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arxiv: 2604.01544 · v2 · submitted 2026-04-02 · 🧮 math.AP

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Solvability of boundary value problem for Schr\"odinger Equations with Reverse H\"older Potentials on L^p and endpoint spaces

Botian Xiao , Lin Tang
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Pith reviewed 2026-05-13 21:29 UTC · model grok-4.3

classification 🧮 math.AP
keywords Schrödinger equationboundary value problemslayer potentialsreverse Hölder potentialsDe Giorgi-Nash-Moser boundsL^p solvabilityNeumann problemregularity problem
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The pith

Solvability of Neumann and regularity problems for Schrödinger equations with reverse Hölder potentials holds on L^p and endpoint spaces when coefficients are small L^∞ perturbations of De Giorgi-Nash-Moser matrices.

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

This paper proves solvability of the Neumann boundary value problem with data in H^p_L(R^n) and the regularity problem with data in H^{1,p}_V(R^n) for the elliptic Schrödinger equation -div(A(x) ∇u) + V u = 0, where V belongs to the reverse Hölder class B_q. The coefficients A are bounded measurable and uniformly elliptic but independent of t, and the proof uses the method of layer potentials. Solvability is obtained precisely when A is a small L^∞ perturbation of a matrix satisfying De Giorgi-Nash-Moser bounds, which allows control of the layer potentials and the required estimates on the boundary data spaces. The work also derives Campanato norm estimates for the double layer potential linked to the Dirichlet problem in appropriate Campanato-type spaces. A sympathetic reader cares because these results extend classical boundary value theory to Schrödinger operators with more singular potentials while staying within the L^p and endpoint regime.

Core claim

The Neumann problem with ∂_ν_A u(x,0) = f in H^p_L(R^n) and the regularity problem with u(x,0) = g in H^{1,p}_V(R^n) are solvable for the Schrödinger-type equation -div(A(x) ∇u(x,t)) + V(x) u(x,t) = 0 when V is in B_q and A is a sufficiently small L^∞ perturbation of a matrix obeying De Giorgi-Nash-Moser bounds; the layer potential method reduces the problems to invertible integral equations on the boundary and yields the stated estimates.

What carries the argument

Layer potentials (single and double) for the Schrödinger operator, which represent the solution and convert the boundary value problems into boundary integral equations whose invertibility follows from the small-perturbation assumption on A.

If this is right

  • The Neumann problem is solvable for f in H^p_L(R^n) and the regularity problem for g in H^{1,p}_V(R^n) under the stated perturbation condition.
  • Campanato-norm estimates hold for the double layer potential when the Dirichlet data lie in suitable Campanato-type spaces.
  • The results apply uniformly to the time-independent elliptic Schrödinger equation with potentials in B_q.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same small-perturbation technique might extend to other classes of potentials or to parabolic Schrödinger equations by treating time as an additional variable.
  • Endpoint space results could be used to obtain well-posedness for semilinear problems whose nonlinearities map between these spaces.
  • Numerical verification of the smallness threshold on concrete matrices (for example, constant plus small oscillation) would give practical bounds on the allowable perturbation size.

Load-bearing premise

The coefficient matrix A must be a small L^∞ perturbation of one that satisfies De Giorgi-Nash-Moser bounds, because this controls the layer potentials and produces the necessary estimates on the boundary spaces.

What would settle it

An explicit matrix A that is not a small L^∞ perturbation of any De Giorgi-Nash-Moser matrix, together with a reverse Hölder V, for which the Neumann or regularity problem fails to have a solution in H^p_L or H^{1,p}_V for some admissible p, would falsify the claim.

read the original abstract

In this paper we discuss the solvability of the Neumann and Regularity boundary value problem of elliptic Schr\"odinger-type equation $-\DIV(A(x)\nabla u(x,t))+V(x)u(x,t)=0$ with bounded measurable uniformly elliptic coefficinets $A(x)$ independent of $t$ and $V$ in Reverse H\"older class $\mathcal{B}_q$, and Neumann boundary data $\partial_{\nu_A}u(x,0)=f(x)\in H^p_{\mathcal{L}}(\rn)$, or Regularity data $u(x,0)=g\in H^{1,p}_V(\rn)$, utilizing the method of layer potential. We prove the solvability when $A$ is a small $L^\infty$ perturbation of a matrix satisfying De Giorgi-Nash-Moser bounds. Besides we also give the Campanato norm estimate of the double layer potential related to the Dirichlet problem with boundary data in certain Campanato-type spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims to prove solvability of the Neumann and regularity boundary value problems for the Schrödinger equation -div(A∇u) + V u = 0, where A is a small L^∞ perturbation of a De Giorgi-Nash-Moser matrix and V lies in the reverse Hölder class B_q, via layer potential methods yielding invertibility on the spaces H^p_L(R^n) and H^{1,p}_V(R^n). It additionally derives Campanato-norm estimates for the double-layer potential in the Dirichlet problem.

Significance. If the central claims hold with the stated quantitative control, the work extends layer-potential techniques for elliptic boundary-value problems to Schrödinger operators with reverse-Hölder potentials under a small-perturbation hypothesis on the leading coefficients. This would furnish new solvability results on L^p and endpoint spaces and supply Campanato estimates that may be useful for further regularity theory.

major comments (1)
  1. [Main theorem and perturbation argument (abstract and §§3–5)] The main solvability theorem (stated in the abstract and presumably proved in §3–§5) asserts that the boundary operators are invertible for sufficiently small ε in A = A_0 + ε B, yet no explicit dependence of the allowable ε on ||V||_{RH_q} is derived. Because the single-layer potential and its adjoint are constructed from the fundamental solution whose estimates incorporate the reverse-Hölder norm of V, the smallness threshold may shrink with ||V||_{RH_q}; without a uniform lower bound on ε independent of this norm, the invertibility statements on H^p_L and H^{1,p}_V rest on an unverified joint-smallness condition.
minor comments (2)
  1. [Abstract] Abstract contains the typo 'coefficinets' (should be 'coefficients').
  2. [Introduction] The spaces H^p_L and H^{1,p}_V are introduced without prior definition of the operators L and the precise role of V; a brief clarification in the introduction would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment below and will revise the manuscript to clarify the dependence of the perturbation size on the reverse Hölder norm of V.

read point-by-point responses

  1. Referee: [Main theorem and perturbation argument (abstract and §§3–5)] The main solvability theorem (stated in the abstract and presumably proved in §3–§5) asserts that the boundary operators are invertible for sufficiently small ε in A = A_0 + ε B, yet no explicit dependence of the allowable ε on ||V||_{RH_q} is derived. Because the single-layer potential and its adjoint are constructed from the fundamental solution whose estimates incorporate the reverse-Hölder norm of V, the smallness threshold may shrink with ||V||_{RH_q}; without a uniform lower bound on ε independent of this norm, the invertibility statements on H^p_L and H^{1,p}_V rest on an unverified joint-smallness condition.

    Authors: We agree that the allowable ε depends on ||V||_{RH_q} through the constants appearing in the fundamental solution estimates and the resulting layer potential bounds. The proof first fixes V (hence fixes its RH_q norm and the associated constants) and then selects ε small enough relative to those constants to ensure the perturbation argument closes. This dependence is implicit in the estimates but not stated explicitly in the theorem. In the revised manuscript we will add an explicit statement of the dependence (ε < ε_0(n,p,q,λ,Λ,||V||_{RH_q})) both in the main theorem and in the opening paragraph of the perturbation argument in §3, thereby making the joint-smallness condition precise and verified. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external DGNM theory and layer potentials

full rationale

The paper proves solvability of the Neumann and regularity problems for the Schrödinger equation via layer potentials precisely when A is a small L^∞ perturbation of a matrix satisfying De Giorgi-Nash-Moser bounds, with V in the reverse Hölder class. This step invokes standard, externally established elliptic regularity (DGNM) and layer-potential invertibility results rather than any self-definition, fitted-input renaming, or load-bearing self-citation that reduces the claim to its own inputs by construction. No equations or sections in the abstract or description exhibit a reduction such as a parameter fitted to data then relabeled as a prediction, or an ansatz smuggled via prior self-work. The central claim therefore retains independent content from the cited external theory and is self-contained against those benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard elliptic regularity theory and layer-potential estimates; no new free parameters, ad-hoc axioms, or invented entities are introduced beyond the small-perturbation hypothesis.

axioms (2)
  • domain assumption Uniform ellipticity and boundedness of the coefficient matrix A(x)
    Invoked throughout to guarantee the existence of layer potentials and De Giorgi-Nash-Moser bounds.
  • domain assumption V belongs to the reverse Hölder class B_q
    Used to control the potential term in the Schrödinger operator and to obtain the required mapping properties on the boundary spaces.

pith-pipeline@v0.9.0 · 5474 in / 1402 out tokens · 30693 ms · 2026-05-13T21:29:40.090912+00:00 · methodology

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Reference graph

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