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REVIEW 5 major objections 4 minor 28 references

On the Schr\"odinger equations with $B_\infty$ potentials in the region above a Lipschitz graph

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Pith's one-line read This paper proves sharp $L^p$ regularity, Neumann, and $W^{1,p}$ estimates for generalized Schrödinger operators $\operatorname{div}(A\nabla)+V$ in Lipschitz graph domains when $A$ is symmetric, independent of the vertical variable, and…

desk verdict Plausible and interesting extension of Shen's Schrödinger BVP results to variable coefficients, but the p>2 regularity proof has an unproved L2 solvability step on truncated domains. read the letter →

arxiv 2411.18458 v1 pith:RYLBKX4U submitted 2024-11-27 math.AP

Jun Geng , Ziyi Xu This is my paper · ORCID
classification math.AP MSC 35J1035J2535B45
keywords SchrödingeroperatorLipschitzgraphLpboundaryvalueproblemregularityNeumannW^{1p}estimateFefferman-Phong-ShenmaximalfunctionB∞potential
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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 aims to prove that the generalized Schrödinger operator $-\operatorname{div}(A\nabla)+V$ in the region above a Lipschitz graph has the same sharp $L^p$ boundary regularity theory as the classical Laplacian, despite rough coefficients and a large positive potential. Specifically, it claims unique solvability of the $L^p$ regularity problem for $1

What carries the argument

The argument is carried by three interacting pieces. The Fefferman-Phong-Shen maximal function $m(x,V)=\inf\{1/r:\psi(x,r)\le 1\}$ with $\psi(x,r)=r^{2-d}\int_{B(x,r)}V\,dy$ encodes the scale at which the potential is critical and appears as a weight in every boundary estimate. The Rellich identity (4.2), valid because $A$ is symmetric and $x_d$-independent, yields the boundary comparability $\int_{\partial\Omega}|\partial u/\partial\nu|^2\,d\sigma \sim \int_{\partial\Omega}(|\nabla_{\tan}u|^2+|u|^2m(x,V)^2)\,d\sigma$, which is the $L^2$ core of the proof. The De Giorgi-Nash type Hölder estimate (Theorem 3.1), obtained by a perturbation argument using fundamental solution bounds with decay $(1+|x-y|m(x,V))^{-k}$, supplies the pointwise control that feeds the weak reverse Hölder inequality and the passage from $L^2$ to the full $p$ ranges.

What would settle it

A concrete check is to solve the regularity problem for $A=I$, $V\equiv 1$ on a wedge-like Lipschitz graph domain with compactly supported boundary data and compare $\|(\nabla u)^*\|_{L^p(\partial\Omega)}$ with $\|\nabla_{\tan}g\|_{L^p}+\|g\|_{L^p}$ for $p$ ranging through $(1,2+\varepsilon)$; a single failure inside the claimed range refutes Theorem 1.3, while uniform success at the endpoint $p=2+\varepsilon$ would show the stated range is not sharp.

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Extended reading notes

Core claim

The central claim is that the $L^p$ regularity problem $Lu+Vu=0$ in $\Omega$, $u=g$ on $\partial\Omega$ is uniquely solvable for $1<p<2+\varepsilon$, with the estimate $\|(\nabla u)^*\|_{L^p(\partial\Omega)}\le C(\|\nabla_{\tan}g\|_{L^p(\partial\Omega)}+\|g\,m(x,V)\|_{L^p(\partial\Omega)})$, under the assumptions that $A$ is elliptic, symmetric, $x_d$-independent and partially Dini continuous and that $0<V\in B_\infty$. The companion $W^{1,p}$ estimate gives $\|\nabla u\|_{L^p(\Omega)}+\|V^{1/2}u\|_{L^p(\Omega)}\le C(\|f\|_{L^p(\Omega)}+\|G\|_{B^{-1/p,p}(\partial\Omega)})$ for $3/2-\varepsilon<p<3+\varepsilon$ for the Neumann problem, and the $L^p$ Neumann problem is solvable for $1<p<2+\varepsilon$. All these ranges are sharp, and the same method yields the Dirichlet version of the $W^{1,p}$ estimate described in Remark 1.2.

Load-bearing premise

The load-bearing premise is that the coefficient matrix $A$ is symmetric and independent of the vertical variable $x_d$, because the Rellich identity that compares normal and tangential boundary integrals requires the term $\partial_d A$ to vanish; the paper notes in Remark 1.4 that without this condition the global estimates fail for every $p$.

Editorial extensions

If this is right

  • The $L^p$ Neumann problem for $Lu+Vu=0$ is uniquely solvable for $1<p<2+\varepsilon$, improving the previously known $1<p\le 2$ range even for the classical operator $-\Delta+V$.
  • For the classical equation $-\Delta u+Vu=0$ on Lipschitz graph domains, the regularity problem is now solvable in the sharp range $1<p<2+\varepsilon$, not just at $p=2$.
  • The $W^{1,p}$ estimate controls both $\|\nabla u\|_{L^p(\Omega)}$ and the potential-weighted term $\|V^{1/2}u\|_{L^p(\Omega)}$ for $3/2-\varepsilon<p<3+\varepsilon$, with a Dirichlet analogue stated in Remark 1.2.
  • The stated ranges $1<p<2+\varepsilon$ and $3/2-\varepsilon<p<3+\varepsilon$ are sharp, so no wider $p$-range can hold under the same structural assumptions.
  • The weighted boundary condition $\|g\,m(x,V)\|_{L^p}$ shows that large potentials contribute a coercive boundary term, so the admissible data space is naturally tied to the potential through the maximal function.

Reading between the lines

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

  • An implication left implicit is that the same Rellich comparability should yield $L^p$ solvability of the Dirichlet problem for $Lu+Vu=0$ for $2-\varepsilon<p<\infty$ by duality with the regularity problem, mirroring the Laplace theory.
  • Because the proof uses local boundary estimates together with a real-variable argument, the same sharp ranges should transfer to bounded Lipschitz domains by standard localization, with the potential condition understood locally.
  • A testable extension is to replace partial Dini continuity of $A$ by a Hölder modulus and quantify the resulting $\varepsilon$; the scheme suggests $\varepsilon$ should depend only on the modulus, the ellipticity constant, and the Lipschitz constant.
  • As $V\to 0$ the weight term $g\,m(x,V)$ disappears and the estimates reduce to classical tangential-gradient control, indicating that the data space introduced here is a natural potential-dependent analogue of the usual regularity data.
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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

5 major / 4 minor

Summary. The paper studies the generalized Schrödinger operator -div(A∇)+V in the region Ω above a Lipschitz graph, under the assumptions that A is elliptic, symmetric, x_d-independent and partially Dini continuous, and that 0<V lies in the class B∞. It claims three main results: unique solvability of the Lp regularity problem for 1<p<2+ε (Theorem 1.3), a W^{1,p} Neumann estimate for 3/2−ε<p<3+ε (Theorem 1.1), and Lp Neumann solvability for 1<p<2+ε (Theorem 7.1). The proofs combine the Fefferman-Phong-Shen maximal function, boundary L∞ and Hölder estimates, Rellich-type identities, layer potentials, and real-variable arguments in the spirit of Shen's work on -Δ+V.

Significance. If the results are correct, they give the first sharp-range Lp boundary estimates for generalized Schrödinger operators with variable coefficients, extending Shen's classical results for -Δ+V in Lipschitz graph domains and complementing the Kenig-Pipher theory for elliptic operators. The structural assumptions (1.2)-(1.4) are natural, and Remark 1.4 correctly identifies the role of x_d-independence. The paper is not circular: it relies on previously published results rather than assuming the target p-ranges. However, the manuscript is not self-contained at several load-bearing points, and the proofs of key truncated-domain estimates are missing.

major comments (5)
  1. [§5.2, around (5.12)-(5.13)] The proof of Theorem 1.3 for 2<p<2+ε is carried out on the truncated domain Ω_R and invokes 'L2 solvability of the regularity problem' on ∂Ω∩∂Ω_R. However, Section 4 establishes L2 regularity solvability only on the full graph domain Ω (Theorem 4.1 and the layer-potential argument for p=2). No L2 regularity result is proved for Ω_R, whose boundary contains lateral faces {|x'|=R} and a top face. On a lateral face n_d=0, the left side of the Rellich identity (4.2) vanishes, so the comparison (4.1) cannot control the normal-derivative integrals on those faces. This missing estimate is load-bearing because it is exactly the mechanism that extends the range beyond p=2.
  2. [Lemma 4.5, inequalities (4.7)-(4.10)] Lemma 4.5 is stated without proof and is described as 'essentially the same' as Lemmas 2.2, 2.6 and 2.7 in [22] with 'slightly modification'. These four inequalities are used repeatedly in the proof of Theorem 4.1, e.g. in (4.11)-(4.15). Since [22] treats -Δ+V while the present operator has variable coefficients A, the modification is not self-evident; a proof or a precise statement of the corresponding results in [22] under assumptions (1.1)-(1.4) is required before Theorem 4.1 can be accepted.
  3. [§6.1, Lemma 6.3] The proof of Lemma 6.3 refers to 'Lemma 3.1' and 'Lemma 7.1', neither of which exists in the manuscript: Lemma 3.1 is not stated (Theorem 3.1 is, but is not a Lipsehitz-gradient estimate), and Section 7 (the appendix) contains no lemmas at all. Moreover, the line (6.15) invokes an L2 estimate on ∂Z_{tr} for the cylinder Z_{tr}, which is never proved. Since Lemma 6.3 is the key boundary reverse Hölder inequality used in Theorem 6.1, the W^{1,p} estimate for 2<p<3+ε rests on missing statements.
  4. [Appendix, Theorem 7.1] The proof of Theorem 7.1 asserts without proof both the L2 invertibility of 1/2 I + K_A on L2(∂Ω) and the atomic H1_at Neumann estimate (with only references [5,10,27] for Hardy-space background). The advertised by-product, Lp Neumann solvability for 1<p<2+ε, is therefore not established by the manuscript as written. Please provide the kernel estimates and invertibility argument, and prove the H1_at estimate or give a precise reference that covers exactly this operator and boundary condition.
  5. [Abstract and Introduction] The statement 'All the ranges of p are sharp' is asserted in the abstract and again in the introduction, but no proof or reference is provided for sharpness under the specific assumptions (1.1)-(1.4) and (1.5). Remark 1.4 concerns failure of global estimates when x_d-independence is dropped, which is not the same as optimality of 2+ε or 3+ε in the admissible class. Either supply a proof or a precise citation, or rephrase the claim.
minor comments (4)
  1. [§6.1, proof of Theorem 6.2] The sentence 'Integrating (7.1) by parts' refers to equation (7.1) in the appendix, but the intended reference is equation (6.5) or (6.9); please correct the cross-reference.
  2. [§2 and §3] There are several grammatical and typographical issues, e.g. 'referring the reader to for a detailed presentation are list below' and 'We list some properties of A∞ weight, as below'; these should be cleaned up.
  3. [Theorem 3.1] Theorem 3.1 states assumptions (1.1)-(1.2) and (1.5), but its proof uses Proposition 3.4 and the fundamental solution estimates that depend on the partial Dini condition (1.4); the theorem's hypotheses should include (1.4).
  4. [§5.1, definition of H1_{1,at}] The normalization of the atom in the definition of H1_{1,at} in §5.1 is stated as ∥∇tan a∥_{L2(∂Ω)} ≤ |B(P,r)∩∂Ω|^{-1/2}; this is standard, but the text should also specify the support condition precisely, as it is used in the proof of Theorem 5.1.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found: the central estimates are derived from external Rellich, layer-potential and real-variable results; the one self-citation is a method citation, not a load-bearing black box.

full rationale

The paper's main theorems are not obtained by fitting parameters or by importing the target p-ranges as assumptions. The L2 Rellich comparison in Theorem 4.1 is proved from the Rellich identities (4.2) and (4.4), Lemma 4.5 quoted from Shen [22], and the boundary estimates of Section 3; the p=2 solvability is then a layer-potential consequence referred to [20] and [22], not to the present authors. The extensions are obtained by the real-variable Theorem 5.3 quoted from Shen [26] and by atomic-Hardy interpolation, neither of which assumes the theorem being proved. The only self-citation is [8] (Geng 2012), used in the introduction as 'Following the method in [8](see also [25])' for the W^{1,p} reduction; the proof of Theorem 6.2 supplies the full verification of the reverse Hölder condition, so [8] is not a load-bearing self-citation. The significant weakness flagged by the skeptic is a missing justification, not circularity: the estimates (5.7) and (5.10) invoke 'by L2 solvability of the regularity problem' on truncated domains Omega_R or Omega\B(y0, sR), whereas the L2 result established in Section 4 is proved for the full graph domain. That is a genuine proof gap indicating that the p>2 extension may rely on an unproved estimate, but it is not a reduction of the conclusion to its own inputs by construction, and no fitted quantity is renamed as a prediction. Accordingly, the circularity score is low.

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

The proof pulls a large amount of background from Shen [22], [24], [26], Coifman-Weiss [2], Dong [4], and Geng [8]. These are treated as axioms rather than reproved: maximal function properties, Fefferman-Phong inequality, real-variable theorems, layer potential invertibility, and the W^{1,p} real-variable reduction. None of these are fitted or target-dependent. No new entities are introduced. The central claim inherits the correctness of these external results.

assumptions (7)
  • domain assumption A is real, elliptic, symmetric, and x_d-independent (Assumptions (1.1)-(1.3)).
    Used in the Rellich identities in Lemma 4.2 and throughout; x_d-independence makes the operator translation-invariant in the vertical direction, which is crucial for the boundary estimates.
  • domain assumption A is partially Dini continuous in x' with modulus η satisfying ∫_0^t η(ρ)/ρ dρ <∞ (Assumption (1.4)).
    Needed for the De Giorgi-Nash type boundary Hölder estimate (Theorem 3.1) and the fundamental solution estimates (Proposition 3.4), both invoked in Section 3.
  • domain assumption 0<V∈B∞, i.e. ∥V∥_{L∞(B)}≤C ⨏_B V for all balls B (Assumption (1.5)).
    Defines the Fefferman-Phong-Shen maximal function m(x,V), which enters all boundary norms and estimates; V>0 is used in the Moser iteration to control signs.
  • standard math Known maximal function properties: Propositions 2.2-2.4 and Lemma 2.5 from [22] are accepted without proof.
    These results on ψ(x,r), m(x,V) comparability, and the Fefferman-Phong inequality are used throughout Sections 3-7 and are quoted from Shen [22].
  • standard math Classical De Giorgi-Nash and interior Lipschitz estimates for elliptic operators with Dini or Hölder coefficients.
    Used in Theorem 3.1 and in the fundamental solution estimates in Section 3; treated as background.
  • standard math Real-variable theorems: Theorem 5.3 from [26, Theorem 4.2.3] and interpolation Theorem 5.2 from [2] are used as black boxes.
    Used to extend L2 estimates to Lp for 2<p<2+ε and for 1<p<2; their proofs are not included.
  • standard math Layer potential invertibility for L2 solvability is asserted from [20, 22].
    The proof of Theorem 1.3 for p=2 and Theorem 7.1 for p=2 says 'omit the details' and refers to [20] or [22], so the L2 solvability is treated as a black box.

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Pith. "Pith review of On the Schr\"odinger equations with $B_\infty$ potentials in the region above a Lipschitz graph." pith.science (2026). https://pith.science/paper/RYLBKX4U

@misc{pith2026241118458,
  author       = {Pith},
  title        = {Pith review of: On the Schr\"odinger equations with $B_\infty$ potentials in the region above a Lipschitz graph},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RYLBKX4U}},
  note         = {Machine review of arXiv:2411.18458}
}
abstract

In this paper we investigate the $L^p$ regularity, $L^p$ Neumann and $W^{1,p}$ problems for generalized Schr\"odinger operator $-\text{div}(A\nabla )+ V $ in the region above a Lipschitz graph under the assumption that $A$ is elliptic, symmetric and $x_d-$independent. Specifically, we prove that the $L^p$ regularity problem is uniquely solvable for $$1<p<2+\varepsilon.$$ Moreover, we also establish the $W^{1,p}$ estimate for Neumann problem for $$\frac{3}{2}-\varepsilon<p<3+\varepsilon.$$ As a by-product, we also obtain that the $L^p$ Neumann problem is uniquely solvable for $1<p<2+\varepsilon.$ The only previously known estimates of this type pertain to the classical Schr\"odinger equation $-\Delta u+ Vu=0$ in $\Omega$ and $\frac{\partial u}{\partial n}=g$ on $\partial\Omega$ which was obtained by Shen [Z. Shen, On the Neumann problem for Schr\"odinger operators in Lipschitz domains, Indiana Univ. Math. J. 43 (1994)] for ranges $1<p\leq 2$. All the ranges of $p$ are sharp.

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Works this paper leans on

28 extracted references · 22 canonical work pages

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