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arxiv: 2605.23248 · v1 · pith:E572APMXnew · submitted 2026-05-22 · 🧮 math.AP · math.OC

Optimal semiconcavity with fractional modulus for Hamilton-Jacobi equations with Neumann boundary conditions

Hiroyoshi Mitake , Panrui Ni This is my paper

Pith reviewed 2026-05-25 04:02 UTC · model grok-4.3

classification 🧮 math.AP math.OC
keywords semiconcavityHamilton-Jacobi equationsNeumann boundary conditionsviscosity solutionsSkorokhod problemfractional modulusoptimal regularity
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The pith

Viscosity solutions to Hamilton-Jacobi equations with Neumann boundary conditions are globally semiconcave with an optimal fractional modulus.

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

The paper proves that viscosity solutions of the Neumann problem for Hamilton-Jacobi equations satisfy a global semiconcavity estimate with a fractional modulus of continuity. The proof proceeds by first establishing a regularity property for solutions of the associated Skorokhod problem and then transferring that property to the PDE solutions. The authors further show that the exponent appearing in the modulus cannot be improved in general. This form of semiconcavity supplies quantitative control on the second derivatives of solutions even when the boundary condition is of Neumann type.

Core claim

Viscosity solutions to the Neumann boundary value problem for Hamilton-Jacobi equations are globally semiconcave with a fractional modulus; the result is obtained by investigating a regularity property of solutions to the Skorokhod problem, and the fractional exponent is shown to be optimal.

What carries the argument

Regularity property of solutions to the Skorokhod problem, which transfers to the viscosity solutions of the Neumann problem to produce the global fractional-modulus semiconcavity.

If this is right

  • Global semiconcavity holds uniformly up to the boundary despite the Neumann condition.
  • The fractional exponent is sharp, so no modulus with a larger exponent works for all solutions.
  • The Skorokhod-problem route supplies an explicit way to obtain the modulus without relying on interior estimates alone.

Where Pith is reading between the lines

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

  • The same Skorokhod regularity may adapt to time-dependent or obstacle problems with similar boundary conditions.
  • Quantitative error bounds for numerical approximations could be derived directly from the fractional modulus.
  • Higher-dimensional or nonconvex Hamiltonians might admit the same transfer argument once the Skorokhod property is verified.

Load-bearing premise

The regularity property established for Skorokhod problem solutions is strong enough to pass to the viscosity solutions of the PDE and produce the claimed fractional modulus globally.

What would settle it

A concrete viscosity solution of a Neumann Hamilton-Jacobi problem whose second derivatives fail to satisfy the claimed fractional modulus of continuity, or for which a strictly better exponent holds in every case.

Figures

Figures reproduced from arXiv: 2605.23248 by Hiroyoshi Mitake, Panrui Ni.

Figure 1
Figure 1. Figure 1: Two optimal trajectories starting from Z. Moreover, noting that AF = AB 2 , we have α˜ = π 2 − 2 arctan 2 − h 2 . Note that the time taken by the optimal trajectory (1) to reach the line {x1 = 0} is π − θ0 + 2, and the time taken by the optimal trajectory (2) to reach the line {x1 = 0} is θ0 + 2˜α + √ r˜ 2 − 1 = θ0 + π − 4 arctan 2 − h 2 + 2 − h. We require π − θ0 + 2 = θ0 + π − 4 arctan 2 − h 2 + 2 − h, w… view at source ↗
read the original abstract

Here, we study the generalized semiconcavity property of viscosity solutions of the Neumann boundary value problem for Hamilton-Jacobi equations. In particular, we establish the global semiconcavity with a fractional modulus by investigating a regularity property of solutions to the Skorokhod problem, and show the optimality of the fractional exponent.

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

2 major / 2 minor

Summary. The manuscript establishes the global semiconcavity with a fractional modulus for viscosity solutions of Hamilton-Jacobi equations with Neumann boundary conditions. This is done by investigating a regularity property of solutions to the Skorokhod problem and proving the optimality of the fractional exponent.

Significance. If the transfer argument holds, the result supplies a sharp global regularity estimate for Neumann problems that improves on local interior semiconcavity results and has potential applications to reflected processes and boundary-value theory for first-order PDEs.

major comments (2)
  1. [Abstract, paragraph 2] Abstract, paragraph 2 and the section containing the main transfer argument: the regularity property proved for Skorokhod solutions is asserted to produce the stated fractional modulus globally for the viscosity solution, but the manuscript must explicitly verify that the boundary reflection does not degrade the exponent or destroy uniformity near the boundary; this step is load-bearing for the central claim.
  2. [Optimality section] The section proving optimality of the fractional exponent: the counter-example or construction showing sharpness must be checked to confirm it respects the Neumann condition and does not rely on interior-only data.
minor comments (2)
  1. Notation for the fractional modulus should be introduced with an explicit formula or reference to its definition early in the paper.
  2. A short comparison table or remark contrasting the new fractional modulus with classical C^{1,1} or Lipschitz moduli would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and valuable suggestions. We address each major comment below and plan to revise the manuscript to incorporate clarifications where needed.

read point-by-point responses

  1. Referee: [Abstract, paragraph 2] Abstract, paragraph 2 and the section containing the main transfer argument: the regularity property proved for Skorokhod solutions is asserted to produce the stated fractional modulus globally for the viscosity solution, but the manuscript must explicitly verify that the boundary reflection does not degrade the exponent or destroy uniformity near the boundary; this step is load-bearing for the central claim.

    Authors: We agree that an explicit verification is important for clarity. In the transfer argument section, the proof already accounts for the Skorokhod reflection by using estimates that control the boundary term without degrading the fractional exponent. However, to make this more transparent, we will add a remark or subsection explicitly bounding the reflection contribution and confirming uniformity up to the boundary. This will strengthen the presentation without altering the result. revision: yes

  2. Referee: [Optimality section] The section proving optimality of the fractional exponent: the counter-example or construction showing sharpness must be checked to confirm it respects the Neumann condition and does not rely on interior-only data.

    Authors: The counterexample in the optimality section is constructed on a domain with Neumann boundary conditions in mind. The initial data and Hamiltonian are chosen so that the solution satisfies the boundary condition, and the singularity or lack of better modulus occurs globally, including near the boundary. We will add a brief explanation in the section to explicitly verify that the construction respects the Neumann condition and is not limited to interior points. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on independent regularity result for Skorokhod problem

full rationale

The abstract states that global semiconcavity with fractional modulus is established by investigating a regularity property of solutions to the Skorokhod problem and transferring it to viscosity solutions of the Neumann problem, with optimality of the exponent shown separately. No quoted equations or steps reduce the claimed modulus to a self-definition, fitted parameter renamed as prediction, or self-citation chain. The transfer is presented as a proved implication rather than an identity by construction, and the paper is self-contained against external benchmarks with no load-bearing self-citations identified in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The central claim rests on an unstated regularity result for the Skorokhod problem whose precise assumptions are not given.

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Reference graph

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