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The zero dispersion limit for the Benjamin–Ono equation on the line

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Comptes Rendus
Mathématique
Thomas A. Courtade, Max Fathi and Dan Mikulincer
Stochastic proof of the sharp symmetrized Talagrand inequality
Volume 362 (2024), p. 1779-1784
Online since: 25 November 2024
https://doi.org/10.5802/crmath.681
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2024, Vol. 362, p. 1779-1784
https://doi.org/10.5802/crmath.681
Research article / Article de recherche
Probability theory / Probabilités
Stochastic proof of the sharp symmetrized
Talagrand inequality
Une preuve stochastique de l’inégalité de Talagrand
symétrisée optimale
Thomas A. Courtade ∗,a, Max Fathi b,c,d and DanMikulincer e
a Department of Electrical Engineering and Computer Sciences, University of
California, Berkeley, CA 94720 USA
b Université Paris Cité and Sorbonne Université, CNRS, Laboratoire Jacques-Louis
Lions and Laboratoire de Probabilités, Statistique et Modélisation, F-75013 Paris,
France
c DMA, École normale supérieure, Université PSL, CNRS, 75005 Paris, France
d Institut Universitaire de France
e Department of Mathematics, MIT, Cambridge, MA 02139 USA
E-mails: courtade@berkeley.edu, mfathi@lpsm.paris, danmiku@mit.edu
Abstract. A new proof of the sharp symmetrized form of Talagrand’s transport-entropy inequality is given.
Compared to stochastic proofs of other Gaussian functional inequalities, the new idea here is a certain
coupling induced by time-reversed martingale representations.
Résumé. Nous donnons une nouvelle preuve de la version symétrisée de l’inégalité de transport-entropie
de Talagrand avec constante optimale. En comparaison avec d’autres preuves stochastiques d’inégalités
fonctionnelles gaussiennes, l’élément nouveau ici est l’utilisation d’un couplage induit par un retournement
du temps sur des représentations de martingales.
Keywords. Transport inequalities, Gaussian inequalities, Blaschke–Santaló inequality, Martingale represen-
tations.
Mots-clés. Inégalités de transport, inégalités Gaussiennes, inégalité de Blaschke–Santaló, représentations de
martingales.
2020 Mathematics Subject Classification. 60H30, 39B62, 52A40.
Funding. T.C. acknowledges NSF-CCF 1750430, the hospitality of the Fondation Sciences Mathématiques
de Paris (FSMP) and the LPSM at Université Paris Cité. M.F. was supported by the Agence Nationale de
la Recherche (ANR) Grant ANR-23-CE40-0003 (Project CONVIVIALITY). D.M. was partially supported by a
Simons Investigator Award (622132, PI: Mossel).
Manuscript received 12 July 2024, revised 22 August 2024, accepted 23 August 2024.
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1780 Thomas A. Courtade, Max Fathi and Dan Mikulincer
1. Introduction
We aim to give a short stochastic proof of the following sharp symmetrized Talagrand inequality:
Theorem 1 ([8, Theorem 1.1]). For Borel probability measures µ,ν on Rn with finite second
moments and µ centered,
W2(µ,ν)
2 ≤ 2D(µ∥γ)+2D(ν∥γ), (1)
where W2 is 2-Wasserstein distance, D is relative entropy, and γ is the standard Gaussian measure
on Rn .
By duality, (1) is formally equivalent to the functional Blaschke–Santaló inequality [13, Theo-
rem 1.2], which states that if Borel functions f , g :Rn →R satisfy∫
Rn
x e− f (x) dx = 0 and f (x)+ g (y) ≥ 〈x, y〉, ∀x, y ∈Rn ,
then (∫
Rn
e− f (x) dx
)(∫
Rn
e−g (x) dx
)
≤ (2π)n . (2)
Equality holds for quadratic f , and g = f ∗, its convex conjugate. Despite the equivalence, (1) may
be regarded as a formal strengthening of (2) in the sense that (2) is recovered from Theorem 1 by
weak duality: briefly, for f , g satisfying the hypotheses demanded by (2), take dµ(x) ∝ e− f (x) dx
and dν(x) ∝ e−g (x) dx in (1) and simplify to obtain (2). The reverse implication corresponds to
strong duality, and is more difficult. See [10, Theorem 11] and [8] for details.
Inequality (2) is a functional generalization of the earlier Blaschke–Santaló inequality for
the volume product of convex sets, earlier proofs of which were accomplished by calculus of
variations [20] and symmetrization arguments [16, 19]. The functional form was proved in [2]
(and earlier in K. Ball’s Ph.D. thesis [3] in a restricted setting of even functions). The original
proof relied on the usual Blaschke–Santaló inequality applied to level sets. Lehec later gave
two alternative proofs; one using induction on the dimension [13], and the other [14] using the
Prekópa–Leindler inequality and the Yao–Yao partition theorem. This last proof actually yields
a more general statement, originally due to [9], but the present work shall be restricted to the
classical setting. More recently, a new semigroup proof of the inequality for even functions
was established in [18] using improved hypercontractive estimates for the heat flow, and then
simplified in [6]. Let us also mention a recent generalization to several functions under a
symmetry assumption, due to Kolesnikov and Werner [12].
Equivalence between integral inequalities of the form (2) and transport inequalities of the
form (1) via duality goes back to [4], where they studied Talagrand quadratic transport-entropy
inequality [21] (which is (1) in the particular case µ = γ). Duality for transport inequalities
involving three measures, such as (1), has been considered in [10] and [11, Proposition 8.2].
Stochastic proofs of functional inequalities, in particular using Brownian motion and Gir-
sanov’s theorem, go back to Borell’s stochastic proof of the Prekópa–Leindler inequality [5]. Our
present work is motivated by Lehec’s short stochastic proofs of various functional inequalities
[15], including in particular Talagrand’s transport-entropy inequality.
2. A Stochastic Proof
We’ll work on the Wiener space (Ω,B,P), where Ω is the set of continuous paths ω : [0,1] → Rn
starting at 0, B is the usual Borel σ-algebra, and P is the Wiener measure. Let Bt (ω) := ω(t ) be
the coordinate process, so that B = (Bt )0≤t≤1 is a standard Brownian motion, and so is the time-
reversed process B̂ t := B1 −B1−t . Let F = (Ft )0≤t≤1 and F+ = (F+t )0≤t≤1 denote the filtrations
Thomas A. Courtade, Max Fathi and Dan Mikulincer 1781
generated by B and B̂ , respectively. For each t ∈ [0,1], Ft and F+1−t are complementary, in the
sense that they are independent and B = σ(Ft ∪F+1−t ). Henceforth, ∥ · ∥ denotes the ℓ2 norm,
and I denotes the identity matrix.
Apart from standard facts in stochastic calculus, we’ll need two lemmas. The first is a
variational representation of entropy, obtained as a consequence of Girsanov’s theorem; it has
been applied to study rigidity and stability of various functional inequalities (see, e.g., [1, 7, 17]).
Lemma 2 ([7]). For a centered probability measure µ on Rn with finite second moments, we have
D(µ∥γ) = inf
F
1
2
∫ 1
0
E[∥Ft − I∥2]
1− t dt , (3)
where the infimum is over F -adaptedRn×n-valued processes F = (Ft )0≤t≤1 such that
∫ 1
0 Ft dBt ∼µ.
Remark 3. The same representation holds if we consider F+-adapted F with
∫ 1
0 Ft dB̂ t ∼µ.
Stochastic proofs of several other functional inequalities use representation formulas for the
entropy and linear couplings of Brownian motions (cf. [7, 15]). Our proof will similarly rely on the
representation formula (3) for the entropy1, but makes use of a new coupling induced by time-
reversed martingale representations. The next lemma is the crucial new ingredient; it relates
martingale representations in terms of (Bt )0≤t≤1 and its time-reversal (B̂ t )0≤t≤1.
Lemma 4. If X ∈ L2(Ω,B,P) is a Rn-valued random vector with martingale representations
X =
∫ 1
0
Ft dBt =
∫ 1
0
Gt dB̂ t ,
then ∫ 1
t
E[∥Fs − I∥2]ds ≥
∫ 1−t
0
E[∥Gs − I∥2]ds, ∀0 ≤ t ≤ 1. (4)
Proof. By the Pythagorean theorem, convexity, and independence of Ft and F+1−t , we have
E[∥X ∥2]−E[∥E[X |Ft ]∥2] = E[∥X −E[X |Ft ]∥2] ≥ E[∥E[X −E[X |Ft ]|F+1−t ]∥2] = E[∥E[X |F+1−t ]∥2].
Since E[X |Ft ] =
∫ t
0 Fs dBs and E[X |F+1−t ] =
∫ 1−t
0 Gs dB̂ s , three applications of Itô’s isometry give∫ 1
t
E[∥Fs∥2]ds = E[∥X ∥2]−E[∥E[X |Ft ]∥2] ≥ E[∥E[X |F+1−t ]∥2] =
∫ 1−t
0
E[∥Gs∥2]ds, ∀0 ≤ t ≤ 1.
Next, by applying Itô’s isometry to each martingale representation of X , we find∫ 1
t
E[Tr(Fs )]ds = E[〈X ,B1 −Bt 〉] = E[〈X , B̂ 1−t 〉] =
∫ 1−t
0
E[Tr(Gs )]ds, ∀0 ≤ t ≤ 1.
Combining the previous two observations gives (4). □
Proof of Theorem 1. The inequality is invariant with respect to translations of ν, so we may also
assume ν is centered. Let F = (Ft )0≤t≤1 be any F -adapted process such that
∫ 1
0 Ft dBt ∼ µ, and
let H = (Ht )0≤t≤1 be any F+-adapted process such that
∫ 1
0 Ht dB̂ t ∼ ν. Let G = (Gt )0≤t≤1 be the
martingale representation of
∫ 1
0 Ft dBt in terms of the time-reversed Brownian motion B̂ ; i.e., G is
F+-adapted, satisfying
∫ 1
0 Gt dB̂ t =
∫ 1
0 Ft dBt ∼µ. By the Tonelli theorem and Lemma 4, we have∫ 1
0
E[∥Gs − I∥2]
s
ds =
∫ 1
0
E[∥Gs − I∥2]ds +
∫ 1
0
1
(1− t )2
(∫ 1−t
0
E[∥Gs − I∥2]ds
)
dt
≤
∫ 1
0
E[∥Fs − I∥2]ds +
∫ 1
0
1
(1− t )2
(∫ 1
t
E[∥Fs − I∥2]ds
)
dt =
∫ 1
0
E[∥Fs − I∥2]
1− s ds.
1The representation (3) is not the same as that used in [15]. However, it is derived from [15, Theorem 4] by
combination with the martingale representation theorem.
1782 Thomas A. Courtade, Max Fathi and Dan Mikulincer
By definition of W2, Itô’s isometry, convexity of ∥ ·∥2, and the previous estimate,
W2(µ,ν)
2 ≤ E
∥∥∥∥∫ 1
0
(Gt −Ht )dB̂ t
∥∥∥∥2 = ∫ 1
0
E[∥Gt −Ht∥2]dt ≤
∫ 1
0
E[∥Gt − I∥2]
t
dt +
∫ 1
0
E[∥Ht − I∥2]
1− t dt
≤
∫ 1
0
E[∥Ft − I∥2]
1− t dt +
∫ 1
0
E[∥Ht − I∥2]
1− t dt .
With the help of Lemma 2, optimizing over F and H completes the proof . □
3. Remarks on the Approach
3.1. Equality cases
The equality cases for (1) are also evident from the given proof. Indeed, if D(µ∥γ) < ∞, then
the infimum in (3) is a.s.-uniquely achieved by an F -adapted process F = (Ft )0≤t≤1. Defining
X := ∫ 10 Ft dBt ∼ µ for this particular F , equality in (1) implies equality in (4) for a.e. t ∈ [0,1],
which requires that X − E[X |Ft ] is F+1−t -measurable for a.e. t ∈ [0,1]. By Proposition 5 below,
this ensures X ∼ µ is Gaussian. By symmetry, any extremal ν is also Gaussian, and explicit
computation shows that µ,ν are extremizers in (1) if and only if µ = N (0,C ) and ν = N (θ,C−1)
for some θ ∈Rn and positive definite C ∈Rn×n .
Proposition 5. Let X ∈ L2(Ω,B,P) admit martingale representation X = ∫ 10 Ft dBt . If X −E[X |Ft ]
is F+1−t -measurable for a.e. t ∈ [0,1], then X is Gaussian.
Proof. Define Mt :=
∫ t
0 Fs dBs . The hypothesis is equivalent to requiring that (M1 −Mt ) is F+1−t -
measurable for each t ∈D, where D is dense in [0,1]. Fix any s, t ∈D, with s ≤ t . Since (M1−Mt ) is
F+1−t -measurable by hypothesis, and (Mt −Ms ) is Ft -measurable by definition, complementarity
ensures (M1−Mt ) and (Mt −Ms ) are independent. Iterating this procedure on the (M1−Mt ) term
allows us to conclude that (Mt )0≤t≤1 has independent increments, provided the endpoints of the
increments are in D. Since X ∈ L2(Ω,B,P), a version of (Mt )0≤t≤1 admits continuous sample
paths, and we conclude by density of D that (Mt )0≤t≤1 has (square-integrable) independent
increments generally, and is thus a Gaussian process. □
3.2. Importance of the coupling induced by time-reversal
With the proof of Theorem 1 in hand and the equality cases characterized, we highlight the impor-
tance of the coupling based on time-reversal. Following previous stochastic proofs of functional
inequalities, one could appeal to martingale representations
∫ 1
0 Ft dB
1
t ∼ µ and
∫ 1
0 Gt dB
2
t ∼ ν
with linearly coupled Brownian motions B 1 and B 2 (equivalently, Brownian motions B 1 and B 2
adapted to a common filtration) to couple µ and ν. This approach cannot work, as we now ex-
plain.
Working in dimension n = 1 for simplicity, recall that when µ = N (0,α) with α > 0, the
minimizer F in (3) has an explicit expression (e.g., [7, Sec. 2]). In particular,∫
Ft dB
1
t ∼µ and D(µ∥γ) =
1
2
∫ 1
0
E[∥Ft − I∥2]
1− t dt −→ Ft =
α
1− t +αt .
Likewise, for ν= N (0,α−1), the “optimal” representation of ν with respect to B 2 satisfies∫ 1
0
Gt dB
2
t ∼ ν and D(ν∥γ) =
1
2
∫ 1
0
E[∥Gt − I∥2]
1− t dt −→ Gt =
1
α(1− t )+ t .
Since B 1 and B 2 are linearly coupled standard Brownian motions, we can write[
B 1t
B 2t
]
=
[
1 σ
σ 1
]1/2
Bt , 0 ≤ t ≤ 1, (5)
Thomas A. Courtade, Max Fathi and Dan Mikulincer 1783
for some |σ| ≤ 1, where (Bt )0≤t≤1 is a 2-dimensional standard Brownian motion. This construc-
tion induces a coupling πσ of X :=
∫ 1
0 Ft dB
1
t ∼µ and Y :=
∫ 1
0 Gt dB
2
t ∼ ν satisfying
Eπσ∥X −Y ∥2 =
∫ 1
0
(
F 2t +G2t −2σFt Gt
)
dt =

(
α+ 1α
)− 4σ logα(
α− 1α
) if α ̸= 1
2(1−σ) if α= 1,
where we made use of Itô’s isometry and (5). A simple calculation reveals that
min
σ:|σ|≤1
Eπσ∥X −Y ∥2 ≥α+
1
α
−2 =W2(µ,ν)2 = 2D(µ∥γ)+2D(ν∥γ),
with equality if and only ifα= 1. So, with the exception of the trivial caseµ= ν= γ, the established
stochastic approach to proving functional inequalities using linearly coupled Brownian motions
fails to produce the requisite optimal coupling between µ and ν in all extremal cases (at least,
in this implementation). This suggests that coupling through time-reversal lends a useful new
degree of freedom to the stochastic program for proving functional inequalities.
Acknowledgments
The authors thank Joseph Lehec and an anonymous referee for their careful reading of the paper
and suggestions for improvement.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
References
[1] E. Aras, T. A. Courtade and A. Zhang, “Equality cases in the Anantharam–Jog–Nair inequal-
ity”, 2022. https://arxiv.org/abs/2206.11809.
[2] S. Artstein-Avidan, B. Klartag and V. Milman, “The Santaló point of a function, and a
functional form of the Santaló inequality”, Mathematika 51 (2004), no. 1-2, pp. 33–48.
[3] K. M. Ball, Isometric problems in ℓp and sections of convex sets, PhD thesis, University of
Cambridge, 1986.
[4] S. G. Bobkov and F. Götze, “Exponential integrability and transportation cost related to
logarithmic Sobolev inequalities”, J. Funct. Anal. 163 (1999), no. 1, pp. 1–28.
[5] C. Borell, “Diffusion equations and geometric inequalities”, Potential Anal. 12 (2000), no. 1,
pp. 49–71.
[6] D. Cordero-Erausquin, N. Gozlan, S. Nakamura and H. Tsuji, “Duality and Heat flow”, 2024.
https://arxiv.org/abs/2403.15357.
[7] R. Eldan and D. Mikulincer, “Stability of the Shannon–Stam inequality via the Föllmer
process”, Probab. Theory Relat. Fields 177 (2020), no. 3-4, pp. 891–922.
[8] M. Fathi, “A sharp symmetrized form of Talagrand’s transport-entropy inequality for the
Gaussian measure”, Electron. Commun. Probab. 23 (2018), article no. 81 (9 pages).
[9] M. Fradelizi and M. Meyer, “Some functional forms of Blaschke–Santaló inequality”, Math.
Z. 256 (2007), no. 2, pp. 379–395.
[10] N. Gozlan, “Characterization of Talagrand’s like transportation-cost inequalities on the real
line”, J. Funct. Anal. 250 (2007), no. 2, pp. 400–425.
1784 Thomas A. Courtade, Max Fathi and Dan Mikulincer
[11] N. Gozlan and C. Léonard, “Transport inequalities. A survey”, Markov Process. Relat. Fields
16 (2010), no. 4, pp. 635–736.
[12] A. V. Kolesnikov and E. M. Werner, “Blaschke–Santaló inequality for many functions and
geodesic barycenters of measures”, Adv. Math. 396 (2022), article no. 108110 (44 pages).
[13] J. Lehec, “A direct proof of the functional Santaló inequality”, C. R. Math. Acad. Sci. Paris
347 (2009), no. 1-2, pp. 55–58.
[14] J. Lehec, “Partitions and functional Santaló inequalities”, Arch. Math. 92 (2009), no. 1,
pp. 89–94.
[15] J. Lehec, “Representation formula for the entropy and functional inequalities”, Ann. Inst.
Henri Poincaré, Probab. Stat. 49 (2013), no. 3, pp. 885–899.
[16] M. Meyer and A. Pajor, “On the Blaschke-Santaló inequality”, Arch. Math. 55 (1990), no. 1,
pp. 82–93.
[17] D. Mikulincer, Universality of High-Dimensional Systems, PhD thesis, Weizmann Institute
of Science, 2021.
[18] S. Nakamura and H. Tsuji, “The functional volume product under heat flow”, 2023. https:
//arxiv.org/abs/2401.00427.
[19] J. Saint-Raymond, “Sur le volume des corps convexes symétriques”, in Initiation Seminar
on Analysis: G. Choquet-M. Rogalski-J. Saint-Raymond, 20th Year: 1980/1981, Univ. Paris VI,
1981. Exp. No. 11, 25 p.
[20] L. A. Santaló, “An affine invariant for convex bodies of n-dimensional space”, Port. Math. 8
(1949), pp. 155–161.
[21] M. Talagrand, “Transportation cost for Gaussian and other product measures”, Geom.
Funct. Anal. 6 (1996), no. 3, pp. 587–600.


Stochastic proof of the sharp symmetrized Talagrand inequality

Lucas Alexis

Comptes Rendus
Mathématique
Lucas Alexis
Purity and almost strict purity of Anderson t -modules
Volume 362 (2024), p. 807-812
Online since: 17 September 2024
https://doi.org/10.5802/crmath.611
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2024, Vol. 362, p. 807-812
https://doi.org/10.5802/crmath.611
Research article / Article de recherche
Number theory / Théorie des nombres
Purity and almost strict purity of Anderson
t-modules
Pureté et presque stricte pureté des t-modules
d’Anderson
Lucas Alexis a
a Normandie Univ, UNICAEN, CNRS, LMNO, 14000 Caen, France
E-mail: alexis.lucas@unicaen.fr
Abstract. We study the relations between the notion of purity of a t-module introduced by Anderson and that
of almost strict purity for a t-module introduced by Namoijam and Papanikolas (concept already mentioned
by G. Anderson and D. Goss).
Résumé. On étudie les relations entre la notion de pureté d’un t-module introduite par Anderson et celle
de presque pureté pour un t-module introduite par Namoijam et Papanikolas (concept déjà mentionné par
G. Anderson et D. Goss).
Keywords. Purity, almost strict purity, Anderson t-modules, t-motive, Newton polygons.
Mots-clés. Pureté, presque stricte pureté, t-modules d’Anderson, t-motifs, polygone de Newton.
Manuscript received 18 November 2022, revised 12 September 2023, accepted 7 January 2024.
1. Introduction
In [1], G.W. Anderson defined t-modules and the notion of purity. Related to t-modules, C.
Namoijam and M. A. Papanikolas defined the notion of almost strict purity in [4, Remark 4.5]
(concept already mentioned by G.Anderson in [1, Section 2.2] and D.Goss in [2, Remark 5.5]),
then they proved that an almost strictly pure t-module is pure. We are interested here in the
reciprocal, and with the help of the work of A. Maurischat in [3] we show that these two notions
are not equivalent by presenting a counter-example (see Theorem 3).
2. Purity
Let K a perfect field containing Fq . We let τ : K → K denote the q-th power Frobenius map and
K {τ} be the ring of twisted polynomials in τ over K , subject to the relation, τa = aqτ for any a ∈ K .
We further consider the skew Laurent series ring over K in σ := τ−1
K ({σ}) :=
{ ∞∑
i=k
aiσ
i
∣∣∣∣∣ k ∈Z, ai ∈ K
}
.
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
808 Lucas Alexis
Consider ℓ : Fq [t ] → K a homomorphism of Fq -algebras and denote σ= τ−1.
A t-module (E ,ϕ) over K of dimension d is by definition an Fq -vector space scheme E over
K isomorphic to Gda together with a homomorphism of Fq -algebras ϕ : Fq [t ] → Endgrp,Fq (E)
into the ring of Fq -vector space scheme endomorphisms of E , such that for all a ∈ Fq [t ], the
endomorphism dϕa on Lie(E) induced by ϕa fulfills the condition that dϕa −ℓ(a) is nilpotent.
We will fix in the following (E ,φ) a t-module on K of dimension d as well as a coordinate
system κ, i.e. an isomorphism of schemes in Fq -vector spaces κ : E ≃ Gda defined on K . With
respect to this coordinate system, we can represent φt by a matrix D ∈ Md (K {τ}).
Let pri :G
d
a →Ga (1 ≤ i ≤ d) be the projection to the i -th component of Gda , and let κi = κ◦pri .
Let κ̂ j :Ga → E be defined by κ̂ j = κ−1 ◦ inj j where inj j :Ga →Gda is the natural injection into the
j−th component.
We say that E is almost strictly pure if there is some integer s ≥ 1 such that
D s = A0 + A1τ+·· ·+ Arτr
with Ar ∈GLd (K ).
The t-motive M(E) of E is the free K {τ}-module of rank d with base {κ1, . . . ,κd } with a t-action
on this base defined by
t .
κ1...
κd
= D
κ1...
κd
 .
We define in a similar way the dual t-motive M as the free K {τ}-module of rank d of basis
{κ̂1, . . . , κ̂d } whose t-action (on the right) on this basis is defined by(
κ̂1 · · · κ̂d
) · · · t = (κ̂1 · · · κ̂d )D.
We say that E is abelian if M(E) is a finitely generated K [t ]-module. In this case, we define w(M)
the weight of M by
w(M) = d
rk(E)
where rk(E) is the rank of M as a K [t ]-module (that is finite because E is abelian).
We moreover consider:
• The ring of formal power series in 1t with coefficients in K denoted by K
[[ 1
t
]]
.
• The field of Laurent series in 1t with coefficients in K denoted by K
(( 1
t
))
(that is the field
of fractions of K
[[ 1
t
]]
).
The t-motive M(E) and the t-module E are called pure if there exists a K
[[ 1
t
]]
-lattice Λ in
K
(( 1
t
))⊗K [t ] M as well as positive integers u, v ∈N such that
t uΛ= τvΛ.
We will use the following result, proved by A. Maurischat in [3, Theorem 6.6, Theorem 7.2],
characterizing the fact of being abelian and being pure using Newton polygons.
Theorem 1 (Maurischat). The t-module E is abelian if and only if the Newton polygon Nλd of the
last invariant factor λd of the matrix D has only positive slopes. In this case, E is pure if and only
if Nλd has exactly one edge. Then we have that the weight of M equals the reciprocal of the slope of
the edge.
Here we recall that the invariant factors of a matrix D ∈ Md (K {τ}) are obtained by diagonalizing
the matrix t Id−D ∈ Md (K ({σ})[t ]) by performing elementary operations on the rows and columns
in K ({σ})[t ] that are the following (denote by Li (resp Ci ) the i -th row (resp the i -th column)):
Lucas Alexis 809
• add to the i -th row Li the j -th row L j multiplied on the left by a ∈ K ({σ})[t ] denote by
Li → Li +a.L j (resp add to i -th column Ci the j -th column C j multiplied on the right by
a ∈ K ({σ})[t ] denote by Ci →Ci +C j ·a),
• multiply on the left i -th the row Li by an element u of K ({σ})∗ denoted by Li → uLi (resp
multiply on the right the i -th column Ci by u denoted by Ci →Ci u),
• exchanging two lines Li and L j (resp two columns Ci and C j ) denote by Li ↔ Li (resp
Ci ↔C j ).
Contrary to the commutative case the invariant factors are only unique up to similarity.
Example. In [3], Maurischat defined the t-module given by the matrix
M :=
(
θ 0
1 θ
)
+
(
0 0
1 0
)
·τ+
(
1 0
0 1
)
·τ2 +
(
0 1
0 0
)
·τ3 ∈ M2(K {τ}).
By diagonalizing the matrix t I2 −M we get the matrix(
1 0
0 λ2
)
where
λ2 =
(
−σ−3 + (θ+θq2 )σ−2 +θq3+1
)
−
(
2σ−2 +θq−3 +θ
)
· t + t 2 ∈ K ({σ}).
If char(K ) = 2 then we represent the Newton polygon of λ2 in Figure 1. It has only one edge of
slope 32 hence by Theorem 1 the t-module is abelian and pure of weight
2
3 .
The authors of [4] showed in the same paper the next result.
Theorem 2 (Namoijam, Papanikolas). With the previous notation, an almost strictly pure t-
module is pure of weight sr .
We now turn our interest to the reciprocal of the above result, and we answer negatively.
Theorem 3. For any integer d ≥ 2 there exists a pure but not almost strictly pure t-module of
dimension d.
Proof. We first consider the case d = 2. Let us note θ = ℓ(t ). Consider the t-module given by the
matrix
D2 =
(
θ 0
1 θ
)
+
(
1 1
θ θ
)
·τ ∈ Md (K {τ}).
Let us diagonalize the matrix t I2 −D2 (the diagonalization is being taken over K ({σ})):(
t −θ−τ −τ
−1−θτ t −θ−θτ
)
L1↔L2−−−−−→
(−1−θτ t −θ−θτ
t −θ−τ −τ
)
C1→C1γ−−−−−−→
(
1 t −θ−θτ
(t −θ−τ)γ −τ
)
(
1 t −θ−θτ
(t −θ−τ)γ −τ
)
L2→L2−(t−θ−τ)γL1−−−−−−−−−−−−−−→
(
1 t −θ−θτ
0 −τ− (t −θ−τ)γ(t −θ−θτ)
)
(
1 t −θ−θτ
0 −τ− (t −θ−τ)γ(t −θ−θτ)
)
C2→C2+C1(−t+θ+θτ)−−−−−−−−−−−−−−−→
(
1 0
0 λ′
)
L2→−γ−1L2−−−−−−−−→
(
1 0
0 λ2
)
where
λ2 =−γ−1λ′
=−γ−1 (−τ− (t −θ−τ)γ(t −θ−θτ))
= t 2 + t · (−θ−θτ−γ−1θγ−γ−1τγ)+γ−1τ+γ−1θγθ+γ−1θγθτ+γ−1τγθ+γ−1τγθτ
and γ= (−1−θτ)−1.
810 Lucas Alexis
We represent the Newton polygon ofλ2 in Figure 2. It has only one edge of positive slope equal
to 1, hence according to Theorem 1 this t-module is pure of weight 1.
An immediate induction shows that for n ≥ 2, the leading coefficient of Dn2 is given by the
matrix
n−1∏
k=1
(1+θqk ) · A
where
A =
(
1 1
θ θ
)
whose determinant is zero, so this t-module is not almost strictly pure. □
Now we consider the general case d > 2. We put m := d − 2 > 0 and consider the t-module
given by the matrix
D2+m =

D2
θ+τ
. . .
θ+τ
 ∈ M2+m(K {τ}).
This t-module is the direct sum of pure t-modules of weight 1 (the t-module associated to D2
and d−2 copies of the Carlitz module), so we can prove it is a pure t-module of weight 1, but here
we give a proof using Maurischat’s algorithm.
For n ≥ 1, the leading coefficient of the matrix Dn2+m is the matrix(
Dn2
Im
)
whose determinant is zero, so this t-module is not almost strictly pure.
Consider (J0) the algorithm that diagonalize as previously the matrix t I2 −D2. Applying (J0)
and exchanging row and columns, we get the matrix:
t I2+m −D2+m −→ S =

1
t −θ−τ
. . .
t −θ−τ
λ2
 ∈ M2+m(K {τ}[t ]).
Consider the euclidean division of λ2 by t −θ−τ:
λ2 = q(t −θ−τ)+ r, r ̸= 0 and degt (r ) = 0.
Let
S′ =
(
t −θ−τ
λ2
)
∈ M2(K {τ}[t ]).
We apply the following operations to the matrix S′ (and denote by (J1) this algorithm):
L2 −→ L2 −qL1
C2 −→C2 +C1
L2 −→ r−1L2
L1 −→ L1 − (t −θ−τ)L2
C2 −→C2 −C1r−1λ2
L1 ←→ L2
L2 −→−L2.
Lucas Alexis 811
We get the matrix: (
λ2
(t −θ−τ)r−1λ2
)
.
By successively applying the algorithm (J1) to the matrices S′ which appear from the matrix S, we
obtain the matrix
S −→

1
λ2
(t −θ−τ)r−1λ2
. . .
(t −θ−τ)r−1λ2
 .
As λ2 and t −θ−τ have Newton polygons consisting of only one edge of slope 1, the Newton
polygon of the last coefficient of the last matrix has only one edge of slope 1. It follows that the
Newton polygon of the last invariant factor of Dm+2 has only one edge of slope 1. Hence Dm+2 is
also a t-module which is pure of weight 1 but not almost strictly pure for all m ≥ 0.
1 2
−3
−2
−1
Figure 1: Newton polygon of the Anderson
module M constructed by Maurischat when
char(K ) = 2 .
1 2
−3
−2
−1
Figure 2: Newton polygon of the Anderson
module D2 in Theorem 3.
Aknowledgements
This result is part of my master-Thesis at University of Caen under the supervision of Floric Tavares
Ribeiro and Tuan Ngo Dac that I would like to thank.
References
[1] Greg W. Anderson. t-motives. Duke Maths. J., 53(2):457-502, 1986.
[2] D. Goss, Basic Structures of Function Field Arithmetic, Springer-Verlag, Berlin, 1996.
[3] Urs Hartl and Ann-Kristin Juschka. Pink’s theory of Hodge structures and the Hodge conjecture
over function fields. t-motives: Hodge structures, transcendence and other motivic aspects, 31–182,
EMS Ser. Congr. Rep., EMS Publ. House, Berlin, [2020], ©2020.
[4] Andreas Maurischat. Abelian equals A-finite for Anderson A-modules. Preprint available from
ArXiv at https://arxiv.org/abs/2110.11114.
[5] Changningphaabi Namoijam and Matthew A. Papanikolas. Hyperderivatives of pe-
riods and quasi-periods for Anderston t-modules. Preprint available from ArXiv at
http://arxiv.org/abs/2103.05836,2021.
Figure 1. Newton polygon of the
Anderson module M constructed by
Maurischat when char(K ) = 2 .
1 2
−3
−2
−1
Figure 1: Newton polygon f the Anderso
module M constructed by Maurischat when
cha (K ) = 2 .
1 2
−3
−2
−1
Figure 2: N wton polygon of the Anderso
module D2 in Theorem 3.
Aknowledgements
This result is part of my master-Thesis at University of Caen und r the supervision of Floric Tavares
Ribeiro and Tu Ngo Dac that I would like to thank.
References
[1] Greg W. Anderso . t-motives. Duke Maths. J., 53(2):4 7-502, 1986.
[2] D. Goss, Basic Structures of Function Field Arithmetic, Springer-Verlag, Berlin, 1996.
[3] Urs Hartl and Ann-Kristin Juschka. Pin ’s theory of Hodge structures and the Hodg c njecture
over function fields. t-motives: Hodge structures, transcendence and other motivic aspects, 31–182,
EMS Ser. Congr. Rep., EMS Publ. House, Berlin, [2020], ©2020.
[4] Andreas Maurischat. Abelian equ ls A-finite for Anderso A-modules. Preprint available from
ArXiv at https://arxiv.org/abs/2110.11114.
[5] Changningphaabi N moijam and Matthew A. Papanikolas. Hyperderivatives of pe-
riods an quasi-periods for Anderston t-modules. Pr print available from ArXiv at
http://arxiv.org/abs/2103.05836,2021.
Figure 2. Newton polygon of the An-
derson module D2 in Theorem 3.
Aknowledgements
This result is part of my master-Thesis at University of Caen under the supervision of Floric
Tavares Ribeiro and Tuan Ngo Dac that I would like to thank.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that c uld benefit fr m this article, and have d clared no affiliations other than their research
organizations.
812 Lucas Alexis
References
[1] G. W. Anderson, “t-motives”, Duke Math. J. 53 (1986), no. 2, pp. 457–502.
[2] D. Goss, Basic structures of function field arithmetic, Springer, 1996, pp. xiv+422.
[3] A. Maurischat, “Abelian equals A-finite for Anderson A-modules”, 2022. https://arxiv.org/
abs/2110.11114v2.
[4] C. Namoijam and M. A. Papanikolas, “Hyperderivatives of periods and quasi-periods for
Anderson t-modules”, 2022. https://arxiv.org/abs/2103.05836.


Purity and almost strict purity of Anderson $t$-modules

Nicolas Forcadel

Cyril Imbert

Régis Monneau

Coercive Hamilton–Jacobi equations in domains: the twin blow-ups method

Filipe Fernandes

Zbigniew Jelonek

Comptes Rendus
Mathématique
Filipe Fernandes and Zbigniew Jelonek
The Pinchuk example revisited
Volume 362 (2024), p. 449-00
Online since: 16 May 2024
https://doi.org/10.5802/crmath.570
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
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Comptes Rendus. Mathématique
2024, Vol. 362, p. 449-00
https://doi.org/10.5802/crmath.570
Research article / Article de recherche
Algebraic geometry / Géométrie algébrique
The Pinchuk example revisited
L’exemple de Pinchuk revisité
Filipe Fernandes a and Zbigniew Jelonek ∗,b,c
a Departamento de Matemática, Universidade Federal do Ceará, 60440-900 Fortaleza, Ceará, Brazil
b Instytut Matematyczny, Polska Akademia Nauk, Śniadeckich 8, 00-656 Warszawa, Poland
c Departamento de Matemática, Universidade Federal do Ceará, Rua Campus do Pici, s/n, Bloco 914,
Pici, 60440-900, Fortaleza-CE, Brazil
E-mails: filipematonb@gmail.com, najelone@cyf-kr.edu.pl
Abstract. In this note we provide two special counterexamples to the strong real Jacobian Conjecture; the first
one is surjective, the second one has non-dense image.
Résumé. Dans cette note, nous fournissons deux contre-exemples spéciaux à la forte conjecture jacobienne
réelle ; le premier est surjectif, le second a une image non dense.
Mathematical subject classification (2010). 14R15, 14P99.
Funding. The first named author was partially supported by the grant 160946/2022-0 of CNPq/INCTMat.
Manuscript received 1 June 2023, revised 14 July 2023 and 3 September 2023, accepted 7 September 2023.
1. Introduction
The Jacobian conjecture is a famous unsolved problem concerning polynomials in several vari-
ables. It states that if the Jacobian of a polynomial function from an n−dimensional space to it-
self is a non-zero constant, then the function has a polynomial inverse (we recall that the Jaco-
bian J (F ) of a holomorphic (or smooth) map F :Cn →Cn (F : Rn → Rn) is the determinant of the
Jacobian matrix of F ). It was first conjectured in 1939 by Ott-Heinrich Keller [K] and widely pub-
licized by Shreeram Abhyankar, as an example of a difficult question in algebraic geometry that
can be understood using little beyond a knowledge of calculus (see for example [bcw, ess]). Here
we consider it in the following form:
Conjecture 1 (Jacobian Conjecture). Let F : C2 → C2 be a polynomial mapping with a constant
non-zero Jacobian. Then F is a bijection.
There is also a real version of this conjecture (see [R]):
Conjecture 2 (Strong Real Jacobian Conjecture.). Let F :R2 →R2 be a polynomial mapping with
a non-vanishing Jacobian. Then F is a bijection.
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
450 Filipe Fernandes and Zbigniew Jelonek
However, over twenty years ago Pinchuk found an example of locally injective polynomial
endomorphism of R2, which is not injective (see [P] for details of his construction). Pinchuk’s
example has topological degree two and it is not a surjective mapping, in fact it omits two points.
Ronen Peretz asked (unpublished, about 2000) whether the Strong Real Jacobian Conjecture is
true whenever F is surjective. A similar question was formulated quite recently by Tiep Dinh (see
the end of this paper). We may also ask whether there exists a counterexample F to this conjecture
such that the image F (R2) is not dense in R2. The aim of this note is to answer both questions.
2. Construction of the examples
The main idea is to modify an existing non-injective polynomial map with non-vanishing Jaco-
bian by composing with special mappings. Here we use the version of Pinchuk map presented
in [BF], because it seems to be the simplest known so far, in the sense that its components have
the lowest degree (however this construction still can be adapted to any of the known examples,
see [F, P]). Letting
p = 4 x6 y3 +12 x5 y2 +12 x4 y +4 x3 y2 +4 x3 +5 x2 y +x + y,
q =−(4 x4 y2 +8 x3 y +4 x2 +2 x y +1)
· (4 x6 y3 +12 x5 y2 +12 x4 y +4 x3 y2 +4 x3 +7 x2 y +3 x + y) , (1)
we obtain the following:
Theorem 3. Let p and q be as in (??) and let F : R2 → R2 be defined by F = (p, q). Then F is a
non-injective polynomial map with non-vanishing Jacobian and (±1,0) are the only points with
no inverse image.
Proof. See [BF]. □
In the following theorem we present a surjective example.
Theorem 4. Let F be as in Theorem ?? and φ : C 7→ z3 − 3z ∈ C, where we treat C as R2. Then
f =φ◦F is a non-injective but surjective polynomial map with non-vanishing Jacobian.
Proof. The only critical points of the map φ are z = ±1. Since (±1,0) ∉ Im(F ), f has non-
vanishing Jacobian. As φ is surjective, f can a priori omit only the points φ(±1) = ∓2. However
φ(±2) =±2, therefore f is surjective. □
We present now an example with non-dense image.
Theorem 5. Let
ψ= ((x y2 +x + y) (x − y), (x y +1)2 +x2) . (2)
and F̃ = (p +1, q), where p and q are as in (??). Then f̃ =ψ◦ F̃ is a non-injective polynomial map
with non-vanishing Jacobian such that f̃ (R2) ⊂R×R+.
Proof. Denote by D(ψ) the Jacobian matrix ofψ. We have
det(D(ψ)) = (2y +2x y2 +x)2 +x2 (2y2 +3)2 ,
thus (0,0) is the only critical point of ψ. From Theorem ?? we have (0,0) ∉ Im(F̃ ). Hence
the Jacobian of f̃ is non-vanishing. Also (x y + 1)2 + x2 > 0 for every (x, y) ∈ R2. Therefore
Im(ψ) ⊂R×R+. Consequently also Im( f̃ ) ⊂R×R+. □
We finish this note answering negatively the following question of professor Tiep Dinh (per-
sonal comunication, February 2023):
Filipe Fernandes and Zbigniew Jelonek 451
Question. 6. Let f :Rn →Rn be a semi-algebraic smooth mapping which is surjective and has a
constant non-zero Jacobian. Is then f a diffeomorphism?
Example 7 (Compare with [LAC]). Let F be as in Theorem ??. Then G(x, y, z) =
(F (x, y), z/J (F (x, y))) is a regular non-injective and surjective local diffeomorphism of R3 with
constant non-zero Jacobian.
Proof. G is well defined for all (x, y, z) ∈R3, since J (F ) > 0. As for J (G), we have
J (G) = J
(
F,
z
J (F )
)
= J (F,1/J (F ))+ J (F, z)/J (F ) = J (F, z)/J (F ) = 1.
Note that G is surjective. Indeed take any (a,b,c) ∈R3 and consider the equation
G(x, y, z) = (a,b,c).
Hence F (x, y) = (a,b) and this equation has a solution (x0, y0) by Theorem ??. Now it is enough
to put z0 = c J (F (x0, y0)) to get the point (x0, y0, z0) such that G(x0, y0, z0) = (a,b,c). Moreover for
generic (a,b) there is another pair (x1, y1) such that F (x1, y1) = (a,b). This means that (x0, y0, z0) ̸=
(x1, y1, z1) but G(x0, y0, z0) =G(x1, y1, z1) = (a,b,c). Hence G is not a diffeomorphism. □
Remark 8. In fact our example is not only semi-algebraic but regular (in the sense of algebraic
geometry, which means that every component fi of F is locally a quotient of two polynomials
P/Q, where Q does not vanish). Moreover in the same way we can show that there exists a regular
mapping F : R3 → R3 with constant non-zero Jacobian such that F (R3) ⊂ R2 ×R+. Indeed, let F
be as in Theorem ??. Then G(x, y, z) = (z/J (F (x, y)),F (x, y)) is a regular non-injective map with
Jacobian equal to one, such that G(R3) ⊂R2×R+. This shows that regular mappings with Jacobian
1 can have non-dense image.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
Acknowledgments
The authors would like to thank an anonymous reviewer for a careful reading of the text and many
helpful comments.
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The Pinchuk example revisited

Andrea Braides

Comptes Rendus
Mathématique
Andrea Braides
A simplified counterexample to the integral representation of the relaxation of
double integrals
Volume 362 (2024), p. 487-491
https://doi.org/10.5802/crmath.558
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
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Comptes Rendus. Mathématique
2024, Vol. 362, p. 487-491
https://doi.org/10.5802/crmath.558
Research article / Article de recherche
Partial differential equations / Équations aux dérivées partielles
A simplified counterexample to the integral
representation of the relaxation of double
integrals
Un contre-exemple simplifié à la représentation
intégrale de la relaxation des intégrales doubles
Andrea Braides a
a SISSA, via Bonomea 265, Trieste, Italy
E-mail: braides@mat.uniroma2.it
Abstract. We show that the lower-semicontinuous envelope of a non-convex double integral may not admit
a representation as a double integral. By taking an integrand with value +∞ except at three points (say −1, 0
and 1) we give a simple proof and an explicit formula for the relaxation that hopefully may shed some light
on this type of problems. This is a simplified version of examples by Mora-Corral and Tellini, and Kreisbeck
and Zappale, who characterize the lower-semicontinuous envelope via Young measures.
Résumé. Nous montrons que l’enveloppe semi-continue inférieure d’une intégrale double non convexe peut
ne pas admettre de représentation sous forme d’intégrale double. En prenant un intégrande avec une valeur
infinie sauf en trois points (disons -1, 0 et 1), nous donnons une preuve simple et une formule explicite pour la
relaxation qui, espérons-le, pourra éclairer ce type de problèmes. Ceci est une version simplifiée des exemples
de Mora-Corral et Tellini, et de Kreisbeck et Zappale, qui caractérisent l’enveloppe semi-continue inférieure
via les mesures de Young.
Manuscript received 11 May 2023, accepted 30 August 2023.
Double-integral functionals defined in Lp spaces of the form
F (u) :=
∫
Ω×Ω
f (u(x)−u(y))d xd y (1)
can be treated using the direct methods of the Calculus of Variations. To that end, necessary
and sufficient conditions for the lower semicontinuity of F with respect to weak Lp topologies
turn out to be the convexity and lower semicontinuity of f , exactly as in the case of single-
integral functionals (see e.g. [1, 5]). In the case of non-convex f the parallel is lost. Indeed,
in [4] it is shown that the lower-semicontinuous envelope of F cannot be represented as a double
integral of the same form when the function f is a simple double-well potential. The proof in [4]
relies on the representation of the relaxed functional in terms of Young measures and on the
study of the optimality conditions satisfied by such measure-valued minimizers. We now give
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
488 Andrea Braides
a simple explanation of the non-representability of the relaxed functional when f is a double-
well potential (or rather a “triple-well” potential with wells in −1, 0 and 1) with “infinite depth”;
namely,
f (z) =

0 if z ∈ {−1,1}
1 if z = 0
+∞ otherwise.
(2)
For simplicity we chose Ω = (0,1). We remark that we can extend this example to everywhere
finite integrands f by approximation.
We note that other examples are shown in [3] when the integrand is of the form f (u(x),u(y)).
In that case the functionals are not invariant by translations, so the parallel with local functionals
would be with integrands depending on (u(x),∇u(x)), for which lower-semicontinuity conditions
are more complex [2].
We now turn to the analysis of the counterexample.
1. Characterization of the lower-semicontinuous envelope
Note preliminarily that a lower bound for the lower-semicontinuous envelope F of F with respect
to the weak L1-convergence is
F0(u) :=
∫
Ω×Ω
f ∗∗(u(x)−u(y))d xd y, (3)
where the lower-semicontinuous convex envelope of f is
f ∗∗(z) =
{
0 if z ∈ [−1,1]
+∞ otherwise; (4)
that is,
F0(u) =
{
0 if ess-supu −ess-infu ≤ 1
+∞ otherwise. (5)
This lower bound implies that F is finite at most on functions u ∈ L∞(0,1) such that
ess-supu −ess-infu ≤ 1. (6)
Let u ∈ L∞(0,1) satisfy (6), and let u j be a sequence weakly converging to u and such that
F (u j ) < +∞ for all j . Note that for fixed j the function u j can take at most two values almost
everywhere and these values are at distance 1. Indeed by Fubini’s theorem for almost all y ∈ (0,1)
we have u j (x) ∈ {u j (y),u j (y)−1,u j (y)+1} for almost every x ∈ (0,1). Hence, there exists z j such
that u j (x) ∈ {z j , z j −1, z j +1} for almost every x ∈ (0,1). If both values z j −1 and z j +1 were taken
on sets of positive measure, then we would have F (u j ) =+∞, and a contradiction. Hence, we can
suppose that there exist z j such that u j (x) ∈ {z j , z j +1} almost everywhere. We can assume, up to
subsequences, that z j → z, so that
z ≤ ess-infu and ess-supu ≤ z +1, (7)
and that, if we let A j := {x : u j (x) = z +1}, there exists t ∈ [0,1] such that lim
j →+∞
|A j | = t . Hence, we
obtain
lim
j →+∞
F (u j ) = lim
j →+∞
(
|A j |2 +
(
1−|A j |
)2)= t 2 + (t −1)2 = 2t 2 −2t +1. (8)
Note that the minimum of t 2 + (1− t )2 is 12 so that (8) implies that F (u) ≥ 12 for all u.
Since by the convergence of
∫
(0,1) u j d x to
∫
(0,1) u d x we have
t =
∫
(0,1)
u d x − z, (9)
Andrea Braides 489
the limit of F (u j ) can be described in terms of
∫
(0,1) u d x and z only, and is independent of the
particular sequence u j .
Note conversely that if u and z are such that (7) holds, then there exist u j with u j ∈ {z, z +1}
and weakly converging to u, so that the value 2t 2+2t+1 is achieved on this sequence with t given
by (9). By optimizing in z we then have a description of F (u) as
F (u) = min
{(∫
(0,1)
u d x − z
)2
+
(∫
(0,1)
u d x − z −1
)2
: z ≤ ess-infu, ess-supu ≤ z +1
}
= min
{
2
(∫
(0,1)
u d x
)2
−2(2z +1)
(∫
(0,1)
u d x
)
+2z2 +2z +1
: z ≤ ess-infu, ess-supu ≤ z +1} .
(10)
We can make this formula more symmetric by the change of variables w = z + 12 , so that
F (u) = min
{
2
(∫
(0,1)
u d x
)2
−4w
(∫
(0,1)
u d x
)
+2w2 + 1
2
: ess-supu − 1
2
≤ w ≤ ess-infu + 1
2
}
. (11)
Furthermore, noting that the functionals are invariant if we add a constant to u, replacing u by
u −∫(0,1) u d x = 0 we also have
F (u) = min
{
2w2 + 1
2
: ess-supu −
∫
(0,1)
u d x − 1
2
≤ w ≤ ess-infu −
∫
(0,1)
u d x + 1
2
}
. (12)
2. Non representability of the lower-semicontinuous envelope
We now prove that there exists no g such that
F (u) =
∫
Ω×Ω
g (u(x)−u(y))d xd y. (13)
Note that g can be assumed to be even, up to replacing g (z) with 12 (g (z)+ g (−z)).
We first describe F (u) more precisely in some “extreme” cases. In the first one the minimiza-
tion does not involve constraint (7), so that F (u) = 12 . To get this, we note that if
ess-supu −ess-infu ≤ 1
2
(14)
then we can take z = ∫(0,1) u d x − 12 , and by (14) we have
ess-supu ≤ ess-infu + 1
2
≤ z +1 and z ≤ ess-supu − 1
2
≤ ess-infu,
and F (u) = 12 by formula (10). As a particular case of a function satisfying (14) we can take u a
constant. In this case (13) would give
g (0) = 1
2
. (15)
The other “extreme” case is when only one z is involved in the minimization in (10); which is
the case when ess-supu − ess-infu = 1, so that z = ess-infu and z +1 = ess-supu. The value of
F (u) is then simply
F (u) =
(∫
(0,1)
u d x −ess-infu
)2
+
(∫
(0,1)
u d x −ess-supu
)2
.
490 Andrea Braides
This can be applied, for fixed t ∈ (0,1), with u given by
u(x) =
{
1 if x ≤ t
0 if x > t ,
for which F (u) = 2t 2 −2t +1. If (13) held true then by (15) we would also have
F (u) = (2t 2 −2t +1)g (0)+2t (1− t )g (1) = 1
2
(
2t 2 −2t +1)+2t (1− t )g (1),
which would give
g (1) = 2t
2 −2t +1
4t (1− t ) =
1
4
(
t
1− t +
1− t
t
)
.
Taking different values for t ∈ (0,1) we get different values for g (1), which is a contradiction.
3. Conclusions and remarks
Formula (10) shows that F (u) is obtained by functions u j weakly converging to u and oscillating
between two values z and z + 1 maximizing the measure of the subset of points (x, y) ∈ Ω×Ω
such that u j (x) = z and u j (y) = z + 1. This operation depends only on z, which satisfies some
constraints due to the convergence of u j to u; minimizing the outcome in z gives the optimal
choice of u j . Minimization in z is unconstrained if ess-supu −ess-infu ≤ 12 , while it is limited to
a single z when ess-supu − ess-infu = 1. The dependence on the quantity ess-supu − ess-infu
highlights the nonlocality of the recovery sequences. An example of this fact is obtained by
considering constants u = c, for which we have minimizing sequences oscillating between c − 12
and c + 12 , while this is not true for piecewise-constant functions: if u takes only two values at
distance 1 then a recovery sequence is u itself, without oscillations.
We remark that from this example we also obtain examples with finite integrand. Indeed, if fn
is a sequence of functions increasingly converging to f given by (2) and
Fn(u) :=
∫
Ω×Ω
fn(u(x)−u(y))d xd y, (16)
then the lower-semicontinuous envelopes F n converge to F . If there existed (convex) functions
gn such that
F n(u) :=
∫
Ω×Ω
gn(u(x)−u(y))d xd y, (17)
then this would hold also for F .
Acknowledgments
The author gratefully acknowledges valuable comments by Carolin Kreisbeck.
Declaration of interests
The author does not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and has declared no affiliations other than his research
organizations.
Andrea Braides 491
References
[1] J. C. Bellido and C. Mora-Corral, “Lower semicontinuity and relaxation via Young measures
for nonlocal variational problems and applications to peridynamics”, SIAM J. Math. Anal.
50 (2018), pp. 779–809.
[2] E. De Giorgi, G. Buttazzo and G. Dal Maso, “On the lower semicontinuity of certain integral
functionals”, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 74 (1983), pp. 274–
282.
[3] C. Kreisbeck and E. Zappale, “Loss of double-integral character during relaxation”, SIAM J.
Math. Anal. 53 (2021), pp. 351–385.
[4] C. Mora-Corral and A. Tellini, “Relaxation of a scalar nonlocal variational problem with a
double-well potential”, Calc. Var. Partial Differ. Equ. 59 (2020), article no. 67 (30 pages).
[5] P. Pedregal, “Weak lower semicontinuity and relaxation for a class of non-local functionals”,
Rev. Mat. Complut. 29 (2016), pp. 485–495.


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