arXiv subjects

Guy Bouchitté

Publications and source records attributed to Guy Bouchitté.

At least 19 recordsLinked to original sources

arXiv2026

The Monge-Kantorovich tangent bundle to a measure

We study the tangent bundle $T_μ$ to a Radon measure $μ$ in Euclidean space, introduced by Bouchitté, Champion and Jimenez (2005). Its construction, inspired by Monge-Kantorovich optimal transport theory, involves the duality between Lipschitz functions and the so-called Arens-Eells space, a Banach subspace of distributions obtained by completing the set of balanced signed measures. Precisely, a velocity field $σ$ is $μ$-tangent when the divergence of $σμ$ lies in the Arens-Eells space. The tangent bundle $T_μ$ defined $μ$-almost everywhere provides a local projection that allows to construct a $μ$-tangential gradient operator on Lipschitz functions that is weakly continuous and enjoys integration by parts. In this paper, we introduce a new quantitative estimate involving the tangential and normal components of a given velocity vector field $σ\in L^1_μ(\mathbb{R}^d)$. Specifically, we show that the tangential condition $σ\in T_μ$ holding $μ$ a.e. is equivalent to each of the following two conditions: the convergence in the Arens-Eells space of $h^{-1} \left((id+h σ)_{\#} μ- μ\right)$ as $h \to 0$, and the differentiability of Lipschitz functions along $σ$. Moreover, given a field of non-tangent directions, we construct a Lipschitz function that is not differentiable on a large set, confirming that $T_μ$ agrees with the decomposability bundle of Alberti and Marchese. We finally relate $T_μ$ to the tangent measures of Preiss and survey further properties of the tangential differential calculus.

math.AP↗

arXiv2025

Calibration for minimal surfaces with free boundary and Cheeger-type problems

We study a problem of minimal surfaces with free boundary written in the form of a non convex minimization problem. Our aim is to characterize optimal solutions by finding a suitable calibration field. A natural upper bound of the infimum is given by a variant of the Cheeger problem that we solve explicitly proving the optimality thanks to the construction of a cut-locus potential. The comparison with the original problem is then discussed in detail.

math.AP↗

arXiv2025

Kantorovich-Rubinstein duality theory for the Hessian

The classical Kantorovich-Rubinstein duality theorem establishes a significant connection between Monge optimal transport and maximization of a linear form on the set of 1-Lipschitz functions. This result has been widely used in various research areas. In particular, it unlocks the optimal transport methods in some of the optimal design problems. This paper puts forth a similar theory when the linear form is maximized over $C^{1,1}$ functions whose Hessian lies between minus and plus identity matrix. The problem will be identified as the dual of a specific optimal transport formulation that involves three-point plans. The first two marginals are fixed, while the third must dominate the other two in the sense of convex order. The existence of optimal plans allows to express solutions of the underlying Beckmann problem as a combination of rank-one tensor measures supported on a graph. In the context of two-dimensional mechanics, this graph encodes the optimal configuration of a grillage that transfers a given load system.

math.OC↗

arXiv2025

Sharp inequalities between Zolotarev and Wasserstein distances in $\mathrm{P}_2(\mathbb{R}^d)$

Based on a new Kantorovich-Rubinstein duality principle for the Hessian that was recently established by the two authors, we extend the Rio inequality to any dimension $d \ge 1$ with an optimal constant. Similarly, we propose an optimal upper bound for the ratio of Zolotarev distance $Z_2(μ,ν)$ to Wasserstein distance $W_2(μ,ν)$ when $μ,ν\in \mathrm{P}_2(\mathbb{R}^d)$ are centred probabilities with prescribed variances.

math.PR↗

arXiv2025

Structural properties of one-dimensional metric currents: SBV-representations, connectedness and the flat chain conjecture

A comprehensive study of one-dimensional metric currents and their relationship to the geometry of metric spaces is presented. We resolve the one-dimensional flat chain conjecture in this general setting, by proving that its validity is equivalent to a simple geometric connectedness property. More precisely, we prove that metric currents can be approximated in the mass norm by normal currents if and only if every $1$-rectifiable set can be covered by countably many Lipschitz curves up to an $\mathscr{H}^1$-negligible set. Building on this, we demonstrate that any $1$-current in a Banach space can be completed into a cycle by a rectifiable current, with the added mass controlled by the Kantorovich--Rubinstein norm of its boundary. We further refine our approximation result by showing that these currents can be approximated by polyhedral currents modulo a cycle. Finally, in arbitrary complete metric spaces, we establish a Smirnov-type decomposition for one-dimensional currents. This decomposition expresses such currents as a superposition, without mass cancellation, of currents associated with curves of bounded variation that have a vanishing Cantor part.

math.AP↗

arXiv2025

Dissociation limits in Density Functional Theory

In this paper we consider the {\it Density Functional Theory} (DFT) framework, where a functional of the form $$F_\eps(ρ)=\eps T(ρ)+bC(ρ)-U(ρ)$$ has to be minimized in the class of non-negative measures $ρ$ which have a prescribed total mass $m$ (the total electronic charge). The parameter $\eps$ is small and the terms $T$, $C$, $U$ respectively represent the kinetic energy, the electronic repulsive correlation, the potential interaction term between electrons and nuclei. Several expressions for the above terms have been considered in the literature and our framework is general enough to include most of them. It is known that in general, when the positive charge of the nuclei is small, the so-called {\it ionization phenomenon} may occur, consisting in the fact that the minimizers of $F_\eps$ can have a total mass lower than $m$; this physically means that some of the electrons may escape to infinity when the attraction of the nuclei is not strong enough. Our main goal, continuing the research we started in \cite{bbcd18}, is to study the asymptotic behavior of the minimizers of $F_\eps$ as $\eps\to0$. We show that the $Γ$-limit functional is defined on sums of Dirac masses and has an explicit expression that depends on the terms $T$, $C$, $U$ that the model takes into account. Some explicit examples illustrate how the electrons are distributed around the nuclei according to the model used.

math-ph↗

arXiv2023

Mean field theory for a general class of short-range interaction functionals

In models of $N$ interacting particles in $\R^d$ as in Density Functional Theory or crowd motion, the repulsive cost is usually described by a two-point function $c_\e(x,y) =\ell\Big(\frac{|x-y|}{\e}\Big)$ where $\ell: \R_+ \to [0,\infty]$ is decreasing to zero at infinity and parameter $\e>0$ scales the interaction distance. In this paper we identify the mean-field energy of such a model in the short-range regime $\e\ll 1$ under the sole assumption that $\exists r_0>0 \ : \ \int_{r_0}^\infty \ell(r) r^{d-1}\, dr <+\infty$. This extends recent results \cite{hardin2021, HardSerfLebl, Lewin} obtained in the homogeneous case $\ell(r) = r^{-s}$ where $s>d$.

math-ph↗

arXiv2022

Relaxed many-body optimal transport and related asymptotics

Optimization problems on probability measures in $\mathbb{R}^d$ are considered where the cost functional involves multi-marginal optimal transport. In a model of $N$ interacting particles, like in Density Functional Theory, the interaction cost is repulsive and described by a two-point function $c(x,y) =\ell(|x-y|)$ where $\ell: \mathbb{R}_+ \to [0,\infty]$ is decreasing to zero at infinity. Due to a possible loss of mass at infinity, non existence may occur and relaxing the initial problem over sub-probabilities becomes necessary. In this paper we characterize the relaxed functional generalizing the results of \cite{bouchitte2020relaxed} and present a duality method which allows to compute the $Γ-$limit as $N\to\infty$ under very general assumptions on the cost $\ell(r)$. We show that this limit coincides with the convex hull of the so-called direct energy. Then we study the limit optimization problem when a continuous external potential is applied. Conditions are given with explicit examples under which minimizers are probabilities or have a mass $<1$ . In a last part we study the case of a small range interaction $\ell_N(r)=\ell (r/\varepsilon)$ ($\varepsilon\ll 1$) and we show how the duality approach can be also used to determine the limit energy as $\varepsilon\to 0$ of a very large number $N_\varepsilon$ of particles.

math.OC↗

arXiv2021

Optimal design versus maximal Monge-Kantorovich metrics

A remarkable connection between optimal design and Monge transport was initiated in the years 1997 in the context of the minimal elastic compliance problem and where the euclidean metric cost was naturally involved. In this paper we present different variants in optimal design of mechanical structures, in particular focusing on the optimal pre-stressed elastic membrane problem. We show that the underlying metric cost is associated with an unknown maximal monotone map which maximizes the Monge-Kantorovich distance between two measures. In parallel with the classical duality theory leading to existence and (in a smooth case) to PDE optimality conditions, we present a general geometrical approach arising from a two-point scheme in which geodesics with respect to the optimal metric play a central role. These two aspects are enlightened by several explicit examples and also by numerical solutions in which optimal structures very often turn out to be truss-like i.e supported by piecewise affine geodesics. In case of a discrete load, we are able to relate the existence of such truss-like solutions to an extension property of maximal monotone maps which is of independent interest and that we propose here as a conjecture.

math.OC↗

arXiv2019

Relaxed multi-marginal costs and quantization effects

We propose a duality theory for multi-marginal repulsive cost that appear in optimal transport problems arising in Density Functional Theory. The related optimization problems involve probabilities on the entire space and, as minimizing sequences may lose mass at infinity, it is natural to expect relaxed solutions which are sub-probabilities. We first characterize the $N$-marginals relaxed cost in terms of a stratification formula which takes into account all $k$ interactions with $k\le N$. We then develop a duality framework involving continuous functions vanishing at infinity and deduce primal-dual necessary and sufficient optimality conditions Next we prove the existence and the regularity of an optimal dual potential under very mild assumptions. In the last part of the paper, we apply our results to a minimization problem involving a given continuous potential and we give evidence of a mass quantization effect for optimal solutions.

math.AP↗

arXiv2019

On the forces that cable webs under tension can support and how to design cable webs to channel stresses

In many applications of Structural Engineering the following question arises: given a set of forces $\mathbf{f}_1,\mathbf{f}_2,\dots,\mathbf{f}_N$ applied at prescribed points $\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_N$, under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points $\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_N$ in the two- and three-dimensional case. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two-dimensions we show that any such web can be replaced by one in which there are at most $P$ elementary loops, where elementary means the loop cannot be subdivided into subloops, and where $P$ is the number of forces $\mathbf{f}_1,\mathbf{f}_2,\dots,\mathbf{f}_N$ applied at points strictly within the convex hull of $\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_N$. In three-dimensions we show that, by slightly perturbing $\mathbf{f}_1,\mathbf{f}_2,\dots,\mathbf{f}_N$, there exists a uniloadable web supporting this loading. Uniloadable means it supports this loading and all positive multiples of it, but not any other loading. Uniloadable webs provide a mechanism for distributing stress in desired ways.

math-ph↗

arXiv2018

Dissociating limit in Density Functional Theory with Coulomb optimal transport cost

In the framework of Density Functional Theory with Strongly Correlated Electrons we consider the so called bond dissociating limit for the energy of an aggregate of atoms. We show that the multi-marginals optimal transport cost with Coulombian electron-electron repulsion may correctly describe the dissociation effect. The variational limit is completely calculated in the case of N=2 electrons. The theme of fractional number of electrons appears naturally and brings into play the question of optimal partial transport cost. A plan is outlined to complete the analysis which involves the study of the relaxation of optimal transport cost with respect to the weak* convergence of measures.

math.AP↗

arXiv2016

A duality theory for non-convex problems in the Calculus of Variations

We present a new duality theory for non-convex variational problems, under possibly mixed Dirichlet and Neumann boundary conditions. The dual problem reads nicely as a linear programming problem, and our main result states that there is no duality gap. Further, we provide necessary and sufficient optimality conditions, and we show that our duality principle can be reformulated as a min-max result which is quite useful for numerical implementations. As an example, we illustrate the application of our method to a celebrated free boundary problem. The results were announced in \cite{BoFr}.

math.OC↗

arXiv2015

Homogenization near resonances and artificial magnetism in 3D dielectric metamaterials

It is now well established that the homogenization of a periodic array of parallel dielectric fibers with suitably scaled high permittivity can lead to a (possibly) negative frequency-dependent effective permeability. However this result based on a two-dimensional approach holds merely in the case of linearly polarized magnetic fields, reducing thus its applications to infinite cylindrical obstacles. In this paper we consider a dielectric structure placed in a bounded domain of $\mathbb{R}^3$ and perform a full 3D asymptotic analysis. The main ingredient is a new averaging method for characterizing the bulk effective magnetic field in the vanishing-period limit. We evidence a vectorial spectral problem on the periodic cell which determines micro-resonances and encodes the oscillating behavior of the magnetic field from which artificial magnetism arises. At a macroscopic level we deduce an effective permeability tensor that we can be make explicit as a function of the frequency. As far as sign-changing permeability are sought after, we may foresee that periodic bulk dielectric inclusions could be an efficient alternative to the very popular metallic split-ring structure proposed by Pendry.

math.AP↗

arXiv2015

A variational method for second order shape derivatives

We consider shape functionals obtained as minima on Sobolev spaces of classical integrals having smooth and convex densities, under mixed Dirichlet-Neumann boundary conditions. We propose a new approach for the computation of the second order shape derivative of such functionals, yielding a general existence and representation theorem. In particular, we consider the p-torsional rigidity functional for p grater than or equal to 2.

math.OC↗

arXiv2015

Optimal design problems for Schrödinger operators with noncompact resolvents

We consider optimization problems for cost functionals which depend on the negative spectrum of Schrödinger operators of the form $-Δ+V(x)$, where $V$ is a potential, with prescribed compact support, which has to be determined. Under suitable assumptions the existence of an optimal potential is shown. This can be applied to interesting cases such as costs functions involving finitely many negative eigenvalues.

math.AP↗

arXiv2013

The Monge-Kantorovich problem for distributions and applications

We study the Kantorovich-Rubinstein transhipment problem when the difference between the source and the target is not anymore a balanced measure but belongs to a suitable subspace $X(Ω)$ of first order distribution. A particular subclass $X_0^\sharp(Ω)$ of such distributions will be considered which includes the infinite sums of dipoles $\sum_k(δ_{p_k}-δ_{n_k})$ studied in \cite{P1, P2}. In spite of this weakened regularity, it is shown that an optimal transport density still exists among nonnegative finite measures. Some geometric properties of the Banach spaces $X(Ω)$ and $X_0^\sharp(Ω)$ can be then deduced.

math.OC↗

arXiv2012

Thin waveguides with Robin boundary conditions

We consider the Laplace operator in a thin three dimensional tube with a Robin type condition on its boundary and study, asymptotically, the spectrum of such operator as the diameter of the tube's cross section becomes infinitesimal. In contrast with the Dirichlet condition case, we evidence different behaviors depending on a symmetry criterium for the fundamental mode in the cross section. If that symmetry condition fails, then we prove the localization of lower energy levels in the vicinity of the minimum point of a suitable function on the tube's axis depending on the curvature and the rotation angle. In the symmetric case, the behavior of lower energy modes is shown to be ruled by a one dimensional Sturm-Liouville problem involving an effective potential given in explicit form.

math-ph↗