H\"older estimates for the Neumann problem in a domain with holes and a relation formula between the Dirichlet and Neumann problems

Duvan Henao; Victor Ca\~nulef-Aguilar

arxiv: 1906.09110 · v1 · pith:FSPKKDPWnew · submitted 2019-06-20 · 🧮 math.AP

H\"older estimates for the Neumann problem in a domain with holes and a relation formula between the Dirichlet and Neumann problems

Victor Ca\~nulef-Aguilar , Duvan Henao This is my paper

Pith reviewed 2026-05-25 19:52 UTC · model grok-4.3

classification 🧮 math.AP

keywords Hölder estimatesNeumann problemdomains with holesDirichlet problemharmonic functionsrelation formula

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The pith

A relation formula connects harmonic extensions to Neumann solutions on the disk and its exterior.

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

The paper investigates how Hölder estimates for the Neumann problem change with the geometry of a domain that has holes. It begins by determining the Hölder regularity of solutions to both the Dirichlet and Neumann problems inside the disk and outside the disk. These calculations produce a relation that links harmonic extensions to functions satisfying a prescribed Neumann condition on the disk boundary, and the same relation is obtained for the exterior problem. The relation is then applied to understand the estimates on more general domains containing holes.

Core claim

From the study of the Hölder regularity of the solutions to the Dirichlet and Neumann problems in the disk (and in the exterior of the disk), we get a relation between harmonic extensions and harmonic functions with prescribed Neumann condition on the boundary of the disk (for both interior and exterior problems).

What carries the argument

The relation formula that equates harmonic extensions arising from the Dirichlet problem with solutions of the Neumann problem on the disk boundary.

Load-bearing premise

The explicit solvability and regularity properties used to derive the relation formula hold only when the domain is exactly the disk or its exterior.

What would settle it

An explicit computation on the unit disk with constant Neumann data that shows the two sides of the proposed relation formula are not equal.

read the original abstract

In this paper we study the dependence of the H\"older estimates on the geometry of a domain with holes for the Neumann problem. For this, we study the H\"older regularity of the solutions to the Dirichlet and Neumann problems in the disk (and in the exterior of the disk), from which we get a relation between harmonic extensions and harmonic functions with prescribed Neumann condition on the boundary of the disk (for both interior and exterior problems).

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.

Desk Editor's Note private letter to a colleague

The disk relation is explicit and clean but the jump to Hölder estimates in domains with holes lacks a visible transfer argument.

read the letter

The paper's main move is to solve the Dirichlet and Neumann problems explicitly in the disk and its exterior, extract a relation between the two, and then claim this helps control Hölder constants for the Neumann problem when the domain has holes. The explicit disk work is the part that actually gets done and looks reproducible; harmonic functions there have closed forms, so deriving a concrete link between the extensions and the Neumann data is feasible and probably correct on its own terms. That is the only new piece visible from the abstract and title. Everything else follows from applying that relation to more general geometry. The difficulty is that the relation is tied to the disk. Conformal maps preserve harmonicity but change the Neumann condition in a non-local way, and approximating a domain with holes by disks would need uniform control on the holes and on the constants. The abstract gives no sign of such a bridging step, so the claimed dependence of the Hölder estimates on the geometry of the holes rests on an unverified extension. If the full text contains a rigorous transfer, that would change the picture; on the evidence supplied it does not appear. This is a paper for people already working on boundary regularity for elliptic equations in domains with holes. A reader who wants to see whether the disk relation can be made to travel would get something out of it, but only after the transfer is written down. It is coherent enough to send to a referee who knows the literature on Neumann problems, provided the authors are asked to supply the missing argument in revision.

Referee Report

1 major / 0 minor

Summary. The paper claims to study the dependence of Hölder estimates on geometry for the Neumann problem in domains with holes. It does so by first analyzing Hölder regularity for the Dirichlet and Neumann problems in the disk and its exterior, from which a relation formula is derived between harmonic extensions and harmonic functions with prescribed Neumann data (both interior and exterior cases).

Significance. If the relation transfers rigorously to general domains with holes, the work could clarify geometry dependence in boundary regularity for Neumann problems, a topic of interest in elliptic PDEs. The explicit solvability in the disk provides a concrete starting point for the relation, which is a methodological strength when properly bridged.

major comments (1)

[Abstract] Abstract: The relation formula is obtained via explicit disk/exterior solutions (Fourier series or Poisson kernel). However, the central claim concerns Hölder estimates whose geometry dependence holds for domains with holes; no mechanism is indicated for transferring the disk relation (e.g., via conformal mapping, which alters Neumann data, or approximation with uniform control). This bridging step is load-bearing for the geometry-dependence result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the scope of our results. We address the major concern point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: The relation formula is obtained via explicit disk/exterior solutions (Fourier series or Poisson kernel). However, the central claim concerns Hölder estimates whose geometry dependence holds for domains with holes; no mechanism is indicated for transferring the disk relation (e.g., via conformal mapping, which alters Neumann data, or approximation with uniform control). This bridging step is load-bearing for the geometry-dependence result.

Authors: We agree that the manuscript derives the relation formula and the associated Hölder estimates explicitly for the disk and its exterior via Fourier series and the Poisson kernel, without providing a rigorous transfer mechanism (such as controlled conformal mappings or uniform approximation) to arbitrary domains with holes. The exterior-disk case is presented as a model for a domain containing a hole, allowing direct comparison of interior versus exterior geometry dependence, but this does not constitute a general bridging argument. In the revision we will (i) clarify in the abstract and introduction that the geometry-dependence statements are established only for these model domains, (ii) add a brief discussion of how the explicit relation might be used locally or via perturbation arguments in more general settings, and (iii) note the limitations of the current approach. revision: yes

Circularity Check

0 steps flagged

No circularity: relation obtained from explicit disk solutions

full rationale

The provided abstract states that the relation is obtained by studying Hölder regularity of Dirichlet and Neumann problems in the disk and exterior, where explicit solvability (Fourier series/Poisson kernel) is available. This is a direct derivation from first-principles solutions in a special geometry rather than a fit, self-definition, or self-citation chain. No load-bearing step reduces by construction to the target result for general domains with holes; the extension step is presented as subsequent application. Per rules, absent any quoted equation or self-citation that forces the central claim, the score is 0 and steps array is empty. The derivation chain remains self-contained against external benchmarks of explicit harmonic functions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the central claim rests on standard facts of harmonic function theory whose precise invocation cannot be audited from the given text.

pith-pipeline@v0.9.0 · 5605 in / 1029 out tokens · 22466 ms · 2026-05-25T19:52:06.357084+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

\ \ W \^ \ 1,p \ \ \ -quasiconvexity and variational problems for multiple integrals

J M Ball and F Murat. \ \ W \^ \ 1,p \ \ \ -quasiconvexity and variational problems for multiple integrals . J. Funct. Anal. , 58(3):225--253, 1984

work page 1984
[2]

Ca \ n ulef-Aguilar and D

V. Ca \ n ulef-Aguilar and D. Henao. A lower bound for the void coalescence load in nonlinearly elastic solids. Preprint available at www.mat.uc.cl/prepublicaciones/download/118

work page
[3]

Partial Differential Equations

E DiBenedetto. Partial Differential Equations . Birkhauser, 2009

work page 2009
[4]

On a partial differential equation involving the Jacobian determinant

B Dacorogna and J Moser. On a partial differential equation involving the Jacobian determinant . Ann. Inst. H. Poincar \' e Anal. Non Lin \' e aire , 7(1):1--26, 1990

work page 1990
[5]

Partial differential equations , volume 19 of Graduate Studies in Mathematics

Lawrence C Evans. Partial differential equations , volume 19 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, second edition, 2010

work page 2010
[6]

Complex Analysis

T Gamelin. Complex Analysis . Springer, 2001

work page 2001
[7]

Energy Estimates and Cavity Interaction for a Critical-Exponent Cavitation Model

Duvan Henao and Sylvia Serfaty. Energy Estimates and Cavity Interaction for a Critical-Exponent Cavitation Model . Comm. Pure Appl. Math. , 66:1028--1101, 2013

work page 2013
[8]

Multiple integrals in the calculus of variations

Charles B Morrey Jr. Multiple integrals in the calculus of variations . Die Grundlehren der mathematischen Wissenschaften, 130. Springer, New York, 1966

work page 1966

[1] [1]

\ \ W \^ \ 1,p \ \ \ -quasiconvexity and variational problems for multiple integrals

J M Ball and F Murat. \ \ W \^ \ 1,p \ \ \ -quasiconvexity and variational problems for multiple integrals . J. Funct. Anal. , 58(3):225--253, 1984

work page 1984

[2] [2]

Ca \ n ulef-Aguilar and D

V. Ca \ n ulef-Aguilar and D. Henao. A lower bound for the void coalescence load in nonlinearly elastic solids. Preprint available at www.mat.uc.cl/prepublicaciones/download/118

work page

[3] [3]

Partial Differential Equations

E DiBenedetto. Partial Differential Equations . Birkhauser, 2009

work page 2009

[4] [4]

On a partial differential equation involving the Jacobian determinant

B Dacorogna and J Moser. On a partial differential equation involving the Jacobian determinant . Ann. Inst. H. Poincar \' e Anal. Non Lin \' e aire , 7(1):1--26, 1990

work page 1990

[5] [5]

Partial differential equations , volume 19 of Graduate Studies in Mathematics

Lawrence C Evans. Partial differential equations , volume 19 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, second edition, 2010

work page 2010

[6] [6]

Complex Analysis

T Gamelin. Complex Analysis . Springer, 2001

work page 2001

[7] [7]

Energy Estimates and Cavity Interaction for a Critical-Exponent Cavitation Model

Duvan Henao and Sylvia Serfaty. Energy Estimates and Cavity Interaction for a Critical-Exponent Cavitation Model . Comm. Pure Appl. Math. , 66:1028--1101, 2013

work page 2013

[8] [8]

Multiple integrals in the calculus of variations

Charles B Morrey Jr. Multiple integrals in the calculus of variations . Die Grundlehren der mathematischen Wissenschaften, 130. Springer, New York, 1966

work page 1966