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arxiv: 2605.18585 · v1 · pith:IYRR4BLXnew · submitted 2026-05-18 · 🧮 math.AP

Constructive solutions of the heat equation with Stieltjes derivatives

Clara Sen\'in , F. Adri\'an F. Tojo This is my paper

Pith reviewed 2026-05-20 08:35 UTC · model grok-4.3

classification 🧮 math.AP
keywords heat equationStieltjes derivativesderivatorsconstructive solutionsinitial value problemboundary conditionsmultivariable derivator
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The pith

Stieltjes calculus yields constructive solutions to the one-dimensional heat equation with fixed derivators.

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

The paper develops a constructive method to prove existence of solutions for the one-dimensional heat equation when formulated using Stieltjes derivatives with respect to two fixed derivators. It treats the associated initial-value problem together with several families of boundary conditions. The authors then define a notion of multivariable derivator suited to higher-dimensional settings and derive explicit solution formulas for relevant classes of these objects. A sympathetic reader would care because the method operates without imposing extra regularity requirements on the derivators or on the solutions themselves.

Core claim

By working in the Stieltjes calculus relative to a pair of fixed derivators, a constructive procedure establishes the existence of solutions to the heat equation; the same framework supplies explicit solutions once the derivators are allowed to be multivariable and to belong to suitable classes.

What carries the argument

The Stieltjes derivative taken with respect to a pair of fixed derivators, which replaces the ordinary derivative and permits the heat equation to be posed and solved constructively in this generalized setting.

If this is right

  • Solutions exist for the initial-value problem under the stated Stieltjes formulation.
  • Boundary conditions of several standard types can be imposed while retaining the constructive character of the solutions.
  • Multivariable derivators admit explicit solution formulas for the heat equation whenever they belong to the relevant classes identified in the paper.
  • The method applies directly to derivators that lack classical differentiability.

Where Pith is reading between the lines

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

  • The same constructive technique could be tried on other linear parabolic equations by replacing the second-order spatial term with its Stieltjes analogue.
  • Numerical verification on explicit non-absolutely-continuous derivators, such as the Cantor function paired with Lebesgue measure, would test the explicit formulas.
  • The multivariable construction suggests a route to heat flow on product spaces equipped with product Stieltjes measures.

Load-bearing premise

The Stieltjes calculus supplies a well-posed framework in which the heat equation can be stated and solved without extra regularity conditions on the derivators or on the solutions.

What would settle it

A concrete pair of derivators together with an initial condition for which the constructed candidate fails to satisfy the heat equation pointwise with respect to the Stieltjes derivative would refute the existence claim.

Figures

Figures reproduced from arXiv: 2605.18585 by Clara Sen\'in, F. Adri\'an F. Tojo.

Figure 4.1
Figure 4.1. Figure 4.1: Derivators g and h [PITH_FULL_IMAGE:figures/full_fig_p017_4_1.png] view at source ↗
read the original abstract

In this work, we investigate the one-dimensional heat equation within the framework of Stieltjes calculus. We first consider the equation associated with two fixed derivators and develop a constructive approach to establish the existence of solutions. We then study the corresponding initial value problem and incorporate several types of boundary conditions. Finally, we introduce a notion of multivariable derivator, suitable for higher-dimensional settings, and obtain explicit solutions of the heat equation for relevant classes of such derivators.

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 paper develops a constructive approach to prove existence of solutions for the one-dimensional heat equation in the Stieltjes calculus with two fixed derivators, studies the associated initial-value problem including several boundary conditions, and introduces a notion of multivariable derivator to obtain explicit solutions for relevant classes in higher-dimensional settings.

Significance. If the constructive existence arguments are complete and the Stieltjes framework is shown to be well-posed under the stated hypotheses, the work would offer a useful extension of classical parabolic theory to irregular derivators, with potential relevance to PDEs on measures or singular structures. The emphasis on explicit constructions and the multivariable extension are positive features.

major comments (2)
  1. [Section developing the constructive method for two fixed derivators] The constructive existence proof for the heat equation with two fixed derivators relies on the Stieltjes derivative of the constructed solution existing and satisfying the equation, yet the manuscript does not impose or verify extra regularity (e.g., bounded variation or continuity) on the derivators that would guarantee the difference-quotient limit exists pointwise or in an integral sense after convolution with the heat kernel.
  2. [Section introducing the multivariable derivator] The newly introduced multivariable derivator is presented without a clear reduction to the one-dimensional Stieltjes derivative or verification that the associated heat equation remains well-posed; it is unclear whether the explicit solutions satisfy the equation in the Stieltjes sense for the claimed classes.
minor comments (2)
  1. [Introduction and preliminary definitions] Notation for the Stieltjes derivative and the derivators should be introduced with explicit definitions and compared to standard references in the literature on Stieltjes integrals.
  2. [Section on the initial-value problem] The treatment of boundary conditions in the initial-value problem would benefit from a statement of the precise function spaces in which the solutions are sought.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised regarding regularity of the derivators and the multivariable extension are helpful for strengthening the rigor of the presentation. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Section developing the constructive method for two fixed derivators] The constructive existence proof for the heat equation with two fixed derivators relies on the Stieltjes derivative of the constructed solution existing and satisfying the equation, yet the manuscript does not impose or verify extra regularity (e.g., bounded variation or continuity) on the derivators that would guarantee the difference-quotient limit exists pointwise or in an integral sense after convolution with the heat kernel.

    Authors: We agree that additional regularity assumptions are needed to rigorously establish the existence of the Stieltjes derivative. In the revised manuscript we will introduce the hypotheses that the two fixed derivators are continuous and of bounded variation. Under these conditions we will verify that the convolution of the initial data with the heat kernel produces a function whose difference quotients converge pointwise (almost everywhere) to the required Stieltjes derivative, thereby confirming that the constructed solution satisfies the equation. This verification will be added explicitly to the section on the constructive method. revision: yes

  2. Referee: [Section introducing the multivariable derivator] The newly introduced multivariable derivator is presented without a clear reduction to the one-dimensional Stieltjes derivative or verification that the associated heat equation remains well-posed; it is unclear whether the explicit solutions satisfy the equation in the Stieltjes sense for the claimed classes.

    Authors: We acknowledge the need for greater clarity on this point. In the revision we will add a subsection that first reduces the multivariable derivator to a collection of one-dimensional Stieltjes derivatives along each coordinate. For the specific classes of derivators treated in the paper (those admitting a product structure with suitable monotonicity), we will then prove directly that the explicit solutions satisfy the heat equation in the Stieltjes sense. A short paragraph on well-posedness within this framework will also be included. revision: yes

Circularity Check

0 steps flagged

Constructive existence proofs for Stieltjes heat equation remain self-contained

full rationale

The paper develops explicit constructive solutions and new definitions for multivariable derivators in the Stieltjes setting. No load-bearing step reduces a claimed prediction or existence result to a fitted parameter or prior self-citation by construction. The framework assumptions are stated upfront and the derivations proceed via direct construction rather than circular redefinition or imported uniqueness theorems from the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work rests on the applicability of Stieltjes calculus to the heat equation and introduces one new concept for higher dimensions; no free parameters or fitted values are mentioned.

axioms (1)
  • domain assumption Stieltjes derivatives exist and allow formulation of the heat equation for the chosen derivators.
    Central to the entire investigation; invoked when setting up the equation with fixed derivators.
invented entities (1)
  • multivariable derivator no independent evidence
    purpose: To extend the one-dimensional Stieltjes framework to higher-dimensional heat equations.
    Newly defined in the paper for relevant classes that admit explicit solutions.

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