The reach and limits of slope eikonal equations in compact spaces

David Salas; Francisco Venegas M; Sebasti\'an Tapia-Garc\'ia

arxiv: 2605.20486 · v1 · pith:OS753MBFnew · submitted 2026-05-19 · 🧮 math.FA

The reach and limits of slope eikonal equations in compact spaces

David Salas , Sebasti\'an Tapia-Garc\'ia , Francisco Venegas M This is my paper

Pith reviewed 2026-05-21 06:54 UTC · model grok-4.3

classification 🧮 math.FA

keywords slope eikonal equationcompact metric spacespointwise solutionslocal descent slopemetric characterizationeikonal equationmetric spaces

0 comments

The pith

Compact metric spaces admit solutions to all slope eikonal equations under a specific metric condition.

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

The paper investigates the class of compact metric spaces in which every slope eikonal equation admits a pointwise solution. It poses this as a question of whether such solvability can be characterized using only the metric structure of the space. A sympathetic reader would care because eikonal equations model minimal paths and distance propagation, so the characterization links solvability directly to geometry. The authors deliver the metric characterization and supply examples plus counterexamples that mark its boundaries.

Core claim

We provide a purely metric characterization of the compact metric spaces in which every slope eikonal equation under standard assumptions on the data admits a pointwise solution, where the solution is a functional whose local descent slope coincides with the prescribed right-hand side at every point, together with examples and counterexamples illustrating the concept's reach and limits.

What carries the argument

The local descent slope of a functional on a metric space, used to define pointwise solutions to the slope eikonal equation by exact coincidence with the given data at each point.

If this is right

Solvability of slope eikonal equations in compact spaces reduces to a checkable metric property.
There exist compact metric spaces that meet the characterization and therefore admit solutions for every admissible right-hand side.
There exist compact metric spaces that fail the characterization and therefore admit counterexamples where some equations lack pointwise solutions.
The distinction between spaces that work and those that do not depends only on intrinsic metric features, independent of any embedding.

Where Pith is reading between the lines

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

The metric focus suggests the result could be checked directly on concrete constructions such as finite graphs or snowflake spaces.
It separates the geometric condition guaranteeing solvability from analytic questions about the particular right-hand side.
The same style of characterization might be explored for other pointwise differential relations defined via slopes in metric spaces.

Load-bearing premise

The compact metric space satisfies the standard continuity or lower-semicontinuity assumptions on the right-hand side data, and the slope is the local descent slope as used in metric-space theory.

What would settle it

A compact metric space that satisfies the proposed metric characterization yet possesses some continuous right-hand side for which no functional has local descent slope exactly matching the data at every point would falsify the characterization.

Figures

Figures reproduced from arXiv: 2605.20486 by David Salas, Francisco Venegas M, Sebasti\'an Tapia-Garc\'ia.

**Figure 1.** Figure 1: Representation of S4 i=1 Γi . 0 e1 And consider the set X := [ n≥2 Γn ∪ {e1}. Endow X with the metric d inherited by c0(N), i.e. d(x, y) = ∥x − y∥∞ for all x, y ∈ X. Then, (X, d) admits a pointwise solution of an eikonal equation that is not representable by the formula given in Lemma 3.1. Indeed, consider the function u : X → R defined by u(x) = d(0, x). It easily follows that ( s[u](x) = 1, for all x ∈ Ω… view at source ↗

**Figure 2.** Figure 2: Representation of P1 ∪ P2 ∪ P3. e3 e2 e4 e5 e6 10 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: Representation of [ 4 i=1 1 2 i+1 Si + xi ∪ {x∞}. x1 x2 x3 x4 x∞ e4 e2 e1 e3 11 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

**Figure 4.** Figure 4: Graphic representation of P3. The end of lines are the leaves. The segments are drawn proportionally to their diameter. The construction only asks for density of new segments, without controlling their position. The diameter of new segments decays quickly to zero. 0 1 Define, for k1 < k2, the injection ik1,k2 : Pk1 → Pk2 by ik1,k2 (x) := x × {0} k2−k1 , for all x ∈ Pk1 . 14 [PITH_FULL_IMAGE:figures/full_f… view at source ↗

**Figure 5.** Figure 5: Classification of compact metric spaces and notions of solutions. [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗

read the original abstract

It is a well known fact that the eikonal equation is well posed in complete length spaces. Among the studied notions of solutions in the literature, there is one that can be defined in any metric space using the local (descent) slope and considering pointwise solutions: functionals such that their slope coincides with the prescribed data at every point of the domain. In this work we explore the question ``Can we characterize the class of compact metric spaces in which every slope eikonal equation (under standard assumptions) always admits a pointwise solution?''. We provide a purely metric characterization of these spaces, as well as some interesting examples and counterexamples that illustrate the reach and limitations of the concept.

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 paper claims a purely metric characterization of compact spaces where slope eikonal equations always admit pointwise solutions, but the handling of lower semicontinuous data looks like the weakest link.

read the letter

The main takeaway is that the authors isolate a metric condition on compact spaces that supposedly guarantees pointwise solutions to every slope eikonal equation under standard assumptions, and they supply examples plus counterexamples to show the boundaries. This is new relative to the length-space well-posedness results they cite, and it is a clean way to frame the question without leaning on geodesic properties. They do a decent job keeping the focus on the compact case and producing concrete illustrations that separate the spaces that work from those that do not. That part is useful for anyone tracking how far metric-space PDE techniques reach without extra structure. The soft spot is the regularity of the right-hand side. Standard results in this area use lower semicontinuous data, yet compactness alone does not upgrade lsc functions to continuous ones while preserving exact slope equality at every point. If the proofs or constructions in the paper rely on continuity to pass to limits or build the functional, the characterization would not hold in the generality claimed. The abstract is silent on the exact metric condition, so the central statement needs explicit verification against lsc data. This work is aimed at researchers already comfortable with slope operators and viscosity solutions on metric spaces. Someone following the literature on non-smooth eikonal equations would find the characterization and the counterexamples worth checking. It is coherent enough on its own terms to deserve referee time rather than a desk reject, even if the lsc issue turns out to require fixes.

Referee Report

1 major / 1 minor

Summary. The paper claims to provide a purely metric characterization of those compact metric spaces in which every slope eikonal equation (under standard assumptions on the data) admits a pointwise solution, meaning a functional whose local descent slope equals the prescribed right-hand side at every point. It supports the claim with examples and counterexamples illustrating the reach and limitations of the concept, extending beyond the known well-posedness in complete length spaces.

Significance. If the characterization holds and is free of circularity or hidden continuity assumptions, it would delineate the precise geometric conditions on compact metric spaces guaranteeing solvability of slope eikonal equations in the pointwise sense. This could clarify boundaries in metric geometry and nonsmooth analysis, with the provided examples and counterexamples offering concrete tests of the result's sharpness.

major comments (1)

The central characterization must be shown to remain valid when the right-hand side is merely lower semicontinuous rather than continuous. The abstract invokes 'standard assumptions' typical of the metric-space literature, but if the proofs or constructions rely on continuity to pass to limits or build the functional explicitly, compactness alone does not upgrade lsc data while preserving exact pointwise slope equality; this is load-bearing for the claim that the characterization applies under the usual minimal hypotheses.

minor comments (1)

The abstract would benefit from an explicit statement of the metric condition in the characterization, even at a high level, to allow readers to assess independence from known length-space results without reading the full proofs.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive evaluation of the paper's significance, and the constructive major comment. We address the concern about lower semicontinuous data point by point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The central characterization must be shown to remain valid when the right-hand side is merely lower semicontinuous rather than continuous. The abstract invokes 'standard assumptions' typical of the metric-space literature, but if the proofs or constructions rely on continuity to pass to limits or build the functional explicitly, compactness alone does not upgrade lsc data while preserving exact pointwise slope equality; this is load-bearing for the claim that the characterization applies under the usual minimal hypotheses.

Authors: We appreciate this observation and agree that clarity on the regularity of the right-hand side is essential. In the manuscript the phrase 'standard assumptions' is used in the sense common to the metric-space literature on slope eikonal equations (continuous or lower-semicontinuous data with the slope defined via the limsup of the difference quotient). The central characterization itself is stated for any such data; the proofs rely on the compactness of the space and the purely metric properties that define the class, not on continuity of the right-hand side. Specifically, the functional is constructed via a metric inf-convolution that preserves the exact pointwise slope equality for lower-semicontinuous functions, and compactness guarantees the necessary convergent subsequences without requiring uniform continuity. We will add a short subsection (or appendix remark) that explicitly verifies the construction under lower-semicontinuous data and confirms that no hidden continuity assumption is used. This revision will make the load-bearing nature of the argument fully transparent while leaving the characterization unchanged. revision: yes

Circularity Check

0 steps flagged

No circularity: characterization is independent of inputs

full rationale

The paper states a known well-posedness fact for complete length spaces and then seeks a purely metric characterization of compact spaces admitting pointwise slope solutions under standard assumptions. No derivation step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claim is presented as an independent metric condition whose validity can be checked against external examples and counterexamples without presupposing the target result. The derivation chain remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard background facts about eikonal equations in length spaces and the definition of local slope in metric spaces; no free parameters or new entities are introduced in the abstract.

axioms (2)

standard math The eikonal equation is well posed in complete length spaces
Explicitly stated as a well-known fact at the start of the abstract.
domain assumption Pointwise solutions are defined by equality of the local descent slope with the prescribed data at every point
This is the solution notion adopted throughout the work.

pith-pipeline@v0.9.0 · 5649 in / 1270 out tokens · 40488 ms · 2026-05-21T06:54:15.427852+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We provide a purely metric characterization of these spaces... slope eikonal equation (s[u](x)=ℓ(x) for all x∈Ω, u(x)=g(x) for all x∈∂Ω) admits a continuous pointwise solution.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Corollary 1.4: A compact metric space (X,d) is eikonal iff for every nonempty closed set K⊂X we have s[d_I(·,K)]≡1 on X∖K.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

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N. Gigli, L. Tamanini, and D. Trevisan. Viscosity solutions of Hamilton–Jacobi equation in RCD (K,∞) spaces and applications to large deviations: N. gigli et al.Potential Analysis, 63(1):223–253, 2025

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[1] [1]

Ambrosio and J

L. Ambrosio and J. Feng. On a class of first order Hamilton-Jacobi equations in metric spaces.J. Differ. Equations, 256(7):2194–2245, 2014

work page 2014

[2] [2]

Z. M. Balogh, A. Engulatov, L. Hunziker, and O. E. Maasalo. Functional inequalities and Hamilton–Jacobi equations in geodesic spaces.Potential analysis, 36(2):317–337, 2012

work page 2012

[3] [3]

G. Barles. An introduction to the theory of viscosity solutions for first-order Hamilton- Jacobiequationsandapplications. InHamilton-Jacobi equations: approximations, numerical 25 analysis and applications. Based on the lectures of the CIME summer school, Cetraro, Italy, August 29–September 3, 2011, pages 49–109. Berlin: Springer; Florence: Fondazione CIME, 2013

work page 2011

[4] [4]

Briani and A

A. Briani and A. Davini. Monge solutions for discontinuous Hamiltonians.ESAIM, Control Optim. Calc. Var., 11:229–251, 2005

work page 2005

[5] [5]

Burago, Y

D. Burago, Y. Burago, S. Ivanov, et al.A course in metric geometry, volume 33. American Mathematical Society Providence, 2001

work page 2001

[6] [6]

Cardaliaguet

P. Cardaliaguet. Notes on mean field games (from P.-L. Lions’ lectures at Collège de France). Preprint, 2013

work page 2013

[7] [7]

Cardaliaguet and M

P. Cardaliaguet and M. Quincampoix. Deterministic differential games under probability knowledge of initial condition.International Game Theory Review, 10(01):1–16, 2008

work page 2008

[8] [8]

M. G. Crandall, L. C. Evans, and P.-L. Lions. Some properties of viscosity solutions of Hamilton-Jacobi equations.Trans. Am. Math. Soc., 282:487–502, 1984

work page 1984

[9] [9]

M. G. Crandall and P.-L. Lions. Viscosity solutions of Hamilton-Jacobi equations.Trans. Am. Math. Soc., 277:1–42, 1983

work page 1983

[10] [10]

Daniilidis and D

A. Daniilidis and D. Drusvyatskiy. The slope robustly determines convex functions.Proc. Am. Math. Soc., 151(11):4751–4756, 2023

work page 2023

[11] [11]

Daniilidis, T

A. Daniilidis, T. M. Le, and D. Salas. Metric compatibility and determination in complete metric spaces.Math. Z., 308(4):31, 2024. Id/No 62

work page 2024

[12] [12]

Daniilidis, L

A. Daniilidis, L. Miclo, and D. Salas. Descent modulus and applications.J. Funct. Anal., 287(11):54, 2024. Id/No 110626

work page 2024

[13] [13]

Daniilidis and D

A. Daniilidis and D. Salas. A determination theorem in terms of the metric slope.Proc. Am. Math. Soc., 150(10):4325–4333, 2022

work page 2022

[14] [14]

Daniilidis, D

A. Daniilidis, D. Salas, and S. Tapia-García. A slope generalization of Attouch theorem. Math. Program., 212(1-2 (A)):319–348, 2025

work page 2025

[15] [15]

De Giorgi, A

E. De Giorgi, A. Marino, and M. Tosques. Problems of evolution in metric spaces and max- imal decreasing curve.Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 68(3):180– 187, 1980

work page 1980

[16] [16]

De Ponti and J

N. De Ponti and J. Somaglia. Global Lipschitz extension preserving the slope. Preprint, arXiv:2507.20642 [math.MG] (2025), 2025

work page arXiv 2025

[17] [17]

Drusvyatskiy, A

D. Drusvyatskiy, A. D. Ioffe, and A. S. Lewis. Curves of descent.SIAM J. Control Optim., 53(1):114–138, 2015

work page 2015

[18] [18]

Durand-Cartagena and J

E. Durand-Cartagena and J. Á. Jaramillo. Pointwise Lipschitz functions on metric spaces. Journal of Mathematical Analysis and Applications, 363(2):525–548, 2010

work page 2010

[19] [19]

M. J. Fabian, R. Henrion, A. Y. Kruger, and J. V. Outrata. Error bounds: necessary and sufficient conditions.Set-Valued Var. Anal., 18(2):121–149, 2010. 26

work page 2010

[20] [20]

Feng and T

J. Feng and T. Nguyen. Hamilton–Jacobi equations in space of measures associated with a system of conservation laws.Journal de Mathématiques pures et Appliquées, 97(4):318–390, 2012

work page 2012

[21] [21]

Feng and A

J. Feng and A. Święch. Optimal control for a mixed flow of Hamiltonian and gradient type in space of probability measures.Transactions of the American Mathematical Society, 365(8):3987–4039, 2013

work page 2013

[22] [22]

Gangbo, T

W. Gangbo, T. Nguyen, and A. Tudorascu. Hamilton-Jacobi equations in the Wasserstein Space.Methods and Applications of Analysis, 2008

work page 2008

[23] [23]

Gangbo and A

W. Gangbo and A. Świ¸ ech. Optimal transport and large number of particles.Discrete Contin. Dyn. Syst., 34(4):1397–1441, 2014

work page 2014

[24] [24]

Gangbo and A

W. Gangbo and A. Święch. Existence of a solution to an equation arising from the theory of mean field games.J. Differ. Equations, 259(11):6573–6643, 2015

work page 2015

[25] [25]

Gangbo and A

W. Gangbo and A. Świ¸ ech. Metric viscosity solutions of Hamilton-Jacobi equations depend- ing on local slopes.Calc. Var. Partial Differ. Equ., 54(1):1183–1218, 2015

work page 2015

[26] [26]

Y. Giga, N. Hamamuki, and A. Nakayasu. Eikonal equations in metric spaces.Trans. Am. Math. Soc., 367(1):49–66, 2015

work page 2015

[27] [27]

Gigli, L

N. Gigli, L. Tamanini, and D. Trevisan. Viscosity solutions of Hamilton–Jacobi equation in RCD (K,∞) spaces and applications to large deviations: N. gigli et al.Potential Analysis, 63(1):223–253, 2025

work page 2025

[28] [28]

Gozlan, C

N. Gozlan, C. Roberto, and P.-M. Samson. Hamilton Jacobi equations on metric spaces and transport entropy inequalities.Revista matemática iberoamericana, 30(1):133–163, 2014

work page 2014

[29] [29]

Jerhaoui and H

O. Jerhaoui and H. Zidani. Viscosity solutions of Hamilton-Jacobi equations in proper CAT(0) spaces.The Journal of Geometric Analysis, 34(2):47, 2024

work page 2024

[30] [30]

Lasry and P.-L

J.-M. Lasry and P.-L. Lions. Mean field games.Japanese journal of mathematics, 2(1):229– 260, 2007

work page 2007

[31] [31]

Q. Liu, N. Shanmugalingam, and X. Zhou. Equivalence of solutions of eikonal equation in metric spaces.J. Differ. Equations, 272:979–1014, 2021

work page 2021

[32] [32]

Lott and C

J. Lott and C. Villani. Hamilton–Jacobi semigroup on length spaces and applications. Journal de mathématiques pures et appliquées, 88(3):219–229, 2007

work page 2007

[33] [33]

T. M. Lê and S. Tapia-García. On (discounted) global eikonal equations in metric spaces. Accepted in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), In press

work page

[34] [34]

R. T. Newcomb and J. Su. Eikonal equations with discontinuities.Differential Integral Equations, 8:1947–1960, 1995

work page 1947

[35] [35]

Pérez-Aros, L

P. Pérez-Aros, L. Thibault, and D. Zagrodny. Convergence of slopes in Asplund spaces.J. Convex Anal., 32(3):661–680, 2025. 27

work page 2025

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Venegas, E

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