The omnidirectional trace in H 1 ($\Omega$)

David Maltese (LAMA); Lucas Oger (LAMA); Robert Eymard (LAMA); Thierry Gallou\"et (AMU)

arxiv: 2605.18056 · v1 · pith:EYOQFYNYnew · submitted 2026-05-18 · 🧮 math.AP

The omnidirectional trace in H 1 (Ω)

Robert Eymard (LAMA) , Thierry Gallou\"et (AMU) , David Maltese (LAMA) , Lucas Oger (LAMA) This is my paper

Pith reviewed 2026-05-20 09:26 UTC · model grok-4.3

classification 🧮 math.AP MSC 46E35

keywords omnidirectional traceSobolev space H1directional derivativeboundary traceintegration by partsvariational problemsirregular domainsclosed subspace

0 comments

The pith

The omnidirectional trace defines a closed subset of H¹(Ω) whose elements possess a consistent boundary value matching all directional traces.

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

All functions in L²(Ω) whose directional derivatives lie in L²(Ω) possess directional traces on the boundary of any bounded open domain, with no regularity required on the boundary. This fact allows definition of the omnidirectional trace as the single boundary function that agrees almost everywhere with the directional trace for every direction according to the directional measure. The set H¹_tr(Ω) of all such elements in H¹(Ω) is closed and always contains the H¹-closure of C₀(Ω) ∩ H¹(Ω), coinciding exactly with that closure in one dimension. The resulting trace satisfies an integration-by-parts formula that pairs values at opposite boundary points and supports variational problems that incorporate explicit boundary values.

Core claim

The paper introduces the omnidirectional trace for elements u of H¹(Ω) that admit a single boundary function g such that, for every direction, the directional trace of u equals g almost everywhere with respect to the directional measure. The collection of all such u, denoted H¹_tr(Ω), is proved closed in H¹(Ω). It always contains the H¹-closure of C₀(Ω) ∩ H¹(Ω) and equals this closure when the dimension is one. The omnidirectional trace satisfies an integration-by-parts identity that combines the values of the trace at opposite boundary points.

What carries the argument

The omnidirectional trace, defined as the single boundary function that equals the directional trace almost everywhere with respect to the directional measure for every direction.

If this is right

The set H¹_tr(Ω) is closed in H¹(Ω).
H¹_tr(Ω) always contains the H¹-closure of C₀(Ω) ∩ H¹(Ω).
In one dimension H¹_tr(Ω) equals the H¹-closure of C₀(Ω) ∩ H¹(Ω).
The omnidirectional trace satisfies an integration-by-parts formula combining values at opposite boundary points.
Variational problems involving explicit boundary values on the domain can be solved using this trace.

Where Pith is reading between the lines

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

In higher dimensions the inclusion of the closure can be strict, so some functions without compact support still possess well-defined omnidirectional traces.
The construction may apply to other function spaces or to domains that are not open or bounded.
Numerical methods for boundary-value problems on non-smooth domains could employ the omnidirectional trace to enforce conditions weakly.
The integration-by-parts identity might yield new weak formulations for elliptic or parabolic equations on arbitrary domains.

Load-bearing premise

There exists a single function on the boundary that coincides almost everywhere with the directional trace for every direction with respect to the directional measure.

What would settle it

A sequence of functions each possessing an omnidirectional trace that converges in the H¹ norm to a limit whose directional traces disagree on a positive-measure set of directions would show the set is not closed.

Figures

Figures reproduced from arXiv: 2605.18056 by David Maltese (LAMA), Lucas Oger (LAMA), Robert Eymard (LAMA), Thierry Gallou\"et (AMU).

**Figure 1.** Figure 1: The domains ΩC and Ω(1) C . Proof. Note that Dn = 2 n[ +1−1 m=2n I (n) m with I (n) m := ]am − yn, bm + yn[, where the length of I (n) m is equal to 2 · 3 −n, hence the measure of Dn is equal to 2(2/3)n and we also have νn(u) = 2−n 2 nX+1−1 m=2n 1 |I (n) m | Z I (n) m u(x, yn) dx. In order to prove the convergence of the sequence (νn)n in H1 (Ω)′ , we compute νn+1(u) − νn(u). We denote by J (n) 2m :=]am − … view at source ↗

**Figure 2.** Figure 2: Domain Ω with H1 0 (Ω) 6= H1 tr,0 (Ω). We consider the same example as [3, Example 5.3]. We let d = 2, and Ω = B(( 1 2 , 0), 2) \ (C1/3 × {0}) (see [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

**Figure 3.** Figure 3: The domain Ω with He 1 (Ω) 6= H1 tr(Ω). Proof. We let d = 2. We define the open set Ω (see [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

**Figure 4.** Figure 4: Cuspidal domain Ω. Hence, for a.e. x2 ∈]0, 1[, µθ(x 3 2 , x2) = 2x 3 p 2 1 + 9x 4 2 H1 (x 3 2 , x2), and µθ is equal to 0 on the remaining of the boundary (in black in [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗

**Figure 5.** Figure 5: Example 4.8. Left: ρ = 1 4 . Right: ρ = 1 3 . Then Ω is connected and H1 tr(Ω) = H1 (Ω), applying Lemma 4.9 below with Ω0 = ]0, 1[×] − 1, 0[ and Ωm =]cm, dm[×]0, 1[. The next lemma gives a simple sufficient condition leading to H1 tr(Ω) = H1 (Ω). Lemma 4.9. Let (Ωi)i∈I be a countable family of open subsets of Ω, such that • for all i ∈ I, H1 tr(Ωi) = H1 (Ωi), • for all θ ∈ S d−1 , ∂θΩ \ S i∈I ∂θΩi is µθ-ne… view at source ↗

read the original abstract

We first prove that all the functions in L 2 whose directional derivative is in L 2 have a directional trace on the boundary of any open bounded domain, without assumptions on its regularity. This enables us to define the omnidirectional trace of the elements of the Sobolev space H 1 ($\Omega$) for which there exists a function on the boundary that is almost everywhere equal, with respect to the directional measure, to the directional trace, regardless of the direction. The set of all these elements of H 1 ($\Omega$), denoted by H 1 tr ($\Omega$), is shown to be closed, and to always contain the closure in H 1 ($\Omega$) of the set C 0 ($\Omega$)$\cap$ H 1 ($\Omega$) (it is always equal to this set in the 1D case, and can be strictly greater in higher dimensions). The omnidirectional trace always satisfies an integration-by-parts formula, which combines the values of the trace on opposite points of the boundary. Examples show that this notion enables the resolution of variational problems involving the values at the boundary of the domain.

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 defines an omnidirectional trace for H1 on arbitrary bounded domains by requiring one boundary function to match all directional traces, but the consistency step across directions is the part that needs verification.

read the letter

This paper introduces the omnidirectional trace for elements of H1(Ω) on arbitrary bounded open sets. The idea is to first establish directional traces for L2 functions whose directional derivative is also in L2, and then look for functions in H1 where there is one boundary function that matches the directional trace almost everywhere no matter which direction you pick. What stands out is the proof that these directional traces exist without any assumption on the boundary. That part seems to build on standard Sobolev definitions and measure theory, and it opens the door to traces on very rough domains like those with cusps or fractal boundaries. The integration-by-parts formula that pairs values at opposite points is a concrete payoff, and the examples indicate it can be used to set up variational problems that incorporate boundary values directly. The potential issue is whether that single boundary function always exists. The definition of the set H1_tr(Ω) requires that such a g exists and coincides with every directional trace with respect to the corresponding measure. If the directional measures for different directions have no overlap or if the traces disagree on a set of positive measure, then no such g would exist. The paper claims this works for any bounded open Ω, but without seeing the details of how they glue the traces together, it's not clear if the argument covers cases where the measures are singular. The closedness of H1_tr and its relation to the closure of C0(Ω) ∩ H1(Ω) depend on this step, so that's where the proofs need to be solid. In one dimension it reduces to the usual thing, which is reassuring, but in higher dimensions it can be strictly larger, which is the interesting part. This work is for researchers focused on Sobolev spaces and weak formulations of PDEs on irregular geometries. Someone looking for new tools to handle boundary conditions without smoothness assumptions would find it relevant. It deserves a serious referee because the directional trace result is a genuine extension, and the omnidirectional version could be useful if the consistency holds. I recommend sending it out for peer review to get the technical details checked, particularly around the existence of the common trace function.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that L² functions with L² directional derivatives admit directional traces on the boundary of any bounded open set Ω, without boundary regularity assumptions. It defines the omnidirectional trace on the subset H¹_tr(Ω) of H¹(Ω) consisting of those functions for which a single boundary function g exists that coincides almost everywhere with the directional trace for every direction with respect to the associated directional measure. The set H¹_tr(Ω) is shown to be closed in H¹(Ω), to contain the H¹-closure of C₀(Ω) ∩ H¹(Ω) (with equality in one dimension and possible strict inclusion in higher dimensions), and to satisfy an integration-by-parts formula that pairs trace values at opposite boundary points. Examples illustrate applications to variational problems with boundary values.

Significance. If the results hold, the construction supplies a trace-like operator on domains too irregular for the classical trace theorem, while preserving key functional-analytic properties such as closedness and an integration-by-parts identity. The observation that H¹_tr(Ω) can properly contain the closure of compactly supported functions in dimensions greater than one identifies a new class of functions with consistent directional boundary values, which may enlarge the scope of variational formulations on non-Lipschitz or fractal domains.

major comments (1)

[Integration-by-parts formula] Integration-by-parts formula (abstract and the section establishing the formula): the claim that the omnidirectional trace combines values at 'opposite points of the boundary' for arbitrary bounded open Ω is load-bearing for the utility of the formula. On domains with inward cusps or fractal boundaries a line parallel to a fixed direction may intersect ∂Ω in more than two points or in a positive-measure set; it is unclear how the pairing of opposite points is defined or whether the directional trace remains a well-defined point value at those intersections. A precise statement of the pairing and a verification that the formula continues to hold under the paper's minimal assumptions are required.

minor comments (2)

[Definition of omnidirectional trace] The definition of the directional measure and the precise sense in which 'almost everywhere with respect to the directional measure' is understood should be stated explicitly before the definition of H¹_tr(Ω) to make the consistency condition across directions unambiguous.
Notation: the symbol H¹_tr(Ω) is introduced without an immediate comparison to the standard trace space H^{1/2}(∂Ω) or to H₀¹(Ω); a short remark clarifying the relationship would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address the major comment regarding the integration-by-parts formula below, and we will incorporate clarifications in the revised version to enhance the precision of our statements.

read point-by-point responses

Referee: [Integration-by-parts formula] Integration-by-parts formula (abstract and the section establishing the formula): the claim that the omnidirectional trace combines values at 'opposite points of the boundary' for arbitrary bounded open Ω is load-bearing for the utility of the formula. On domains with inward cusps or fractal boundaries a line parallel to a fixed direction may intersect ∂Ω in more than two points or in a positive-measure set; it is unclear how the pairing of opposite points is defined or whether the directional trace remains a well-defined point value at those intersections. A precise statement of the pairing and a verification that the formula continues to hold under the paper's minimal assumptions are required.

Authors: We appreciate the referee's observation concerning the integration-by-parts formula on general domains. In our construction, the directional traces are defined with respect to the directional measures on the boundary, and the integration-by-parts identity is obtained by applying the one-dimensional fundamental theorem of calculus along almost every line in a given direction. For each such line, the intersection with Ω is a countable union of open intervals, and the boundary terms arise from the endpoints of these intervals, which we refer to as 'opposite points' in the sense of the entry and exit points for each segment. The omnidirectional trace is a single function g on ∂Ω that matches these directional traces a.e. with respect to the respective measures. We acknowledge that the manuscript's presentation of this pairing could be made more explicit, particularly for domains where a line may intersect the boundary in multiple points or sets of positive measure. In the revised manuscript, we will provide a detailed definition of the pairing, clarifying that the formula accounts for all such paired contributions along each line, and include a proof sketch or reference to the one-dimensional case that verifies the identity holds under our assumptions of Ω being bounded and open. This revision will not alter the main results but will improve readability and address the referee's concern directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper first proves existence of directional traces on arbitrary bounded open domains for L² functions with directional derivative in L², using direct arguments. It then defines H¹_tr(Ω) as the subset of H¹(Ω) for which a single boundary function g exists that coincides a.e. with each directional trace w.r.t. the corresponding measure. Closedness of this set, its containment of the H¹-closure of C₀(Ω) ∩ H¹(Ω), equality in 1D, and the integration-by-parts formula are all established by subsequent proofs relying on standard Sobolev and measure-theoretic tools. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear; the central claims do not reduce to their inputs by construction and remain independent of the definition step.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claims rest on the standard definition of H1(Ω) as functions with square-integrable weak derivatives, the existence of directional derivatives in L2, and the notion of directional measure on the boundary; no free parameters or new physical entities are introduced.

axioms (2)

standard math H1(Ω) consists of L2 functions whose distributional derivatives are in L2
Invoked in the opening sentence of the abstract when discussing directional derivatives.
domain assumption Directional traces exist for L2 functions with directional derivative in L2 on any bounded open set
Stated as the first result proved in the abstract; used to define the omnidirectional trace.

invented entities (1)

omnidirectional trace no independent evidence
purpose: A single boundary function that equals the directional trace for every direction with respect to the directional measure
Newly defined object that is the central contribution; no independent existence proof outside the paper is given in the abstract.

pith-pipeline@v0.9.0 · 5742 in / 1471 out tokens · 30926 ms · 2026-05-20T09:26:50.972211+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Arendt, I

W. Arendt, I. Chalendar, and R. Eymard. Extensions of dissipat ive and symmetric operators. Semigroup Forum, 106(2):339–367, 2023

work page 2023
[2]

Arendt and A

W. Arendt and A. F. M. ter Elst. The Dirichlet-to-Neumann opera tor on rough domains. J. Diﬀerential Equations, 251(8):2100–2124, 2011

work page 2011
[3]

Arendt, A

W. Arendt, A. F. M. ter Elst, and M. Sauter. The Perron solution for elliptic equations without the maximum principle. Math. Ann., 390(1):763–810, 2024

work page 2024
[4]

Arendt and M

W. Arendt and M. Warma. The Laplacian with Robin boundary condit ions on arbitrary domains. Potential Anal., 19(4):341–363, 2003

work page 2003
[5]

Boccardo, T

L. Boccardo, T. Gallou¨ et, and L. Orsina. Existence and uniquen ess of entropy solutions for nonlinear elliptic equations with measure data. Ann. Inst. H. Poincar´ eC Anal. Non Lin´ eaire, 13(5):539–551, 1996

work page 1996
[6]

Br´ ezis

H. Br´ ezis. Analyse fonctionnelle. Th´ eorie et applications. Paris: Masson, 199 4

work page
[7]

Bucur and A

D. Bucur and A. Giacomini. A variational approach to the isoperime tric inequality for the Robin eigenvalue problem. Arch. Ration. Mech. Anal., 198(3):927–961, 2010

work page 2010
[8]

D. L. Cohn. Measure Theory. Birkh¨ auser Advanced Texts Basler Lehrb¨ ucher. Birkh¨ auser New York, NY, 2013

work page 2013
[9]

B. E. J. Dahlberg. Estimates of harmonic measure. Arch. Rational Mech. Anal., 65(3):275–288, 1977

work page 1977
[10]

D. E. Edmunds and W. D. Evans. Spectral theory and diﬀerential operators. Oxford Mathemati- cal Monographs. The Clarendon Press, Oxford University Press, New York, 1987. Oxford Science Publications

work page 1987
[11]

L. C. Evans and R. F. Gariepy. Measure theory and ﬁne properties of functions. Textbooks in Mathematics. CRC Press, Boca Raton, FL, revised edition, 2015

work page 2015
[12]

Falconer

K. Falconer. Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, 1990

work page 1990
[13]

Gagliardo

E. Gagliardo. Caratterizzazioni delle tracce sulla frontiera rela tive ad alcune classi di funzioni in n variabili. Rendiconti del Seminario Matematico della Universit ˜A di Padova, 27:284–305, 1957

work page 1957
[14]

Gallou¨ et and R

T. Gallou¨ et and R. Herbin.Mesure, int´ egration,probabilit´ es. Collection R´ ef´ erences sciences. Ellipses, Paris, 2022

work page 2022
[15]

Kazaniecki and M

K. Kazaniecki and M. Wojciechowski. Trace Operator on von Ko ch’s Snowﬂake. Potential Anal., 61(4):659–684, 2024

work page 2024
[16]

J. L. Lewis. Approximation of Sobolev functions in Jordan domain s. Ark. Mat., 25(2):255–264, 1987

work page 1987
[17]

V. Maz’ya. Sobolev Spaces with Applications to Elliptic Partial Diﬀerential Equations, volume 342 of A series of comprehensive studies in mathematics. Springer, 1985

work page 1985
[18]

W. F. Pfeﬀer. The Gauss-Green theorem. Advances in Mathematics, 87(1):93–147, 1991

work page 1991
[19]

Poborchi

S. Poborchi. Sobolev spaces for domains with cusps. In The Maz’ya anniversary collection, Vol. 1 (Rostock, 1998), volume 109 of Oper. Theory Adv. Appl., pages 175–185. Birkh¨ auser, Basel, 1999

work page 1998
[20]

M. Sauter. Uniqueness of the approximative trace. Indiana Univ. Math. J., 69(1):171–204, 2020. 29

work page 2020
[21]

Shvartsman

P. Shvartsman. On the boundary values of Sobolev W 1 p -functions. Adv. Math., 225(4):2162–2221, 2010

work page 2010
[22]

W. P. Ziemer. Weakly Diﬀerentiable Functions: Sobolev Spaces and Functions of Bounded Variation. Springer-Verlag, 1989. 30

work page 1989

[1] [1]

Arendt, I

W. Arendt, I. Chalendar, and R. Eymard. Extensions of dissipat ive and symmetric operators. Semigroup Forum, 106(2):339–367, 2023

work page 2023

[2] [2]

Arendt and A

W. Arendt and A. F. M. ter Elst. The Dirichlet-to-Neumann opera tor on rough domains. J. Diﬀerential Equations, 251(8):2100–2124, 2011

work page 2011

[3] [3]

Arendt, A

W. Arendt, A. F. M. ter Elst, and M. Sauter. The Perron solution for elliptic equations without the maximum principle. Math. Ann., 390(1):763–810, 2024

work page 2024

[4] [4]

Arendt and M

W. Arendt and M. Warma. The Laplacian with Robin boundary condit ions on arbitrary domains. Potential Anal., 19(4):341–363, 2003

work page 2003

[5] [5]

Boccardo, T

L. Boccardo, T. Gallou¨ et, and L. Orsina. Existence and uniquen ess of entropy solutions for nonlinear elliptic equations with measure data. Ann. Inst. H. Poincar´ eC Anal. Non Lin´ eaire, 13(5):539–551, 1996

work page 1996

[6] [6]

Br´ ezis

H. Br´ ezis. Analyse fonctionnelle. Th´ eorie et applications. Paris: Masson, 199 4

work page

[7] [7]

Bucur and A

D. Bucur and A. Giacomini. A variational approach to the isoperime tric inequality for the Robin eigenvalue problem. Arch. Ration. Mech. Anal., 198(3):927–961, 2010

work page 2010

[8] [8]

D. L. Cohn. Measure Theory. Birkh¨ auser Advanced Texts Basler Lehrb¨ ucher. Birkh¨ auser New York, NY, 2013

work page 2013

[9] [9]

B. E. J. Dahlberg. Estimates of harmonic measure. Arch. Rational Mech. Anal., 65(3):275–288, 1977

work page 1977

[10] [10]

D. E. Edmunds and W. D. Evans. Spectral theory and diﬀerential operators. Oxford Mathemati- cal Monographs. The Clarendon Press, Oxford University Press, New York, 1987. Oxford Science Publications

work page 1987

[11] [11]

L. C. Evans and R. F. Gariepy. Measure theory and ﬁne properties of functions. Textbooks in Mathematics. CRC Press, Boca Raton, FL, revised edition, 2015

work page 2015

[12] [12]

Falconer

K. Falconer. Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, 1990

work page 1990

[13] [13]

Gagliardo

E. Gagliardo. Caratterizzazioni delle tracce sulla frontiera rela tive ad alcune classi di funzioni in n variabili. Rendiconti del Seminario Matematico della Universit ˜A di Padova, 27:284–305, 1957

work page 1957

[14] [14]

Gallou¨ et and R

T. Gallou¨ et and R. Herbin.Mesure, int´ egration,probabilit´ es. Collection R´ ef´ erences sciences. Ellipses, Paris, 2022

work page 2022

[15] [15]

Kazaniecki and M

K. Kazaniecki and M. Wojciechowski. Trace Operator on von Ko ch’s Snowﬂake. Potential Anal., 61(4):659–684, 2024

work page 2024

[16] [16]

J. L. Lewis. Approximation of Sobolev functions in Jordan domain s. Ark. Mat., 25(2):255–264, 1987

work page 1987

[17] [17]

V. Maz’ya. Sobolev Spaces with Applications to Elliptic Partial Diﬀerential Equations, volume 342 of A series of comprehensive studies in mathematics. Springer, 1985

work page 1985

[18] [18]

W. F. Pfeﬀer. The Gauss-Green theorem. Advances in Mathematics, 87(1):93–147, 1991

work page 1991

[19] [19]

Poborchi

S. Poborchi. Sobolev spaces for domains with cusps. In The Maz’ya anniversary collection, Vol. 1 (Rostock, 1998), volume 109 of Oper. Theory Adv. Appl., pages 175–185. Birkh¨ auser, Basel, 1999

work page 1998

[20] [20]

M. Sauter. Uniqueness of the approximative trace. Indiana Univ. Math. J., 69(1):171–204, 2020. 29

work page 2020

[21] [21]

Shvartsman

P. Shvartsman. On the boundary values of Sobolev W 1 p -functions. Adv. Math., 225(4):2162–2221, 2010

work page 2010

[22] [22]

W. P. Ziemer. Weakly Diﬀerentiable Functions: Sobolev Spaces and Functions of Bounded Variation. Springer-Verlag, 1989. 30

work page 1989