We consider a nonlinear elliptic equation of the formdiv [a(∇u)] + F[u] = 0on a domain Ω, subject to a Dirichlet boundary conditiontru = φ. We do not assume that the higher order terma satisfies growth conditions from above. We prove the existence ofcontinuous solutions either when Ω is convex and φ satisfies a one-sidedbounded slope condition, or when a is radial:\hbox{$a(\xi)=\fr{l(|\xi|)}{|\xi|} \xi$}$\mathit{a}\mathrm{\left(}\mathit{\xi }\mathrm{\right)}\mathrm{=}\frac{\mathit{l}\mathrm{\left(}\mathrm{\right|}\mathit{\xi }\mathrm{\left|}\mathrm{\right)}}{\mathrm{\left|}\mathit{\xi }\mathrm{\right|}}\mathit{\xi }$ for some increasingl:ℝ+ → ℝ+.

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.

An approximation procedure for time optimal control problems for the linear wave equation is analyzed. Its asymptotic behavior is investigated and an optimality system including the maximum principle and the transversality conditions for the regularized and unregularized problems are derived.

In this paper, we consider two-scale limits obtained with increasing homogenization periods, each period being an entire multiple of the previous one. We establish that, up to a measure preserving rearrangement, these two-scale limits form a martingale which is bounded: the rearranged two-scale limits themselves converge both strongly in L2 and almost everywhere when the period tends to  +∞. This limit, called the Two-Scale Shuffle limit, contains all the information present in all the two-scale limits in the sequence.

Any two-input left-invariant control affine system of full rank, evolving on theEuclidean group SE (2), is (detached) feedback equivalent to one ofthree typical cases. In each case, we consider an optimal control problem which is thenlifted, via the Pontryagin Maximum Principle, to a Hamiltonian system onthe dual space 𝔰𝔢 (2)*. These reduced Hamilton − Poisson systems are the maintopic of this paper. A qualitative analysis of each reduced system is performed. Thisanalysis includes a study of the stability nature of all equilibrium states, as well asqualitative descriptions of all integral curves. Finally, the reduced Hamilton equationsare explicitly integrated by Jacobi elliptic functions. Parametrisations for all integralcurves are exhibited.

In this paper we study a controllability problem for a simplified one dimensional model for the motion of a rigid body in a viscous fluid. The control variable is the velocity of the fluid at one end. One of the novelties brought in with respect to the existing literature consists in the fact that we use a single scalar control. Moreover, we introduce a new methodology, which can be used for other nonlinear parabolic systems, independently of the techniques previously used for the linearized problem. This methodology is based on an abstract argument for the null controllability of parabolic equations in the presence of source terms and it avoids tackling linearized problems with time dependent coefficients.

The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described by an auxiliary optimal control problem free of state constraints, and for which the value function is Lipschitz continuous and can be characterized, without any additional assumptions, as the unique viscosity solution of a Hamilton-Jacobi equation. The idea introduced in this paper bypasses the regularity issues on the value function of the constrained control problem and leads to a constructive way to compute its epigraph by a large panel of numerical schemes. Our approach can be extended to more general control problems. We study in this paper the extension to the infinite horizon problem as well as for the two-player game setting. Finally, an illustrative numerical example is given to show the relevance of the approach.

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.

In this paper, we study a class of Initial-Boundary Value Problems proposed by Colin andGhidaglia for the Korteweg-de Vries equation posed on a bounded domain(0,L). We show that this class of Initial-Boundary Value Problems islocally well-posed in the classical Sobolev spaceHs(0,L) for s > -3/4, which provides a positive answer to one of the openquestions of Colin and Ghidaglia [Adv. Differ. Equ. 6 (2001)1463–1492].

Optimal nonanticipating controls are shown to exist in nonautonomous piecewisedeterministic control problems with hard terminal restrictions. The assumptions needed arecompletely analogous to those needed to obtain optimal controls in deterministic controlproblems. The proof is based on well-known results on existence of deterministic optimalcontrols.

Our concern is the computation of optimal shapes in problems involving(−Δ)1/2. We focus on the energyJ(Ω) associated to the solution uΩ of thebasic Dirichlet problem( − Δ)1/2uΩ = 1in Ω, u = 0 in Ωc. We show that regularminimizers Ω of this energy under a volume constraint are disks. Our proof goes throughthe explicit computation of the shape derivative (that seems to be completely new in thefractional context), and a refined adaptation of the moving plane method.

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 study here the impulse control minimax problem. We allow the cost functionals anddynamics to be unbounded and hence the value functions can possibly be unbounded. We provethat the value function of the problem is continuous. Moreover, the value function ischaracterized as the unique viscosity solution of an Isaacs quasi-variational inequality.This problem is in relation with an application in mathematical finance.

We consider a Canham − Helfrich − type variational problem defined over closed surfacesenclosing a fixed volume and having fixed surface area. The problem models the shape ofmultiphase biomembranes. It consists of minimizing the sum of the Canham − Helfrichenergy, in which the bending rigidities and spontaneous curvatures are nowphase-dependent, and a line tension penalization for the phase interfaces. By restrictingattention to axisymmetric surfaces and phase distributions, we extend our previous resultsfor a single phase [R. Choksi and M. Veneroni, Calc. Var. Partial Differ. Equ.(2012). DOI:10.1007/s00526-012-0553-9] and prove existence of a globalminimizer.

The least Steklov eigenvalue d1 for the biharmonic operatorin bounded domains gives a bound for the positivity preserving property for the hingedplate problem, appears as a norm of a suitable trace operator, and gives the optimalconstant to estimate the L2-norm of harmonic functions. Theseapplications suggest to address the problem of minimizing d1in suitable classes of domains. We survey the existing results and conjectures about thistopic; in particular, the existence of a convex domain of fixed measure minimizingd1 is known, although the optimal shape is still unknown. Weperform several numerical experiments which strongly suggest that the optimal planar shapeis the regular pentagon. We prove the existence of a domain minimizingd1 also among convex domains having fixed perimeter andpresent some numerical results supporting the conjecture that, among planar domains, thedisk is the minimizer.

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.

This paper is devoted to the general problem of reconstructing the cost from theobservation of trajectories, in a problem of optimal control. It is motivated by thefollowing applied problem, concerning HALE drones: one would like them to decide bythemselves for their trajectories, and to behave at least as a good human pilot. Thisapplied question is very similar to the problem of determining what is minimized in humanlocomotion. These starting points are the reasons for the particular classes of controlsystems and of costs under consideration. To summarize, our conclusion is that in general,inside these classes, three experiments visiting the same values of the control are neededto reconstruct the cost, and two experiments are in general not enough. The method isconstructive.

The proof of these results is mostly based upon the Thom’s transversality theory.

This study is partly supported by FUI AAP9 project SHARE, and by ANR Project GCM, program“blanche”, project number NT09-504490.

This paper is concerned with the stochastic linear quadratic optimal control problems (LQproblems, for short) for which the coefficients are allowed to be random and the costfunctionals are allowed to have negative weights on the square of control variables. Wepropose a new method, the equivalent cost functional method, to deal with the LQ problems.Comparing to the classical methods, the new method is simple, flexible and non-abstract.The new method can also be applied to deal with nonlinear optimization problems.

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.

A two-person zero-sum differential game with unbounded controls is considered. Underproper coercivity conditions, the upper and lower value functions are characterized as theunique viscosity solutions to the corresponding upper and lower Hamilton–Jacobi–Isaacsequations, respectively. Consequently, when the Isaacs’ condition is satisfied, the upperand lower value functions coincide, leading to the existence of the value function of thedifferential game. Due to the unboundedness of the controls, the corresponding upper andlower Hamiltonians grow super linearly in the gradient of the upper and lower valuefunctions, respectively. A uniqueness theorem of viscosity solution to Hamilton–Jacobiequations involving such kind of Hamiltonian is proved, without relying on theconvexity/concavity of the Hamiltonian. Also, it is shown that the assumed coercivityconditions guaranteeing the finiteness of the upper and lower value functions are sharp insome sense.