Annales de la Faculté des Sciences de Toulouse

Renata Bunoiu

Lucas Chesnel

Karim Ramdani

Mahran Rihani

Homogenization of Maxwell’s equations and related scalar problems with sign-changing coefficients

Damien Gaboriau

Camille Noûs

On the top-dimensional $\ell ^2$-Betti numbers

Camille Laurent-Gengoux

Mohsen Masmoudi

Infinis morphismes de Leibniz pour les crochets dérivés

Binru Li

Cyclic covers of Stable curves and their moduli spaces

Christiane Rousseau

Two-parameter unfolding of a parabolic point of a vector field in $\protect \mathbb{C}$ fixing the origin

Shaoyu Dai

Yifei Pan

A right inverse of Cauchy–Riemann operator $\protect \bar{\partial }^k+a$ in the weighted Hilbert space $L^2(\protect \mathbb{C},e^{-|z|^2})$

Bruno Schapira

Daniel Valesin

Exponential rate for the contact process extinction time

Max Fathi

Yuxin Ge

Mihai Maris

Xavier Lamy

Publication membre du centre
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ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
MAX FATHI, YUXIN GE, MIHAI MARIS AND XAVIER LAMY
Volume spécial à l’occasion du semestre thématique “Calculus of Variations
and Probability”
Tome XXX, no 2 (2021), p. i–iii.
https://doi.org/10.5802/afst.1672
© Université Paul Sabatier, Toulouse, 2021.
L’accès aux articles de la revue « Annales de la faculté des sciences de Toulouse
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articles sont publiés sous la license CC-BY 4.0.
Preface
A thematic semester entitled Calculus of Variations and Probability spon-
sored by the International Center for Mathematics and Computer Science
in Toulouse (CIMI) was organized from February to June 2019.
Calculus of variations is a branch of mathematical analysis that aims
at finding and analyzing extrema of functionals via studying the effects of
small perturbations in their argument. Many concrete problems from natural
sciences can be recast in terms of calculus of variations. For example, equi-
librium configurations of physical systems are often critical points of some
energy functional, while evolutions of many physical systems follow some
least action principle (optics, Newtonian mechanics, general relativity, super-
fluidity, magnetic theories, liquid crystals. . . ). Techniques from calculus of
variations have found applications several branches of mathematics: partial
differential equations, geometry (minimal surfaces, isoperimetric problems),
numerical analysis, probability (large deviations, statistical physics), statis-
tics, linear algebra (eigenvalue problems) or dynamical systems (Hamiltonian
systems).
The thematic program consisted of three conferences, several intensive
courses and lectures, and a winter school aimed at promoting modern and
active research areas linked to Calculus of Variations, with a particular focus
on topics at the boundary with probability theory. The main topics included
the study of singularities in physical problems such as micromagnetics or
liquid crystals; optimal transport, Monge-Ampère equation and gradient flow
structures for evolution equations; convex integration and fluid mechanics;
quantitative stochastic homogenization; geometric flows; minimal surfaces;
shape optimization; isoperimetric and functional inequalities.
This volume collects several research and expository articles by some of
the plenary lecturers and their coauthors, aiming at presenting an overview
of some topics covered during the semester program.
The contribution of Y. Ge and J. Ye investigates the smoothness of the
optimal transport map between the smooth densities with respect to the
squared Riemannian distance cost. The optimal transport map is character-
ized by exp(∇u), where the potential function u satisfies a Monge–Ampère
type equation. To study the regularity of u, Ma–Trudinger–Wang introduced
for the first time the MTW tensor. Later on, Kim–McCann interpreted the
MTW tensor as a curvature tensor of some pseudo-Riemannian metric. The
– i –
authors study the regularity issue on Riemannian manifolds with curvature
sufficiently close to curvature of round sphere in C2 norm in all dimensions.
They prove that the MTW tensor on such Riemannian manifolds satisfies
an improved Ma–Trudinger–Wang condition. As a consequence, they imply
the smoothness of the optimal transport map by the continuity method.
Optimal transport methods have found many applications to geometric
functional inequalities, and the extension to a discrete setting has recently
seen a lot of activity. D. Halikias, B. Klartag and B. A. Slomka study dis-
crete versions of the classical Brunn–Minkowski inequality on the volume of
the Minkowski sum of sets. They give a unified proof of the discrete Brunn–
Minkowski inequality of Klartag and Lehec and of the four function theo-
rem of Ahlswede and Daykin, and extend them to finitely-generated abelian
groups.
One of the important applications of optimal transport in mathematics is
that it can be used to study geometry of metric spaces, including non-smooth
situations. F. Cavalletti, N. Gigli and F. Santarcangelo consider synthetic
notions of Ricci curvature lower bounds in terms of optimal transport. They
show that for non-branching metric-measure spaces, the Lott–Sturm–Villani
characterization in terms of convexity of entropy along geodesics for the L2
optimal transport distance is equivalent to a notion of curvature defined
from L1 optimal transport. The latter is formulated in terms of convexity
along one-dimensional curves defined from an L1 transport map, rather than
L2. In particular, they show that the Lott–Sturm–Villani conditon can be
equivalently expressed using L1 optimal transport, instead of L2.
G. Canevari and G. Orlandi briefly review the distributional Jacobian
and the oriented coarea formula, which are classical tools for the study of
vector-valued Sobolev maps, and indicate some applications to variational
problems with emphasis on the Ginzburg-Landau energy. Then they propose
new tools — flat chains with coefficients in an abelian group — for the
study of singularities of manifold-valued Sobolev maps and apply them to
variational models arising in material science. The lifting of BV maps with
values into manifolds is also discussed.
A. Pisante reviews very recent results obtained with F. Dipasquale and
V. Millot on minimizers of a Landau–De Gennes functional appearing in
the modeling of three-dimensional nematic liquid crystals. The main goal of
that series of results is to provide new insight into the structure of axially
symmetric minimizers under a pointwise norm constraint. He describes how
minimizers encode topological information on the domain and presents sev-
eral regimes and scenarios, such as the emergence of torus-like solutions and
of split solutions.
– ii –
P. Mironescu and J. Van Schaftingen consider the problem of character-
izing the traces of functions in the Sobolev space W 1,p(Bm−1 × (0, 1);N ),
where Bm−1 is the unit ball in Rm−1 and N is a compact manifold. The
classical trace theory implies that traces must belong to W 1−
1
p ,p(Bm−1;N ).
A natural question is whether any function in the latter space is the trace
of a function in the former one. If the answer is affirmative, it is also nat-
ural to ask whether the trace operator has a continuous right inverse; in
other words, whether it is possible to find an extension of any function in
W 1−
1
p ,p(Bm−1;N ) whose norm can be estimated in terms of the norm of the
trace. The article provides a review of the current state of the art on these
topics, and presents new, original results. In particular, the authors find a
new analytical obstruction for the existence of an extension, and generalize
a result of R. Hardt and F. H. Lin on the existence of an extension under a
topological condition on the target manifold N .
The contribution of S. Dipierro, O. Savin, and E. Valdinoci deals with
the notion of divergent fractional Laplacian previously introduced by the
authors. They show that any function u defined on RN with polynomial
growth at infinity can be locally approximated by s-harmonic functions:
given ε > 0, there exist Rε > 0 and an s-harmonic function uε such that
‖u − uε‖Cm(B1) 6 ε and u = uε on RN \ B(0, Rε). Of course, the price
to pay is that uε may oscillate heavily and may be far away from u on
B(0, Rε) \B(0, 1). Similarly, it is shown that any function with mild growth
at infinity can be locally approximated by solutions of nonlinear equations
involving the divergent fractional Laplacian. The solvability of the Dirichlet
problem for the divergent fractional Laplacian in a ball is also proven.
We are very grateful to the Labex CIMI for funding and for administrative
support to organize the semester, to the Annales de la Faculté des Sciences
de Toulouse for the opportunity to publish this special issue, to the authors
and to the anonymous referees.
Max Fathi, Yuxin Ge, Mihai Maris, Xavier Lamy
– iii –


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Homogenization of Maxwell’s equations and related scalar problems with sign-changing coefficients

On the top-dimensional $\ell ^2$ -Betti numbers

Infinis morphismes de Leibniz pour les crochets dérivés

Cyclic covers of Stable curves and their moduli spaces

Two-parameter unfolding of a parabolic point of a vector field in \protect \mathbb{C} fixing the origin

A right inverse of Cauchy–Riemann operator \protect \bar{\partial }^k+a in the weighted Hilbert space L^2(\protect \mathbb{C},e^{-|z|^2})

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