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arxiv: 2310.12578 · v2 · pith:5MPG26EKnew · submitted 2023-10-19 · 🧮 math.AP · math.FA

Maximal L^(q)-regularity for the Laplacian on manifolds with edges

Nikolaos Roidos This is my paper

Pith reviewed 2026-05-24 06:49 UTC · model grok-4.3

classification 🧮 math.AP math.FA
keywords maximal L^q-regularityLaplacianmanifolds with edgesweighted Sobolev spacesporous medium equationR-sectorialityH^infty functional calculusconical singularities
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The pith

An R-sectoriality perturbation technique for non-commuting operators in Bochner spaces yields maximal L^q-regularity for the Laplacian on manifolds with edges.

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

The paper develops a perturbation method to prove R-sectoriality when operators do not commute and act in Bochner spaces. It combines the new technique with existing bounded H^∞ functional calculus results known for conical singularities to obtain maximal L^q-regularity of the Laplacian on edge manifolds inside appropriate weighted Sobolev spaces. The regularity statement is then used to establish short-time existence, uniqueness, and maximal regularity for solutions of the porous medium equation on these spaces, together with explicit space asymptotics near the edges that depend on local geometry.

Core claim

We introduce an R-sectoriality perturbation technique for non-commuting operators defined in Bochner spaces. Based on this and on bounded H^∞-functional calculus results for the Laplacian on manifolds with conical singularities, we show maximal L^q-regularity for the Laplacian on manifolds with edge type singularities in appropriate weighted Sobolev spaces. As an application, we consider the porous medium equation on manifolds with edges and show short time existence, uniqueness and maximal regularity for the solution. We also provide space asymptotics near the singularities in terms of the local geometry.

What carries the argument

R-sectoriality perturbation technique for non-commuting operators in Bochner spaces, which transfers sectoriality properties from conical to edge settings.

If this is right

  • Short-time existence and uniqueness hold for the porous medium equation on manifolds with edges.
  • Solutions of the porous medium equation satisfy maximal regularity in the weighted spaces.
  • Space asymptotics for solutions near the edges are controlled by the local geometry of the singularity.

Where Pith is reading between the lines

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

  • The same perturbation approach could be tested on other nonlinear parabolic equations that require maximal regularity on edge manifolds.
  • The weighted-space framework may make it possible to track how edge geometry influences long-time behavior or stability of solutions.
  • If the technique generalizes, it could supply regularity for initial-value problems on manifolds that combine several types of singularities.

Load-bearing premise

The R-sectoriality perturbation technique remains valid and combines with the conical functional calculus results without new obstructions created by the edge geometry.

What would settle it

A concrete manifold with edges on which the Laplacian fails to satisfy maximal L^q-regularity in the stated weighted Sobolev spaces, or on which the perturbation step does not preserve the required sectoriality.

read the original abstract

We introduce an $R$-sectoriality perturbation technique for non-commuting operators defined in Bochner spaces. Based on this and on bounded $H^{\infty}$-functional calculus results for the Laplacian on manifolds with conical singularities, we show maximal $L^{q}$-regularity for the Laplacian on manifolds with edge type singularities in appropriate weighted Sobolev spaces. As an application, we consider the porous medium equation on manifolds with edges and show short time existence, uniqueness and maximal regularity for the solution. We also provide space asymptotics near the singularities in terms of the local geometry.

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 / 1 minor

Summary. The paper introduces an R-sectoriality perturbation technique for non-commuting operators defined in Bochner spaces. Based on this technique together with known bounded H^∞-functional calculus results for the Laplacian on manifolds with conical singularities, the authors establish maximal L^q-regularity for the Laplacian on manifolds with edge-type singularities in appropriate weighted Sobolev spaces. As an application, they prove short-time existence, uniqueness, and maximal regularity for the porous medium equation on such manifolds and derive space asymptotics near the singularities in terms of the local geometry.

Significance. If the new perturbation argument is valid and closes without additional obstructions arising from the edge stratification, the result would extend maximal regularity theory from conical to edge singularities, providing a useful tool for linear and nonlinear PDEs on singular manifolds. The concrete application to the porous medium equation and the derivation of asymptotics add practical value. The work builds explicitly on prior H^∞-calculus results rather than re-deriving them.

major comments (1)
  1. [The perturbation technique and its application to the edge Laplacian (as described in the abstract and main claim)] The central reduction from the edge case to the conical case rests on the R-sectoriality perturbation preserving the necessary sectoriality and R-boundedness constants uniformly near the edge. Without explicit verification that the commutator estimates in the Bochner-space setting close without new obstructions from the positive-dimensional singular set, the load-bearing step of the argument remains unconfirmed.
minor comments (1)
  1. The abstract is concise but would benefit from a brief statement of the precise weight range and the geometric assumptions on the edge (e.g., the structure of the link).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for the positive assessment of the paper's significance and for identifying the central technical point requiring clarification. We respond to the major comment as follows.

read point-by-point responses

  1. Referee: The central reduction from the edge case to the conical case rests on the R-sectoriality perturbation preserving the necessary sectoriality and R-boundedness constants uniformly near the edge. Without explicit verification that the commutator estimates in the Bochner-space setting close without new obstructions from the positive-dimensional singular set, the load-bearing step of the argument remains unconfirmed.

    Authors: We thank the referee for this observation. The perturbation technique is developed in Section 3 for general non-commuting operators in Bochner spaces, with the main result being the preservation of R-sectoriality under suitable commutator conditions (see Theorem 3.2 and the estimates in (3.8)--(3.20)). In the application to manifolds with edges in Section 4, the edge Laplacian is treated as a perturbation of the conical Laplacian in the Bochner space setting over the edge. The commutator estimates are verified explicitly using the product structure and Mellin transform techniques, showing that the constants are uniform and independent of the position along the edge. The positive-dimensional nature of the singular set is accounted for by the Fourier analysis in the edge directions, reducing to conical singularities with parameters, without introducing new obstructions beyond those already controlled in the conical case. These verifications are provided in the proof of the main result, Theorem 4.5. We maintain that the argument is complete as presented. revision: no

Circularity Check

0 steps flagged

New perturbation technique independent; relies on external conical results without load-bearing self-citation loop

full rationale

The paper introduces an original R-sectoriality perturbation technique for non-commuting operators in Bochner spaces as its core contribution. This is then combined with pre-existing bounded H^∞-functional calculus results on conical singularities to extend to edge manifolds. No self-definitional reductions, fitted inputs renamed as predictions, or ansatz smuggling appear in the derivation chain. Reliance on prior conical results is standard external input rather than a self-citation chain that forces the outcome by construction. The central claim retains independent mathematical content from the new perturbation step.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; the central claim rests on the validity of a new perturbation technique and on previously established functional calculus for conical singularities. No explicit free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption Bounded H^∞-functional calculus holds for the Laplacian on manifolds with conical singularities
    Invoked explicitly as the base on which the new perturbation result is built.

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