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Stability of $L^p$ Dirichlet problem under small bi-Lipschitz transformations of domains

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Pith's one-line read This paper proves that small bi-Lipschitz transformations of a Lipschitz graph domain preserve the solvability of the L^p Dirichlet problem for the Laplacian for the same exponent p > 1, even when the base domain has large Lipschitz constan

desk verdict Major stability theorem with a novel Green-function frame, but the proof that ρ is a diffeomorphism is genuinely underdeveloped. read the letter →

arxiv 2602.08115 v2 pith:QNSNRFS2 submitted 2026-02-08 math.AP

Joseph Feneuil , Linhan Li , Jinping Zhuge This is my paper · ORCID
classification math.AP MSC 35J0535J2542B37
keywords L^pDirichletproblemLaplacianbi-LipschitztransformationsLipschitzdomainsGreenfunctionCarlesonmeasuresconvexstronglyquasiconvex
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 establishes a domain-perturbation stability result: if the L^p Dirichlet problem for the Laplacian is solvable in an unbounded Lipschitz graph domain, then it remains solvable for the same p > 1 in any bi-Lipschitz image of that domain whose Jacobian is uniformly close to the identity. This removes the previous restrictions that either the base domain be nearly flat or that the perturbation be confined to the transversal direction. As a byproduct, the paper defines a class of 'strongly quasiconvex' domains — small bi-Lipschitz deformations of convex graph domains — and proves the L^p Dirichlet problem is solvable there for every 1 < p < ∞, unifying two previously separate settings: convex domains and C^1 (vanishing chord-arc) domains. The key novelty is a change of variables built from the Green function with pole at infinity in the base domain, using a non-constant orthonormal basis that encodes the domain's geometry.

What carries the argument

The carrying object is the Green function G for the Laplacian with pole at infinity in the base Lipschitz graph domain Ω0. Since G vanishes exactly on ∂Ω0 and its level sets foliate Ω0, the paper uses v_n = ∇G/|∇G| as a position-dependent 'vertical' direction and completes it to a full orthonormal frame {v_1,...,v_n} by a Gram-Schmidt process that controls all derivatives through Carleson measures. The change of variables ρ is then defined by ρ(X) = X − t(X)v_n(X) + λ(X) + a(X)t(X)v_n(X)O(X), where t = G/|∇G| plays the role of the transversal variable, λ is a smoothed version of the boundary perturbation, a is a determinant factor, and O is a rotation slightly tilting the frame to match the

What would settle it

Compute, for a cone of aperture close to π, the Carleson norm of δ²|∇²G|/G where G is the Green function with pole at infinity; if it grows with the aperture faster than the stated dependence on M (or is infinite), Theorem 2.20 is false and the stability theorem would need a different input. Alternatively, find a small bi-Lipschitz deformation of a convex domain for which the L^p Dirichlet problem fails for some p<2; that would contradict Corollary 1.10.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.3: for any Lipschitz graph domain Ω0 (with possibly large Lipschitz constant M) where the L^p Dirichlet problem for −Δ is solvable for some p > 1, there is an ε0 depending only on n, p, the reverse-Hölder constant of the elliptic measure, and M such that every bi-Lipschitz map Φ with ‖∇Φ−I‖∞ ≤ ε0 carries Ω0 to a domain Ω = Φ(Ω0) where the same L^p Dirichlet problem is solvable for the same p. The proof constructs an explicit diffeomorphism ρ from Ω0 to Ω, designed so that the pull-back Lρ of the Laplacian is a small Carleson perturbation of an intermediate operator L0 that shares the Green function (with pole at infinity) with the

Load-bearing premise

The whole construction depends on a Carleson measure estimate for the second derivatives of the Green function at infinity in any Lipschitz graph domain; if that estimate fails for domains with large Lipschitz constant, the argument collapses.

Editorial extensions

If this is right

  • For any bounded strongly quasiconvex domain, the L^p Dirichlet problem for the Laplacian is solvable for all p ∈ (1,∞) (Corollary 1.10).
  • The same range of p holds for the L^p Regularity problem in these domains (Corollary 1.11).
  • The class of strongly quasiconvex domains is stable under C^1 diffeomorphisms, so any C^1 image of a convex domain is covered (Remark 1.9).
  • The allowed perturbations include, but are strictly larger than, small transversal Lipschitz perturbations of the graph (Remark 1.5).
  • The stability is for the same exponent p, including the previously inaccessible range 1 < p < 2.

Reading between the lines

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

  • If the Carleson-measure estimate for δ²|∇²G|/G (Theorem 2.20) is as robust as stated, the same change-of-variables strategy should extend to elliptic operators satisfying the DKP condition, since the Green-function machinery is already available for such operators.
  • The strong-quasiconvexity condition suggests a candidate geometric characterization for the open Question 1.1: domains that are locally small bi-Lipschitz deformations of convex graphs; one could test whether the condition is also necessary by looking for domains where (D)_p fails for some p despite being locally close to convex.
  • The method's reliance on a non-constant basis indicates that the natural 'flat' direction for a Lipschitz domain is supplied by the Green function itself; this may provide a route to extend the results to domains with lower-dimensional boundaries, following the spirit of higher-codimension harmonic measure 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 / 4 minor

Summary. The paper proves a stability theorem for the L^p Dirichlet problem for the Laplacian under small bi-Lipschitz perturbations of Lipschitz graph domains. If (D)_p is solvable in a Lipschitz graph domain Ω0, possibly with large Lipschitz constant, then for any bi-Lipschitz Φ with ∇Φ uniformly close to the identity, (D)_p is solvable in Ω = Φ(Ω0) with the same exponent p > 1, and ε0 depends only on n, p, the reverse-Hölder constant, and the Lipschitz constant of Ω0. The proof constructs a non-constant orthonormal basis from the Green function with pole at infinity, builds a deformation ρ: Ω0 → Ω, and shows that the pulled-back Laplacian is a small Carleson perturbation of an intermediate operator L0 for which the Green function G is a solution. The paper also derives Corollary 1.10, asserting that all bounded strongly quasiconvex domains are L^p-solvable for every p ∈ (1,∞), thereby proposing a class unifying convex and C^1 domains.

Significance. If the proof is completed, this is a significant contribution: it provides the first domain-perturbation result preserving the same exponent p for the full range 1 < p < ∞, including the difficult range 1 < p < 2, and it introduces a genuinely new tool — a non-constant basis built from the Green function — to encode the geometry of the base domain. The paper is largely transparent: constants are tracked, there are no fitted parameters, and the main Carleson estimates are based on published results [FL23], [DLM22], [Azz19]. However, the final topological step asserting that ρ is a global diffeomorphism is not fully justified; this step is load-bearing for the main theorem and requires a substantive repair. I therefore regard the central claim as plausible and probably correct, but not yet rigorously established in the present text.

major comments (2)
  1. [Theorem 3.43, Step 3 (after (3.50)–(3.51))] The local constancy of Nρ(Z) = #{X ∈ Ω0 : ρ(X)=Z} is asserted on the basis of local invertibility and the bound ∥∇ρ−I∥∞ ≤ 1/2. This is not valid without a properness or boundary-control argument: a map such as the inclusion of a bounded interval has everywhere invertible Jacobian, yet the preimage count changes at the endpoints. The authors need to prove that ρ is proper on Ω0, or restrict to Ω0 ∩ B_R and establish degree invariance as R → ∞. Without this, the proof does not establish that ρ(Ω0)=Ω or that ρ is one-to-one, and Proposition 2.17 cannot be invoked in the final chain of Theorem 1.3.
  2. [Theorem 3.43, Step 3, homotopy argument] The continuity of θ ↦ N_{ρθ}(Φθ(X)) is asserted but not demonstrated. To use homotopy invariance of the degree, one must know that Φθ(X) stays at a positive distance from ∂Ωθ, that the relevant preimage sets are uniformly bounded, and that no preimages appear or disappear at infinity along the homotopy. The manuscript gives no such uniform control. A repair will likely require quantitative bounds showing that ρθ is proper and has no zeros on a large sphere, uniformly in θ ∈ [0,1].
minor comments (4)
  1. [Abstract and Theorem 1.3] The abstract and introduction speak of 'small bi-Lipschitz transformations', but Theorem 1.3 assumes Φ is differentiable with ∥∇Φ−I∥∞ ≤ ε0. This is stronger than having a small bi-Lipschitz constant alone. Please state this hypothesis explicitly in the abstract and in Question 1.2.
  2. [Lemma 3.2, Step 1a] There is a typographical error in the displayed convergence of gradients: the term 'G_{Y_i}(Z)/G_{Y_i}(Y_0) − G_{Y_i}(Z)/G_{Y_i}(Y_0)' should presumably be 'G_{Y_i}(Z)/G_{Y_i}(Y_0) − G(Z)/G(Y_0)'.
  3. [Theorem 2.20 and constants] The paper imports the estimate δ²|∇²G|/G ∈ CM_sup from [FL23]/[DLM22]. Since ε0 in Theorem 1.3 depends on the constant in this estimate, the precise dependence on the Lipschitz constant M should be stated explicitly, or at least quoted verbatim from the source, so that the dependency claim in Theorem 1.3 is verifiable.
  4. [Section 4.2 and Appendix B] The proof of Corollary 1.10 is concise and relies on a localized comparison-principle argument plus a covering argument. It would be helpful to spell out the covering argument for the intermediate scales and to state more carefully how the uniform reverse-Hölder constant in Theorem B.6 is independent of x0.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction found; self-cited Green-function estimates are independent inputs, and the sole weak point is a non-circular degree-theoretic gap.

full rationale

The claimed derivation does not reduce to its own inputs. Starting from the assumed (D_p)_{-Δ,Ω0}, Theorem 2.12 converts solvability into a reverse-Hölder bound for κ=G/δ. The intermediate operator L0 is constructed in Lemma 4.2 with A0∇G=∇G, so G is also the L0-Green function and the same reverse-Hölder bound is inherited; no parameter is fitted to force the conclusion. Theorem 2.16 then transfers solvability to Lρ because |Aρ−A0|∈CM_sup(C2ε0), and Proposition 2.17 transfers back to −Δ on Φ(Ω0). Each arrow uses a stated theorem whose hypotheses do not include the desired conclusion. The Green-function estimates (Theorem 2.20, Lemma 2.11) are imported from [FL23], [DLM22], [Azz19], and [DFM23]; despite author overlap, these are published, parameter-free theorems not derived from the present stability claim, so they constitute independent evidence rather than circularity. The real weakness is a non-circular proof gap in Theorem 3.43 Step 3: local invertibility plus the assertion that the degree Nρ is locally constant, and the subsequent homotopy argument, require properness/positive-distance control that is not supplied. This threatens correctness but is not a self-referential reduction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof is a chain of published results plus a new construction; the ledger records the imported theorems that carry the heaviest load.

assumptions (6)
  • domain assumption Theorem 2.20: δ²|∇²G|/G ∈ CM_sup for the Green function with pole at infinity on Lipschitz graph domains.
    Imported from [FL23]/[DLM22]; every later Carleson estimate (Thm 3.10, Cor 3.15, Thm 3.23) and hence the whole frame construction starts from this.
  • standard math Theorem 2.16: small Carleson perturbations of symmetric uniformly elliptic operators preserve L^p Dirichlet solvability.
    Quoted from [CHM19, Thm 1.1] plus Thm 2.12; used at the final step to pass from L0 to Lρ.
  • standard math Theorem 2.12 and Lemma 2.11: reverse-Hölder condition on G/δ characterizes (D)_p; Green function and elliptic measure with pole at infinity are comparable.
    Used to transfer solvability of −Δ in Ω0 to L0 and back; proof of Thm 2.12 is sketched in §2.2.
  • standard math Dorronsoro's theorem: affine approximation error β of a Cϵ-Lipschitz function satisfies β² dy dt/t ∈ Carleson with norm ≤ Cϵ.
    Used in §3.2 to control λ and its derivatives (Thm 3.23).
  • standard math L² Dirichlet solvability and uniqueness for (possibly unbounded) Lipschitz domains.
    Used in Lemma 3.2 to identify ∂_n G_Y via Poisson extension and in Proposition 2.17.
  • domain assumption The perturbing map Φ has a Jacobian with ||∇Φ−I||∞≤ε; in particular the theorem does not cover small-distortion bi-Lipschitz maps with large derivative oscillation.
    This is the stated hypothesis of Theorem 1.3; it is stronger than merely having bi-Lipschitz constant close to 1, and the paper's title/abstract phrase 'small bi-Lipschitz' should be read with this C^1-smallness.

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Pith. "Pith review of Stability of $L^p$ Dirichlet problem under small bi-Lipschitz transformations of domains." pith.science (2026). https://pith.science/paper/QNSNRFS2

@misc{pith2026260208115,
  author       = {Pith},
  title        = {Pith review of: Stability of $L^p$ Dirichlet problem under small bi-Lipschitz transformations of domains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QNSNRFS2}},
  note         = {Machine review of arXiv:2602.08115}
}
abstract

We show that small bi-Lipschitz deformations of a Lipschitz domain (with possibly large Lipschitz constant) preserve the solvability of the Dirichlet problem for the Laplacian with boundary data in $L^p$, for the same value of $p>1$. As a consequence, for all $p\in(1,\infty)$, we obtain the solvability of the $L^p$ Dirichlet problem for small Lipschitz perturbations of convex domains, thereby unifying two fundamentally different settings in which such results were previously known: convex and $C^1$ domains. The key ingredient and novelty of our approach is a construction of a change of variables based on a non-constant basis derived from the Green function, which encodes the geometry of the base domain.

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    2023 arXiv

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.