We consider a multi-polaron model obtained by coupling the many-body Schrödinger equationfor N interacting electrons with the energy functional of a mean-fieldcrystal with a localized defect, obtaining a highly non linear many-body problem. Thephysical picture is that the electrons constitute a charge defect in an otherwise perfectperiodic crystal. A remarkable feature of such a system is the possibility to form a boundstate of electrons via their interaction with the polarizable background. We prove firstthat a single polaron always binds, i.e. the energy functional has aminimizer for N = 1. Then we discuss the case of multi-polaronscontaining N ≥ 2 electrons. We show that their existence is guaranteedwhen certain quantized binding inequalities of HVZ type are satisfied.

In this paper, starting from classical non-convex and nonlocal3D-variational model of the electric polarization in a ferroelectricmaterial, via an asymptotic process we obtain a rigorous2D-variational model for a thin film. Depending on the initial boundaryconditions, the limit problem can be either nonlocal or local.

We study a form of optimal transportation surplus functions which arise in hedonicpricing models. We derive a formula for the Ma–Trudinger–Wang curvature of thesefunctions, yielding necessary and sufficient conditions for them to satisfy(A3w). We use this to give explicit new examples of surplus functionssatisfying (A3w), of the formb(x,y) = H(x + y)where H is a convex function on ℝn. We alsoshow that the distribution of equilibrium contracts in this hedonic pricing model isabsolutely continuous with respect to Lebesgue measure, implying that buyers are fullyseparated by the contracts they sign, a result of potential economic interest.

We characterize generalized Young measures, the so-called DiPerna–Majda measures whichare generated by sequences of gradients. In particular, we precisely describe thesemeasures at the boundary of the domain in the case of the compactification of ℝm × n by the sphere. We show that this characterization is closely related to the notion of quasiconvexity at the boundary introduced by Ball and Marsden [J.M. Ball and J. Marsden, Arch. Ration. Mech.Anal. 86 (1984) 251–277]. As a consequence we get new results onweak W1,2(Ω; ℝ3) sequentialcontinuity ofu → a· [Cof∇u] ϱ,where Ω ⊂ ℝ3 has a smooth boundary and a,ϱare certain smooth mappings.

The notion of adequate (resp. strongly adequate) function has been recently introduced tocharacterize the essentially strictly convex (resp. essentially firmly subdifferentiable)functions among the weakly lower semicontinuous (resp. lower semicontinuous) ones. In thispaper we provide various necessary and sufficient conditions in order that the lowersemicontinuous hull of an extended real-valued function on a reflexive Banach space isessentially strictly convex. Some new results on nearest (farthest) points are derivedfrom this approach.

This article is the starting point of a series of works whose aim is the study of deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space ℝN. As a consequence, the dynamic and running cost present discontinuities at the boundary of these domains and this is the main difficulty of this type of problems. We address these questions by using a Bellman approach: our aim is to investigate how to define properly the value function(s), to deduce what is (are) the right Bellman Equation(s) associated to this problem (in particular what are the conditions on the set where the dynamic and running cost are discontinuous) and to study the uniqueness properties for this Bellman equation. In this work, we provide rather complete answers to these questions in the case of a simple geometry, namely when we only consider two different domains which are half spaces: we properly define the control problem, identify the different conditions on the hyperplane where the dynamic and the running cost are discontinuous and discuss the uniqueness properties of the Bellman problem by either providing explicitly the minimal and maximal solution or by giving explicit conditions to have uniqueness.

We establish some new results about the Γ-limit, with respect to the L1-topology, of two different (but related) phase-field approximations \hbox{$\{\mathcal E_\eps\}_\eps,\,\{\widetilde{\mathcalE}_\eps\}_\eps$}$\mathrm{\{}{\mathrm{ℰ}}_{\mathit{\epsilon }}{\mathrm{\}}}_{\mathit{\epsilon }}\mathit{,}\hspace{0.17em}\mathrm{\{}\begin{array}{}{}_{\mathrm{􏽥}}\\ {\mathrm{ℰ}}_{\mathit{\epsilon }}\end{array}{\mathrm{\}}}_{\mathit{\epsilon }}$ of the so-called Euler’s Elastica Bending Energy for curves in the plane. In particular we characterize the Γ-limit as ε → 0 of ℰε, and show that in general the Γ-limits of ℰε and \hbox{$\widetilde{\mathcal E}_\eps$}$\begin{array}{}{\mathrm{􏽥}}_{}\\ {\mathrm{ℰ}}_{\mathit{\epsilon }}\end{array}$ do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of the Γ-limit of \hbox{$\widetilde{\mathcal E}_\eps$}$\begin{array}{}{\mathrm{􏽥}}_{}\\ {\mathrm{ℰ}}_{\mathit{\epsilon }}\end{array}$ strictly contains the domain of the Γ-limit of ℰε.

The adjoint method, recently introduced by Evans, is used to study obstacle problems,weakly coupled systems, cell problems for weakly coupled systems of Hamilton − Jacobiequations, and weakly coupled systems of obstacle type. In particular, new results aboutthe speed of convergence of some approximation procedures are derived.

We address in this article the computation of the convex solutions of the Dirichletproblem for the real elliptic Monge − Ampère equation for general convex domains in twodimensions. The method we discuss combines a least-squares formulation with a relaxationmethod. This approach leads to a sequence of Poisson − Dirichlet problems and anothersequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finiteelement approximations with a smoothing procedure are used for the computer implementationof our least-squares/relaxation methodology. Domains with curved boundaries are easilyaccommodated. Numerical experiments show the convergence of the computed solutions totheir continuous counterparts when such solutions exist. On the other hand, when classicalsolutions do not exist, our methodology produces solutions in a least-squares sense.

The paper studies discrete/finite-difference approximations of optimal control problemsgoverned by continuous-time dynamical systems with endpoint constraints. Finite-differencesystems, considered as parametric control problems with the decreasing step ofdiscretization, occupy an intermediate position between continuous-time and discrete-time(with fixed steps) control processes and play a significant role in both qualitative andnumerical aspects of optimal control. In this paper we derive an enhanced version of theApproximate Maximum Principle for finite-difference control systems, which is new even forproblems with smooth endpoint constraints on trajectories and occurs to be the firstresult in the literature that holds for nonsmooth objectives and endpoint constraints. Theresults obtained establish necessary optimality conditions for constrained nonconvexfinite-difference control systems and justify stability of the Pontryagin MaximumPrinciple for continuous-time systems under discrete approximations.

The partially observed optimal control problem is considered for forward-backward doubly stochastic systems with controls entering into the diffusion and the observation. The maximum principle is proven for the partially observable optimal control problems. A probabilistic approach is used, and the adjoint processes are characterized as solutions of related forward-backward doubly stochastic differential equations in finite-dimensional spaces. Then, our theoretical result is applied to study a partially-observed linear-quadratic optimal control problem for a fully coupled forward-backward doubly stochastic system.

In this paper, we consider the approximation of second order evolution equations. It is well known that the approximated system by finite element or finite difference is not uniformly exponentially or polynomially stable with respect to the discretization parameter, even if the continuous system has this property. Our goal is to damp the spurious high frequency modes by introducing numerical viscosity terms in the approximation scheme. With these viscosity terms, we show the exponential or polynomial decay of the discrete scheme when the continuous problem has such a decay and when the spectrum of the spatial operator associated with the undamped problem satisfies the generalized gap condition. By using the Trotter–Kato Theorem, we further show the convergence of the discrete solution to the continuous one. Some illustrative examples are also presented.

By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem constrained to a fixed transport class is equivalent to an abstract Monge problem over a Wasserstein space of probability measures. Concerning solvability of this kind of constrained problems, it turns out that in some sense the Monge problem corresponds to a lucky case.

This paper is concerned with the construction of local observers for nonlinear systems without inputs, satisfying an observability rank condition. The aim of this study is, first, to define an homogeneous approximation that keeps the observability property unchanged at the origin. This approximation is further used in the synthesis of a local observer which is proven to be locally convergent for Lyapunov-stable systems. We compare the performance of the homogeneous approximation observer with the classical linear approximation observer on an example.