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 ≥ 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∈𝒫_2(ℝ^d) are centred probabilities with prescribed variances.

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.

Abstract 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 $$d \ge 1$$ d ≥ 1 with an optimal constant. Similarly, we propose an optimal upper bound for the ratio of Zolotarev distance $$Z_2(\mu ,\nu )$$ Z 2 ( μ , ν ) to Wasserstein distance $$W_2(\mu ,\nu )$$ W 2 ( μ , ν ) when $$\mu ,\nu \in \mathcal {P}_2(\mathbb {R}^d)$$ μ , ν ∈ P 2 ( R d ) are centred probabilities with prescribed variances.

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.

We derive a convex relaxation principle for a large class of non convex variational problems where the functional to be minimized involves a one homogeneous gradient energy. This applies directly to free boundary or multiphase problems in the case of the classical total variation or of some anisotropic variants. The underlying argument is an exclusion principle which states that any global minimizer avoids taking values in the intervals where the lower order potential is nonconvex. This allows using duality methods and deriving a saddle point characterization of the global minimizers. A numerical validation of our principle is presented in the case of several free boundary and multiphase problems that we treat through a primal-dual algorithm. The accuracy of the interfaces and the convergence of the algoritm benefit in a large way of a new epigraphical projection method that we introduced to tackle the non differentiability of the convexified Lagrangian.

In this paper we consider the Density Functional Theory (DFT) framework, where a functional of the form F_(ρ)= 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 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 ionization phenomenon may occur, consisting in the fact that the minimizers of F_ 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 , is to study the asymptotic behavior of the minimizers of F_ as →0. 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.

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(epsilon)(x,y)=& ell;(|x-y|/epsilon) where & ell;:R+->[0,infinity] is decreasing to zero at infinity and parameter epsilon>0 scales the interaction distance. In this paper we identify the limit energy of such a model in the short-range regime epsilon << 1 under the sole assumption that there exists r(0)>0 : integral r(0)(infinity)& ell;(r)r(d-1)dr<+infinity. This extends recent results [D. Hardin, E. B. Saff, and O. Vlasiuk, Asymptotic Properties of Short-Range Interaction Functionals, preprint, https://arxiv.org/abs/2010.11937, 2021], [D. P. Hardin, T. Lebl & eacute;, E. B. Saff, and S. Serfaty, Constr. Approx., 48 (2018), pp. 61-100], [M. Lewin, J. Math. Phys., 63 (2022), 061101] obtained in the homogeneous case & ell;(r)=r(-s) where s>d.

Abstract Optimization problems on probability measures in ℝ d {\mathbb{R}^{d}} are considered where the cost functional involves multi-marginal optimal transport. In a model of N interacting particles, for example in Density Functional Theory, the interaction cost is repulsive and described by a two-point function c ⁢ ( x , y ) = ℓ ⁢ ( | x - y | ) {c(x,y)=\ell(\lvert x-y\rvert)} where ℓ : ℝ + → [ 0 , ∞ ] {\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 [4] and present a duality method which allows to compute the Γ-limit as N → ∞ {N\to\infty} under very general assumptions on the cost ℓ ⁢ ( r ) {\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 {<1} . In a last part, we study the case of a small range interaction ℓ N ⁢ ( r ) = ℓ ⁢ ( r / ε ) {\ell_{N}(r)=\ell(r/\varepsilon)} ( ε ≪ 1 {\varepsilon\ll 1} ) and we show how the duality approach can also be used to determine the limit energy as ε → 0 {\varepsilon\to 0} of a very large number N ε {N_{\varepsilon}} of particles.

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 $s>d$.

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.

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.

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.

We consider the shape optimization problem which consists in placing a given mass $m$ of elastic material in a design region so that the compliance is minimal. Having in mind optimal light structures, our purpose is to show that the problem of finding the stiffest shape configuration simplifies as the total mass $m$ tends to zero: we propose an explicit relaxed formulation where the compliance appears after rescaling as a convex functional of the relative density of mass. This allows us to write necessary and sufficient optimality conditions for light structures following the Monge-Kantorovich approach developed recently in [5].

We establish the uniqueness of the solutions for a degenerate scalar problem in the multiple integrals calculus of variations. The proof requires as a preliminary step the study of the regularity properties of the solutions and of their level sets. We exploit the uniqueness and the regularity results to explore some of their qualitative properties. In particular, we emphasize the link between the supports of the solutions and the Cheeger problem.

In many applications of structural engineering, the following question arises: given a set of forces f1, f2, …, fN applied at prescribed points x1, x2, …, xN, 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 x1, x2, …, xN in the two- and three-dimensional cases. 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 that the loop cannot be subdivided into subloops, and where P is the number of forces f1, f2, …, fN applied at points strictly within the convex hull of x1, x2, …, xN. In three dimensions, we show that, by slightly perturbing f1, f2, …, fN, 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 channelling stress in desired ways.

We show that the compliance functional in elasticity is differentiable with respect to horizontal variations of the load term, when the latter is given by a possibly concentrated measure; moreover, we provide an integral representation formula for the derivative as a linear functional of the deformation vector field. The result holds true as well for the p-compliance in the scalar case of conductivity. Then we study the limit problem as \(p \rightarrow + \infty \), which corresponds to differentiate the Wasserstein distance in optimal mass transportation with respect to horizontal perturbations of the two marginals. Also in this case, we obtain an existence result for the derivative, and we show that it is found by solving a minimization problem over the family of all optimal transport plans. When the latter contains only one element, we prove that the derivative of the p-compliance converges to the derivative of the Wasserstein distance in the limit as \(p \rightarrow + \infty \).

We study a class of optimal transport planning problems where the reference cost involves a non-linear function G ( x, p ) representing the transport cost between the Dirac measure δ x and a target probability p . This allows to consider interesting models which favour multi-valued transport maps in contrast with the classical linear case ( $G(x,p)=\int c(x,y)dp$ ) where finding single-valued optimal transport is a key issue. We present an existence result and a general duality principle which apply to many examples. Moreover, under a suitable subadditivity condition, we derive a Kantorovich–Rubinstein version of the dual problem allowing to show existence in some regular cases. We also consider the well studied case of Martingale transport and present some new perspectives for the existence of dual solutions in connection with Γ-convergence theory.

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}.

The aim of this paper is to present a general convexification recipe that can be useful for studying non-convex variational problems. In particular, this allows us to treat such problems by using a powerful primal-dual scheme. Possible further developments and open issues are given. (C) 2017 Academie des sciences. Published by Elsevier Masson SAS.

In this chapter the structures under study are made of a basic cell whose elements show a resonant behavior The paradigmatic example is a dielectric rod with a sufficiently high index to have present a Mie resonance at a wavelength that is large with respect to the period. Silicon material could open the route toward the realization of all-dielectric metamaterials operating at optical frequencies. It is worth noting that very similar structures have been fabricated and characterized some years ago. he magnetic dipole term is, responsible for the first-order spatial dispersion. Since non-local effects are very sensitive to the symmetry of structure, the coupling between the rods plays an important role in both the spectral and spatial responses. The underlying origin of their dispersion properties actually relies on the collective response of the resonant rods, which defined them as true metamaterials on the same level as metallic metamaterials, with advantages that dielectric structures are scale-invariant and exhibit no intrinsic loss.