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Equivalent Characterizations and Their Applications of Solvability of $L^p$ Poisson--Robin(-Regularity) Problems on Rough Domains

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Pith's one-line read On bounded one-sided chord arc domains, the solvability of the $L^p$ Poisson–Robin problem is equivalent to the solvability of the $L^p$ Poisson–Robin-regularity problem, and the two sit in a chain with the classical $L^p$ Robin and…

desk verdict Genuinely new Robin analogues of the Poisson-Dirichlet/Neumann equivalence theory, but the key Green's function estimate as printed is dimensionally wrong and comes from an unpublished sibling paper, so read it as a strong conditional. read the letter →

arxiv 2507.11103 v1 pith:GOEE3ATF submitted 2025-07-15 math.AP math.CAmath.FA

Xuelian Fu , Dachun Yang , Sibei Yang This is my paper · ORCID
classification math.APmath.CAmath.FA MSC 35J2535J1535J0542B3542B2535B65
keywords LpRobinproblemPoisson–RobinPoisson–Robin-regularityone-sidedchordarcdomainLipschitzdivergence-formellipticoperatorGreen'sfunctiontentspaces
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

This paper studies inhomogeneous Robin boundary value problems for uniformly elliptic divergence-form operators on bounded one-sided chord arc domains. It tries to establish that the solvability of the $L^p$ Poisson–Robin problem is the same as the solvability of the $L^p$ Poisson–Robin-regularity problem, and that both sit in a chain with the classical homogeneous $L^p$ Robin and $L^{p'}$ Dirichlet problems. The chain is: solvability of the Robin problem plus solvability of the Dirichlet problem implies solvability of the Poisson–Robin-regularity problem, which is equivalent to solvability of the Poisson–Robin problem, which in turn implies solvability of the Robin problem. The paper also proves that the local Robin estimate self-improves, implies the weak Poisson–Robin problem when $n\ge3$, and gives an extrapolation theorem for Robin solvability. The payoff is sharp solvability ranges for the Laplace operator on Lipschitz domains: $p\in(2-\varepsilon_1,\infty)$ for the Poisson–Robin problem and $q\in(1,2+\varepsilon_2)$ for the Poisson–Robin-regularity problem.

What carries the argument

The carrying object is the duality between the modified non-tangential maximal function $\widetilde{N}$ (a boundary maximal function built from cones and local $L^2$ averages) and the averaged Carleson functional $\widetilde{C}_1$ (the corresponding measurement of the data), stated in Lemma 2.3. This duality converts the desired estimates on $\widetilde{N}(u)$ or $\widetilde{N}(\nabla u)$ into pairings with compactly supported test functions, which are then evaluated through the Green's function representation of Robin solutions. The specifically Robin ingredients are Caccioppoli's inequality, the Moser estimate, the boundary Hölder estimate, and the pointwise and local bounds on the Robin Green's function (Lemma 2.9); Proposition 4.1 (equivalence of the Robin problem with its enhanced version) and a good-$\lambda$ argument close the implication chain.

What would settle it

Construct, for $n\ge3$, a bounded one-sided chord arc domain and a uniformly elliptic operator for which the local Robin estimate $(LocR_p)_L$ holds but the weak Poisson–Robin problem $(wPR_{p'})_{L^*}$ fails; Theorem 1.15(iii) asserts this cannot happen, so such an example would settle the central equivalence chain negatively. A more direct test is to check whether the Robin Green's function on any such domain obeys the decay in Lemma 2.9(v), since that estimate is the point where the proof of Lemma 3.3 could break.

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Extended reading notes

Core claim

On a bounded one-sided chord arc domain with a uniformly elliptic divergence-form operator $L:=-\operatorname{div}(A\nabla\cdot)$ and Robin coefficient $\alpha$ as in (1.3), the paper's central assertion is Theorem 1.12: for any $p\in(1,\infty)$, solvability of the $L^{p'}$ Poisson–Robin problem for the adjoint $L^*$ is equivalent to solvability of the $L^p$ Poisson–Robin-regularity problem for $L$, and also to the same regularity problem with data $F=0$; the same holds for the weak versions. Together with Theorem 1.17 this yields the chain $(R_p)_L+(D_{p'})_{L^*}\Rightarrow (PRR_p)_L\Leftrightarrow (PR_{p'})_{L^*}\Rightarrow (R_p)_L$. For the Laplacian on bounded Lipschitz domains the solvable ranges are $p\in(2-\varepsilon_1,\infty)$ for the Poisson–Robin problem and $q\in(1,2+\varepsilon_2)$ for the Poisson–Robin-regularity problem, and the paper proves these ranges are sharp.

Load-bearing premise

The load-bearing premise is that the Robin Green's function on every bounded one-sided chord arc domain satisfies the pointwise bound (2.12) and the local decay estimates of Lemma 2.9(v); these bounds are imported from references [10] and [50] and are used essentially to control the non-local part of the solution in Lemma 3.3, so if they failed on some such domain the weak Poisson–Robin solvability result and the Lipschitz applications would not follow.

Editorial extensions

If this is right

  • To solve the $L^p$ Poisson–Robin problem on a bounded one-sided chord arc domain, it is enough to solve the $L^p$ Poisson–Robin-regularity problem (or its $F=0$ case) for the adjoint operator, and conversely.
  • Whenever the homogeneous Robin problem $(R_p)_L$ and the homogeneous Dirichlet problem $(D_{p'})_{L^*}$ are solvable, the Poisson–Robin-regularity problem $(PRR_p)_L$ and the Poisson–Robin problem $(PR_{p'})_{L^*}$ become solvable automatically.
  • Solvability of the Poisson–Robin problem is monotone in $p$: if $(PR_p)_L$ holds, then $(PR_q)_L$ holds for every $q\in[p,\infty)$.
  • Under the additional assumption that the Dirichlet problem for the adjoint is solvable, the Robin problem $(R_p)_L$ extrapolates upward to $q\in(1,p+\varepsilon)$ for some $\varepsilon>0$.
  • For the Laplacian on bounded Lipschitz domains, the ranges $p\in(2-\varepsilon_1,\infty)$ and $q\in(1,2+\varepsilon_2)$ are optimal; the equivalence chain transfers the known sharp Robin and Dirichlet ranges to the inhomogeneous problems.

Reading between the lines

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

  • Editorial inference: the equivalence chain implies that on domains where the Dirichlet problem is already solved on a large $p$-range, the entire Poisson–Robin theory inherits that range for free; the new content of the paper is really the Robin-to-enhanced-Robin equivalence and the local Robin estimate.
  • Editorial inference: the proof leans on Green's function estimates taken from a submitted companion paper, so the unconditional status of the theorems depends on those estimates; if a bounded one-sided chord arc domain violated Lemma 2.9(v), the weak Poisson–Robin step would need a different argument even though the main equivalence might survive.
  • Editorial inference: a natural stress test would be to check whether the equivalence persists for Robin coefficients $\alpha$ outside $L^p$, where the local estimate (1.9) and the boundary Hölder estimate may degenerate.
  • Editorial inference: for the Laplacian on Lipschitz domains, the sharpness argument transfers the known sharp Robin range upward; if the Robin range on some Lipschitz domain were larger than $(1,2+\varepsilon)$, the Poisson–Robin ranges in Theorem 5.2 would widen correspondingly.
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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

3 major / 5 minor

Summary. The paper studies weak L^p Poisson–Robin and Poisson–Robin-regularity problems for uniformly elliptic divergence-form operators on bounded one-sided chord arc domains. The main results are: (i) Theorem 1.12, an equivalence between solvability of the Poisson–Robin problem for the adjoint operator at exponent p' and the Poisson–Robin-regularity problem at p (including the case F=0), with a weak-form analogue; (ii) Theorem 1.15, relating the local property (LocR_p)_L to the weak Poisson–Robin problem; (iii) Theorem 1.17, giving the chain (R_p)_L + (D_{p'})_{L*} ⇒ (PRR_p)_L ⇔ (PR_{p'})_{L*} ⇒ (R_p)_L, together with an extrapolation result; and (iv) applications to the Laplacian on bounded Lipschitz domains yielding sharp ranges p∈(2−ε_1,∞) for the Poisson–Robin problem and q∈(1,2+ε_2) for the Poisson–Robin-regularity problem. The proofs use duality between the modified non-tangential maximal function and the averaged Carleson functional, Green's function estimates, and techniques adapted from Mourgoglou–Poggi–Tolsa and Feneuil–Li.

Significance. If the missing foundational estimates are supplied, this is a substantial contribution. It provides the first systematic Robin analogue of the Poisson–Dirichlet and Poisson–Neumann solvability equivalences, and the chain (1.11) is a clean and useful formulation. The paper is honest about the differences from the Neumann case (Remark 1.18(ii)), and the extrapolation and sharpness statements for Lipschitz domains are valuable. The equivalence proofs are based on genuine duality arguments rather than tautologies, and no parameter-fitting is present. The main weakness is the dependence on unpublished or submitted sources for the Robin Green's function estimates and for several technical bounds used in Lemma 3.3.

major comments (3)
  1. [Lemma 2.9(v), used in Lemma 3.3 (3.38)] The printed estimate [⨍_{B∩Ω} |∇_x G_R(x,y)| dx]^{1/2} ≤ C r^{1-n} is dimensionally inconsistent: squaring would give an L1 average bounded by C r^{2-2n}, whereas the pointwise gradient bound |∇G| ≲ |x-y|^{1-n} yields the plain L1 average bound ⨍_{B∩Ω} |∇G| dx ≤ C r^{1-n}. More importantly, the proof of Lemma 3.3 at (3.38) uses exactly the plain L1 average bound for |∇_x G_R| + |G_R|, not the square-root version. The proof of Lemma 2.9(v) is a single sentence citing [10,50], and [50] is a submitted manuscript by two of the present authors. Since Theorem 1.15(iii), the equivalence Theorem 1.12, and the Lipschitz applications depend on this estimate, the statement must be corrected (either to the L1 average form or to the L2-average form with exponent 1−n) and proved in the present paper or cited to a published source.
  2. [Lemma 3.3, equations (3.31)–(3.38)] The proof of Lemma 3.3 imports several key technical bounds verbatim from the unpublished preprint [18] (the cone inclusion on [18, p. 34], the covering estimate (3.35), and the maximal-function bound for eC_1^{5/8}(δF 1_{B_k}) on [18, p. 35]) without proofs. Because [18] is not yet peer-reviewed and the present paper's central equivalence relies on these bounds, the argument is conditional. Please either include self-contained proofs of the imported estimates or cite a published version of [18].
  3. [Section 5, Proposition 5.1 and Theorem 5.2] The statements in Section 5 assume n≥3, whereas the abstract and the main theorems (Theorems 1.12, 1.15, 1.17) are stated for n≥2. The restriction to n≥3 in the Lipschitz applications is not explained, and the abstract's unqualified claim for n≥2 is misleading. Please clarify whether the applications are only for n≥3 and, if so, state this restriction in the abstract.
minor comments (5)
  1. [Proof of Theorem 1.12(a)⇒(b), displayed problem] The displayed Robin problem for u contains a boundary condition ∂u/∂ν + αu = f + F·ν, but no f has been introduced in this context; it should be F·ν (or the sentence should define f=0).
  2. [Lemma 2.9(v)] If the intended estimate is (⨍_{B∩Ω} |∇G|^2 dx)^{1/2} ≤ C r^{1-n}, please state it with the square inside the average; the current formula with the outer square root on an L1 average is ambiguous and does not match the use in (3.38).
  3. [Remark 5.3] The sharpness deduction for the Poisson–Robin range (2−ε,∞) from the sharpness of the Robin range (1,2+ε) is stated in one sentence; please spell out the contrapositive using Theorems 1.12(i) and 1.17(i) so the reader can verify the implication.
  4. [Throughout] There are several typos, including 'Propostion' for 'Proposition', 'H ¨older' with an unwanted space, and a stray '+' in the first line of the display in the proof of Lemma 2.5 before (2.3).
  5. [Equation (3.38)] The notation eC_1^{5/8} is used without definition; please define it as the averaged Carleson functional with parameter c=5/8, or refer to the definition in Section 2.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction found; the main equivalence chain is proved by duality and good-lambda arguments, while the reliance on the submitted same-group manuscript [50] and the dimensionally suspect printed form of Lemma 2.9(v) are verifiability and correctness concerns, not circularity.

full rationale

The central equivalences in Theorem 1.12 are proved directly by pairings with the duality estimate (2.1), the density of L∞_c in tent spaces, and the assumed solvability of one of the problems; the implication (c)=> (a) uses (PRR_p)_L exactly as the hypothesis and derives (PR_{p'})_{L*} by duality, while (a)=> (b) uses (PR_{p'})_{L*} as the hypothesis and reduces the estimate on ∇u to an estimate on the dual solution v. These are standard reduction arguments, not a renaming or a fitted-input prediction. Theorem 1.17 is proved by constructing the boundary datum g_D and using the already-cited Poisson-Dirichlet-regularity equivalence from [42], together with the solvability assumptions (R_p)_L and (D_{p'})_{L*}; again the conclusions are not identical to the hypotheses by construction. The only same-group citation is [50], a submitted manuscript by J. Wang, D. Yang, and S. Yang, used in Lemma 2.9 for existence, representation, and pointwise bounds of the Robin Green function. However, the paper itself attributes the same facts also to independent sources: 'The existence of Green's function G_R, (i), (ii), and (2.10) was obtained in [10, Theorem 5.6] and [50, Theorem 1.5]' and '(iv) ... essentially obtained in [5, Theorem 4.1] (see also [50, Theorem 1.5])'. Thus the self-citation is not the sole load-bearing support, and the main results of the paper—Theorems 1.12, 1.17, and the Lipschitz application Theorem 5.2—do not depend on Theorem 1.15(iii) or on Lemma 2.9(v). I also note a serious non-circularity concern: Lemma 2.9(v) is printed as '(∫_{B∩Ω} |∇_x G_R(x,y)| dx)^{1/2} ≤ C r^{1-n}', which is dimensionally inconsistent for the usual Green function gradient; the proof of Lemma 3.3 appears to need an L2-average gradient bound. This is a correctness/typo issue, not a circularity, because it does not make the derived claim equivalent to its input by construction. Overall, no step reduces a 'prediction' to a fitted parameter or a self-citation to the claimed result.

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

No fitted parameters. The proof is built on imported theorems; the principal sources of epistemic burden are [50] (submitted, same group) and [18] (preprint).

assumptions (7)
  • domain assumption Existence and bounds for the Robin Green's function on one-sided chord arc domains (Lemma 2.9), including pointwise bound (2.12) and the local L2/L1 decay estimates for n>=3.
    Load-bearing in Lemma 3.3; cited to [10, Theorem 5.6] and [50, Theorem 1.5], with [50] a submitted manuscript by two of the present authors.
  • ad hoc to paper Feneuil-Li technical estimates for the Poisson-Neumann problem, used in Lemma 3.3 (estimates on pp.31, 34, 35 of [18]).
    [18] is an arXiv preprint and these estimates are not proved in the present paper.
  • standard math Duality between the modified non-tangential maximal function and the averaged Carleson functional (Lemma 2.3), from [6,18,42].
    Basis for the equivalence proofs in Theorem 1.12.
  • standard math Varopoulos extension theorem with eN/eC1 bounds as in (4.11), from [45].
    Published in Adv. Math.; used in Lemma 4.3 to construct extensions for duality.
  • domain assumption Lanzani-Shen solvability and sharpness of the Robin problem for Laplace on Lipschitz domains (Proposition 5.1 and Remark 5.3, [41]).
    Provides the base ranges and sharpness used in Theorem 5.2.
  • domain assumption Equivalence of Poisson-Dirichlet regularity solvability and D_{p'} solvability, used in Lemma 4.3 via [42, Theorem 1.22].
    A published/forthcoming theorem of Mourgoglou, Poggi and Tolsa.
  • domain assumption Boundary Holder regularity for Robin weak solutions (Lemma 2.8), from [48] and [10, Theorem 4.5].
    Used in Lemma 3.3 and the local property arguments.

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Pith. "Pith review of Equivalent Characterizations and Their Applications of Solvability of $L^p$ Poisson--Robin(-Regularity) Problems on Rough Domains." pith.science (2026). https://pith.science/paper/GOEE3ATF

@misc{pith2026250711103,
  author       = {Pith},
  title        = {Pith review of: Equivalent Characterizations and Their Applications of Solvability of $L^p$ Poisson--Robin(-Regularity) Problems on Rough Domains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GOEE3ATF}},
  note         = {Machine review of arXiv:2507.11103}
}
abstract

Let $n\ge2$, $\Omega\subset\mathbb{R}^n$ be a bounded one-sided chord arc domain, and $p\in(1,\infty)$. In this article, we study the (weak) $L^p$ Poisson--Robin(-regularity) problem for a uniformly elliptic operator $L:=-\mathrm{div}(A\nabla\cdot)$ of divergence form on $\Omega$, which considers weak solutions to the equation $Lu=h-\mathrm{div}\boldsymbol{F}$ in $\Omega$ with the Robin boundary condition $A\nabla u\cdot\boldsymbol{\nu}+\alpha u=\boldsymbol{F}\cdot\boldsymbol{\nu}$ on the boundary $\partial\Omega$ for functions $h$ and $\boldsymbol{F}$ in some tent spaces. Precisely, we establish several equivalent characterizations of the solvability of the (weak) $L^p$ Poisson--Robin(-regularity) problem and clarify the relationship between the $L^p$ Poisson--Robin(-regularity) problem and the classical $L^p$ Robin problem. Moreover, we also give an extrapolation property for the solvability of the classical $L^p$ Robin problem. As applications, we further prove that, for the Laplace operator $-\Delta$ on the bounded Lipschitz domain $\Omega$, the $L^p$ Poisson--Robin and the $L^q$ Poisson--Robin-regularity problems are respectively solvable for $p\in(2-\varepsilon_1,\infty)$ and $q\in(1,2+\varepsilon_2)$, where $\varepsilon_1\in(0,1]$ and $\varepsilon_2\in(0,\infty)$ are constants depending only on $n$ and the Lipschitz constant of $\Omega$ and, moreover, these ranges $(2-\varepsilon_1,\infty)$ of $p$ and $(1,2+\varepsilon_2)$ of $q$ are sharp. The main results in this article are the analogues of the corresponding results of both the $L^p$ Poisson--Dirichlet(-regularity) problem, established by M. Mourgoglou, B. Poggi and X. Tolsa [J. Eur. Math. Soc. 2025], and of the $L^p$ Poisson--Neumann(-regularity) problem, established by J. Feneuil and L. Li [arXiv: 2406.16735], in the Robin case.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Solvability of the Poisson-Dirichlet problem in domains with boundaries of mixed dimension

    math.AP 2026-08 conditional novelty 6.0 of 10

    For degenerate elliptic operators on domains with mixed-dimensional boundaries, L^p solvability of the Dirichlet problem is shown equivalent to Poisson-Dirichlet and Poisson-regularity solvability.

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