In this paper, we study a connection between disintegration of measures and geometric properties of probability spaces. We prove a disintegration theorem, addressing disintegration from the perspective of an optimal transport problem. We look at the disintegration of transport plans, which are used to define and study disintegration maps. Using these objects, we study the regularity and absolute continuity of disintegration of measures. In particular, we exhibit conditions for which the disintegration map is weakly continuous and one can obtain a path of measures given by this map. We show a rigidity condition for the disintegration of measures to be given into absolutely continuous measures.

In this paper, we give necessary and sufficient conditions for the rigidity of the perimeter inequality under Schwarz symmetrization. The term rigidity refers to the situation in which the equality cases are only obtained by translations of the symmetric set. In particular, we prove that the sufficient conditions for rigidity provided in M. Barchiesi, F. Cagnetti and N. Fusco [Stability of the Steiner symmetrization of convex sets. J. Eur. Math. Soc. 15 (2013), 1245-1278.] are also necessary.

$c$-cyclical monotonicity is the most important optimality condition for an optimal transport plan. While the proof of necessity is relatively easy, the proof of sufficiency is often more difficult or even elusive. We present here a new approach, and we show how known results are derived in this new framework and how this approach allows to prove sufficiency in situations previously not treatable.

We investigate the convergence rate of multi-marginal optimal transport costs that are regularized with the Boltzmann–Shannon entropy, as the noise parameter $\varepsilon $ tends to $0$. We establish lower and upper bounds on the difference with the unregularized cost of the form $C\varepsilon \log (1/\varepsilon )+O(\varepsilon )$ for some explicit dimensional constants C depending on the marginals and on the ground cost, but not on the optimal transport plans themselves. Upper bounds are obtained for Lipschitz costs or locally semiconcave costs for a finer estimate, and lower bounds for $\mathscr {C}^2$ costs satisfying some signature condition on the mixed second derivatives that may include degenerate costs, thus generalizing results previously in the two marginals case and for nondegenerate costs. We obtain in particular matching bounds in some typical situations where the optimal plan is deterministic.

In this paper, we review some recent results on nonlocal interaction problems. The focus is on interaction kernels that are anisotropic variants of the classical Coulomb kernel. In other words, while preserving the same singularity at zero of the Coulomb kernel, they present preferred directions of interaction. For kernels of this kind and general confinement we will prove existence and uniqueness of minimizers of the corresponding energy. In the case of a quadratic confinement we will review a recent result by Carrillo and Shu about the explicit characterization of minimizers, and present a new proof, which has the advantage of being extendable to higher dimensions. In light of this result, we will re-examine some previous works motivated by applications to dislocation theory in materials science. Finally, we will discuss some related results and open questions.

We show that the energy–momentum equations arising from inner variations whose Lagrangian satisfies a generic symmetry condition are ill-posed. This is done by proving that there exists a subclass of Lipschitz solutions that are also solutions to a differential inclusion into the orthogonal group and in particular these solutions can be nowhere $C^1$. We prove that these solutions are not stationary points if the Lagrangian $W$ is $C^1$ and strictly rank-one convex. In view of the Lipschitz regularity result of Iwaniec, Kovalev and Onninen for solution of the energy–momentum equation in dimension 2, we give a sufficient condition for the non-existence of a partial $C^1$ -regularity result even under the condition that the mappings satisfy a positive Jacobian determinant condition. Finally, we consider a number of well-known functionals studied in non-linear elasticity and geometric function theory and show that these do not satisfy this obstruction to partial regularity.

We make some remarks on the Euler–Lagrange equation of energy functional $I(u)=\int _\Omega f(\det Du)\,{\rm d}x,$ where $f\in C^1(\mathbb {R}).$ For certain weak solutions $u$ we show that the function $f'(\det Du)$ must be a constant over the domain $\Omega$ and thus, when $f$ is convex, all such solutions are an energy minimizer of $I(u).$ However, other weak solutions exist such that $f'(\det Du)$ is not constant on $\Omega.$ We also prove some results concerning the homeomorphism solutions, non-quasimonotonicity and radial solutions, and finally we prove some stability results and discuss some related questions concerning certain approximate solutions in the 2-Dimensional cases.

We propose an extension of the anisotropic interaction model which allows for collision avoidance in pairwise interactions by a rotation of forces (Totzeck (2020) Kinet. Relat. Models 13(6), 1219–1242.) by including the agents’ body size. The influence of the body size on the self-organisation of the agents in channel and crossing scenarios as well as the fundamental diagram is studied. Since the model is stated as a coupled system of ordinary differential equations, we are able to give a rigorous well-posedness analysis. Then we state a parameter calibration problem that involves data from real experiments. We prove the existence of a minimiser and derive the corresponding first-order optimality conditions. With the help of these conditions, we propose a gradient descent algorithm based on mini-batches of the data set. We employ the proposed algorithm to fit the parameter of the collision avoidance and the strength parameters of the interaction forces to given real data from experiments. The results underpin the feasibility of the method.

We provide a geometric interpretation of the well-known A3w condition for regularity in optimal transport.

This paper is concerned with the p-Ginzburg–Landau (p-GL) type model with $p\neq 2$ . First, we obtain global energy estimates and energy concentration properties by the singularity analysis. Next, we give a decay rate of $1-|u_\varepsilon |$ in the domain away from the singularities when $\varepsilon \to 0$ , where $u_\varepsilon $ is a minimizer of p-GL functional with $p \in (1,2)$ . Finally, we obtain a Liouville theorem for the finite energy solutions of the p-GL equation on $\mathbb {R}^2$ .

Let $\boldsymbol{f}$ be a square-integrable, zero-mean, random vector with observable realizations in a Hilbert space H, and let $\boldsymbol{g}$ be an associated square-integrable, zero-mean, random vector with realizations which are not observable in a Hilbert space K. We seek an optimal filter in the form of a closed linear operator X acting on the observable realizations of a proximate vector $\boldsymbol{f}_{\epsilon } \approx \boldsymbol{f}$ that provides the best estimate $\widehat{\boldsymbol{g}}_{\epsilon} = X \boldsymbol{f}_{\epsilon}$ of the vector $\boldsymbol{g}$ . We assume the required covariance operators are known. The results are illustrated with a typical example.

The propagation of gradient flow structures from microscopic to macroscopic models is a topic of high current interest. In this paper, we discuss this propagation in a model for the diffusion of particles interacting via hard-core exclusion or short-range repulsive potentials. We formulate the microscopic model as a high-dimensional gradient flow in the Wasserstein metric for an appropriate free-energy functional. Then we use the JKO approach to identify the asymptotics of the metric and the free-energy functional beyond the lowest order for single particle densities in the limit of small particle volumes by matched asymptotic expansions. While we use a propagation of chaos assumption at far distances, we consider correlations at small distance in the expansion. In this way, we obtain a clear picture of the emergence of a macroscopic gradient structure incorporating corrections in the free-energy functional due to the volume exclusion.

Within the framework of the generalised Landau-de Gennes theory, we identify a Q-tensor-based energy that reduces to the four-constant Oseen–Frank energy when it is considered over orientable uniaxial nematic states. Although the commonly considered version of the Landau-de Gennes theory has an elastic contribution that is at most cubic in components of the Q-tensor and their derivatives, the alternative offered here is quartic in these variables. One clear advantage of our approach over the cubic theory is that the associated minimisation problem is well-posed for a significantly wider choice of elastic constants. In particular, this quartic energy can be used to model nematic-to-isotropic phase transitions for highly disparate elastic constants. In addition to proving well-posedness of the proposed version of the Landau-de Gennes theory, we establish a rigorous connection between this theory and its Oseen–Frank counterpart via a Г-convergence argument in the limit of vanishing nematic correlation length. We also prove strong convergence of the associated minimisers.

We revisit the problem of approximating minimizers of certain convex functionals subject to a convexity constraint by solutions of fourth order equations of Abreu type. This approximation problem was studied in previous articles of Carlier–Radice (Approximation of variational problems with a convexity constraint by PDEs of Abreu type. Calc. Var. Partial Differential Equations58 (2019), no. 5, Art. 170) and the author (Singular Abreu equations and minimizers of convex functionals with a convexity constraint, arXiv:1811.02355v3, Comm. Pure Appl. Math., to appear), under the uniform convexity of both the Lagrangian and constraint barrier. By introducing a new approximating scheme, we completely remove the uniform convexity of both the Lagrangian and constraint barrier. Our analysis is applicable to variational problems motivated by the original 2D Rochet–Choné model in the monopolist's problem in Economics, and variational problems arising in the analysis of wrinkling patterns in floating elastic shells in Elasticity.

We consider a deformation $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ of the Dedekind eta function depending on two $d$-dimensional simple lattices $(L,\unicode[STIX]{x1D6EC})$ and two parameters $(m,t)\in (0,\infty )$, initially proposed by Terry Gannon. We show that the minimisers of the lattice theta function are the maximisers of $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ in the space of lattices with fixed density. The proof is based on the study of a lattice generalisation of the logarithm, called the lattice logarithm, also defined by Terry Gannon. We also prove that the natural logarithm is characterised by a variational problem over a class of one-dimensional lattice logarithms.

We compute the optimal investment strategy for an individual who wishes to minimize her probability of lifetime ruin. The financial market in which she invests consists of two riskless assets. One riskless asset is a money market, and she consumes from that account. The other riskless asset is a bond that earns a higher interest rate than the money market, but buying and selling bonds are subject to proportional transaction costs. We consider the following three cases. (1) The individual is allowed to borrow from both riskless assets; ruin occurs if total imputed wealth reaches zero. Under the optimal strategy, the individual does not sell short the bond. However, she might wish to borrow from the money market to fund her consumption. Thus, in the next two cases, we seek to limit borrowing from that account. (2) We assume that the individual pays a higher rate to borrow than she earns on the money market. (3) The individual is not allowed to borrow from either asset; ruin occurs if both the money market and bond accounts reach zero wealth. We prove that the borrowing rate in case (2) acts as a parameter connecting the two seemingly unrelated cases (1) and (3).

It is proved that c-cyclical monotonicity is a sufficient condition for optimality in the multi-marginal optimal transport problem with Coulomb repulsive cost. The notion of c-splitting set and some mild regularity property are the tools. The result may be extended to Coulomb like costs.

We consider a multimarginal transport problem with repulsive cost, where the marginals are all equal to a fixed probability $\unicode[STIX]{x1D70C}\in {\mathcal{P}}(\mathbb{R}^{d})$. We prove that, if the concentration of $\unicode[STIX]{x1D70C}$ is less than $1/N$, then the problem has a solution of finite cost. The result is sharp, in the sense that there exists $\unicode[STIX]{x1D70C}$ with concentration $1/N$ for which the cost is infinite.

In this note, we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation $-\Delta _p u = f(u)$ in bounded Steiner symmetric domains $ \Omega \subset {{\open R}^N} $ under the zero Dirichlet boundary conditions. The nonlinearity f is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet p-Laplacian in Ω. We show that the nodal set of any least energy sign-changing solution intersects the boundary of Ω. The proof is based on a moving polarization argument.

Minimizing movements are investigated for an energy which is the superposition of a convex functional and fast small oscillations. Thus a minimizing movement scheme involves a temporal parameter τ and a spatial parameter ε, with τ describing the time step and the frequency of the oscillations being proportional to 1/ε. The extreme cases of fast time scales τ ≪ ε and slow time scales ε ≪ τ have been investigated in [4]. In this paper, the intermediate (critical) case of finite ratio ε/τ > 0 is studied. It is shown that a pinning threshold exists, with initial data below the threshold being a fixed point of the dynamics. A characterization of the pinning threshold is given. For initial data above the pinning threshold, the equation and velocity describing the homogenized motion are determined.