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arxiv: 2604.27219 · v1 · submitted 2026-04-29 · 🧮 math.AP

Anchored Peskin Problem

Achyuta Telekicherla Kandalam , Daniel Spirn This is my paper

Pith reviewed 2026-05-07 09:41 UTC · model grok-4.3

classification 🧮 math.AP
keywords Peskin problemanchored filamentfractional LaplacianDirichlet boundary conditionsimmersed boundary methodelastic filamentwell-posednesshydrodynamic interactions
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The pith

The leading-order motion of an elastic filament anchored to a wall follows a fractional Laplacian with homogeneous Dirichlet conditions at the anchors.

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

This paper generalizes the Peskin problem to an elastic filament whose ends are fixed to a no-slip wall in the half-plane. Using a boundary-symmetric approach to the fluid forces, the work shows that the filament's evolution is led by a fractional Laplacian operator with zero values imposed at the anchored points. The authors establish that stationary shapes are always circular arcs joining the two fixed ends, and this holds across many forms of the filament's bending energy. They further prove local existence and uniqueness of solutions along with immediate smoothing to infinite differentiability. Such results offer a solid mathematical ground for analyzing biological filaments tethered to cell membranes or other surfaces.

Core claim

We prove that the leading-order evolution of the anchored filament is governed by a fractional Laplacian equipped with homogeneous Dirichlet boundary conditions. We characterize the stationary states of the system, proving that all equilibria are circular arcs connecting the anchor points, a result that holds for a broad class of elastic energy densities. By framing the non-local dynamics in weighted little Hölder spaces, we establish local well posedness and prove that the filament exhibits instantaneous C^∞ regularization in both space and time.

What carries the argument

The boundary-symmetric formulation of hydrodynamic interactions, which splits the flow into a free-space part and a reflected correction to derive the fractional Laplacian dynamics without hypersingular terms.

Load-bearing premise

The leading-order hydrodynamic approximation remains valid under the boundary-symmetric formulation when the filament is anchored to the wall.

What would settle it

Simulating the full immersed boundary equations for a small perturbation around a circular arc and comparing the resulting velocity field to that predicted by the fractional Laplacian operator applied to the same shape.

Figures

Figures reproduced from arXiv: 2604.27219 by Achyuta Telekicherla Kandalam, Daniel Spirn.

Figure 1
Figure 1. Figure 1: Schematic of an elastic filament anchored at two points on a rigid wall. The rigorous study of this problem in R 2 was initiated in [10, 12], with recent progress on tension determination given in [7]. 2.1. Peskin Problem in R 2 +. We now formulate the Peskin problem for a one￾dimensional membrane anchored on a wall in the half-space R 2 +. The half-space is divided into two domains, Ω1 and Ω2, separated b… view at source ↗
Figure 2
Figure 2. Figure 2: Equilibrium configuration: a circular arc anchored at two points on the wall. Assume the equilibrium arc is part of a circle centered at (0, c), c ∈ R, passing through the anchoring points (±1, 0). The radius is r = p c 2 + 1, and the maximum height of the arc is h = r + c > 0. A convenient parametrization of the membrane, especially in our numerical calculations, is X(s) =    r exp i 2 − 2 π arctan 1… view at source ↗
Figure 3
Figure 3. Figure 3: Sample evolution of an asymmetric initial profile defined by X(s) = cos(s) and Y (s) = sin(s) − a exp[−(s − d) 2/(2w 2 )] sin2 (s) with a = 0.5, d = 0.4π, w = 0.12π. Simula￾tion parameters are ∆t = 0.01 and N = 512. The filament evolves to its circular-arc equilibrium (light gray) with the area conserved to spatial-quadrature accuracy view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of a deeply notched obtuse arc toward its circular-arc equilibrium. We define the initial curves, parametriz￾ing by Φ ∈ view at source ↗
Figure 5
Figure 5. Figure 5: log / log plots of the error of the convergence of the isoperimetric ratio R to 1 of the asymmetric arc and deeply notched obtuse arc over 1000 time steps. We observe consistent convergence trends in both cases. Appendix A. Reflected Stokeslet We recall the derivation of the half-space Stokeslet following [6]. We begin with the free-space Stokeslet in R 2 , S[f](x, y) = 1 4π  − log |x − y|f + (f · (x − y)… view at source ↗
read the original abstract

The Immersed Boundary Method has long served as a robust computational framework for fluid-structure interactions, yet the rigorous analysis of 1D Peskin filaments anchored to rigid boundaries remains sparse. In this paper, we generalize the classical Peskin problem to the half-plane by considering an elastic filament whose endpoints are anchored to a no-slip wall. Moving beyond the algebraic complexity of the traditional Blake image system, we utilize the boundary-symmetric formulation of Gimbutas, Greengard, and Veerapaneni. This representation allows for a transparent decomposition of the hydrodynamic interactions into a free space principal part and a regularizing reflected component without resorting to hypersingular integral operators. Through this framework, we prove that the leading-order evolution of the anchored filament is governed by a fractional Laplacian equipped with homogeneous Dirichlet boundary conditions. We characterize the stationary states of the system, proving that all equilibria are circular arcs connecting the anchor points, a result that holds for a broad class of elastic energy densities. By framing the non-local dynamics in weighted little H\"older spaces, we establish local well posedness and prove that the filament exhibits instantaneous $C^\infty$ regularization in both space and time. This work provides a rigorous analytical foundation for anchored filaments in bounded domains and suggests a spectrally accurate numerical path for simulating tethered biological structures.

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

0 major / 2 minor

Summary. The manuscript generalizes the classical Peskin problem to an elastic filament anchored to a no-slip wall in the half-plane. Using the boundary-symmetric formulation, it decomposes the hydrodynamics into a free-space principal part (yielding a fractional Laplacian) and a regularizing reflected part. The paper proves that the leading-order evolution is governed by this fractional Laplacian with homogeneous Dirichlet boundary conditions, that all equilibria are circular arcs for a broad class of elastic energies, and that the system is locally well-posed in weighted little Hölder spaces with instantaneous C^∞ smoothing in space and time.

Significance. If the central claims hold, the work supplies a rigorous analytical foundation for anchored filaments in bounded domains. The clean reduction to a Dirichlet fractional Laplacian via the boundary-symmetric formulation is a technical strength that avoids hypersingular operators. The result that equilibria are circular arcs independent of the specific energy density (within the stated class) is a notable geometric characterization. The well-posedness and smoothing theorem in weighted little Hölder spaces strengthens the regularity theory for such nonlocal fluid-structure models and supports spectrally accurate numerics for tethered biological structures.

minor comments (2)
  1. Abstract: the phrase 'broad class of elastic energy densities' is used for the circular-arc equilibria result; a one-sentence indication of the precise assumptions (e.g., convexity or growth conditions) would make the claim self-contained.
  2. The decomposition into free-space principal part and reflected component is central; adding one or two explicit lines in the introduction showing how the boundary-symmetric kernel produces the fractional Laplacian (without hypersingular terms) would improve transparency for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our manuscript. We appreciate the recognition of the boundary-symmetric formulation's role in reducing the problem to a Dirichlet fractional Laplacian, the geometric result on circular-arc equilibria, and the well-posedness with instantaneous smoothing. The recommendation for minor revision is noted, and we will prepare the revised version accordingly.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation begins from the external boundary-symmetric formulation of Gimbutas-Greengard-Veerapaneni (distinct authors), decomposes the hydrodynamics into a free-space principal part plus regularizing reflection, and obtains the fractional Laplacian with Dirichlet conditions as the leading-order operator on the anchored filament. Stationary states (circular arcs) and local well-posedness in weighted little Hölder spaces follow from standard nonlocal PDE theory applied to this operator; neither step reduces to a self-definition, a fitted parameter renamed as prediction, nor a load-bearing self-citation. The argument remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard mathematical background and one external formulation; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption Boundary-symmetric formulation of Gimbutas, Greengard, and Veerapaneni decomposes hydrodynamic interactions into free-space and reflected parts.
    Invoked to obtain the leading-order evolution equation without hypersingular operators.
  • standard math Fractional Laplacian with homogeneous Dirichlet boundary conditions generates the stated well-posedness and smoothing properties in weighted little Hölder spaces.
    Central to the evolution, equilibria, and regularization results.

pith-pipeline@v0.9.0 · 5526 in / 1230 out tokens · 60746 ms · 2026-05-07T09:41:32.750907+00:00 · methodology

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Reference graph

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