
  3. [Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central regularity theorems are proved by direct estimates, and the self-cited technical lemmas are independent computational tools whose hypotheses do not include the target conclusions.

full rationale

This paper does not exhibit circular reasoning. The central regularity claims (Theorems 2.2, 2.3, 2.5, and 2.6) are established through direct Caccioppoli estimates, De Giorgi-type iteration, explicit barrier constructions with powers of the distance function, and a perturbative comparison with frozen-coefficient p-harmonic problems. The only self-citations are technical lemmas imported from the authors' previous work: [35, Eqn (5.23)] and [37, Theorem 3.11] for the fractional q-Laplacian of d^alpha, and [34] for comparison arguments. These are parameter-free computations whose stated hypotheses (such as alpha > qs/(q-1)) do not include the boundary regularity conclusion; they are not fitted to the target solution and do not redefine the target. The main interior C^{1,theta} input is the external result [20, Theorem 6.1]. No fitted parameter is renamed as a prediction, no known pattern is presented under new coordinates, and no uniqueness theorem is imported from same-author work to force the choice of barrier exponents. A possible internal gap noted by a skeptical reader -- the barrier requires an alpha with qs/(q-1) < alpha < 1, which is empty when q <= 1/(1-s) -- would be a correctness failure in the proof as written, not a circularity, because the claimed conclusion is not assumed through those lemmas. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claims rest on: the standing assumption p > sq; smoothness of Ω and the weight Kγ; imported regularity and comparison results from [2], [20], and [35]; and the additional restrictive assumption q > 1/(1-s) for the strong comparison principle. There are no free parameters fitted to data and no newly postulated physical entities.

assumptions (7)
  • domain assumption p > sq
    Standing assumption of the paper (stated after the Notations section) that ensures the local operator dominates the nonlocal part.
  • domain assumption Ω is a bounded open set with smooth boundary and Kγ is comparable to d^{-γ} in Ω
    Used throughout for boundary regularity, barrier constructions, and the distance function.
  • domain assumption Hopf lemma and global gradient regularity for the mixed local-nonlocal operator hold for the class of solutions considered, as established in [2, Theorems 1.1 and 1.2]
    Boundary lower bounds and the perturbative boundary gradient estimates in Theorems 2.3, 2.5, and 2.6 depend on these results.
  • domain assumption The fractional q-Laplacian of the distance power d^α is bounded in L∞ near the boundary when α > qs/(q-1), from [35, Eqn (5.23)]
    Barrier constructions in Section 3.3.1 and the proof of Theorem 2.5 rely on this estimate.
  • domain assumption The interior C^{1,α} regularity theorem of De Filippis-Mingione [20, Theorem 6] applies to the local solutions of the mixed operator, quoted as Theorem 3.7
    Used to upgrade local boundedness to interior C^{1,α} regularity in the proof of Theorem 2.2.
  • domain assumption Weak solutions satisfy the integrability and positivity conditions of Definition 2.4 (u ∈ X(Ω) for local solutions, or u^θ ∈ W^{1,p}_0 ∩ W^{s,q}_0 for some θ ≥ 1 and ess inf_K u > 0 for every K compact)
    The regularity results are only asserted for solutions satisfying these conditions.
  • ad hoc to paper q > 1/(1-s) for Theorem 2.8
    The strong comparison principle and Lemma 5.4 require this condition, which is more restrictive than the standing assumption p > sq.

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Cite this review

Pith. "Pith review of Interior and Boundary Regularity of Mixed Local Nonlocal Problem with Singular Data and Its Applications." pith.science (2026). https://pith.science/paper/4LLHLA7I

@misc{pith2026241118505,
  author       = {Pith},
  title        = {Pith review of: Interior and Boundary Regularity of Mixed Local Nonlocal Problem with Singular Data and Its Applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4LLHLA7I}},
  note         = {Machine review of arXiv:2411.18505}
}
abstract

In this article, we examine the H\"older regularity of solutions to equations involving a mixed local-nonlocal nonlinear nonhomogeneous operator $\fp + \fqs$ with singular data, under the minimal assumption that $p> sq$. The regularity result is twofold: we establish interior gradient H\"older regularity for locally bounded data and boundary regularity for singular data. We prove both boundary H\"older and boundary gradient H\"older regularity depending on the degree of singularity. Additionally, we establish a strong comparison principle for this class of problems, which holds independent significance. As the applications of these qualitative results, we further study sublinear and subcritical perturbations of singular nonlinearity.

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