We consider an evolution equation similar to that introduced by Vese in [Comm.Partial Diff. Eq. 24 (1999) 1573–1591] and whose solutionconverges in large time to the convex envelope of the initial datum. We give a stochasticcontrol representation for the solution from which we deduce, under quite generalassumptions that the convergence in the Lipschitz norm is in fact exponential in time.

We consider an equilibrium problem with equilibrium constraints (EPEC) arising from the modeling of competition in an electricity spot market (under ISO regulation). For a characterization of equilibrium solutions, so-called M-stationarity conditions are derived. This first requires a structural analysis of the problem, e.g., verifying constraint qualifications. Second, the calmness property of a certain multifunction has to be verified in order to justify using M-stationarity conditions. Third, for stating the stationarity conditions, the coderivative of a normal cone mapping has to be calculated. Finally, the obtained necessary conditions are made fully explicit in terms of the problem data for one typical constellation. A simple two-settlement example serves as an illustration.

In this paper we study a one dimensional model of ferromagnetic nano-wires of finite length. First we justify the model by Γ-convergence arguments. Furthermore we prove the existence of wall profiles. These walls being unstable, we stabilize them by the mean of an applied magnetic field.

In this paper, using direct and inverse images for fractional stochastic tangent sets, weestablish the deterministic necessary and sufficient conditions which control that thesolution of a given stochastic differential equation driven by the fractional Brownianmotion evolves in some particular sets K. As a consequence, a comparisontheorem is obtained.

The stabilization with time delay in observation or control represents difficult mathematical challenges in the control of distributed parameter systems. It is well-known that the stability of closed-loop system achieved by some stabilizing output feedback laws may be destroyed by whatever small time delay there exists in observation. In this paper, we are concerned with a particularly interesting case: Boundary output feedback stabilization of a one-dimensional wave equation system for which the boundary observation suffers from an arbitrary long time delay. We use the observer and predictor to solve the problem: The state is estimated in the time span where the observation is available; and the state is predicted in the time interval where the observation is not available. It is shown that the estimator/predictor based state feedback law stabilizes the delay system asymptotically or exponentially, respectively, relying on the initial data being non-smooth or smooth. Numerical simulations are presented to illustrate the effect of the stabilizing controller.

We study the regularity of finite energy solutions to degeneraten-harmonic equations. The functionK(x), which measures the degeneracy, is assumed to besubexponentially integrable, i.e. it verifies the condition exp(P(K)) ∈ Lloc1. The function P(t) is increasing on [0,∞[  and satisfies the divergence condition \begin{equation}\int_1^\infty\frac{P(t)}{t^2}\,{\rm d}t=\infty.\end{equation}${\mathrm{\int }}_{\mathrm{1}}^{\mathrm{\infty }}\frac{\mathit{P}\mathrm{\left(}\mathit{t}\mathrm{\right)}}{{\mathit{t}}^{\mathrm{2}}}\hspace{0.17em}\mathrm{d}\mathit{t}\mathrm{=}\mathrm{\infty }\mathit{.}$

In this paper, we study the multiplicity of solutions for a class of noncooperativep-Laplacian operator elliptic system. Under suitable assumptions, weobtain a sequence of solutions by using the limit index theory.

In this paper, we consider the back and forth nudging algorithm that has been introduced for data assimilation purposes. It consists of iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We consider the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (Burgers’ equation). Our aim is to prove some theoretical results on the convergence, and convergence properties, of this algorithm. We show that for non viscous equations (both linear transport and Burgers), the convergence of the algorithm holds under observability conditions. Convergence can also be proven for viscous linear transport equations under some strong hypothesis, but not for viscous Burgers’ equation. Moreover, the convergence rate is always exponential in time. We also notice that the forward and backward system of equations is well posed when no nudging term is considered.

Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinearterm, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem onthe Sierpiński gasket is proved. Our approach is based on variational methods and on someanalytic and geometrical properties of the Sierpiński fractal. The abstract results areillustrated by explicit examples.

In the nonconvex case, solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate-independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential that is a viscous regularization of a given rate-independent dissipation potential. The resulting definition of “BV solutions” involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play crucial roles in the description of the associated jump trajectories. We shall prove general convergence results for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting BV solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rate-independent systems.

We consider the problem of approximating a probability measure defined on a metric space by a measure supported on a finite number of points. More specifically we seek the asymptotic behavior of the minimal Wasserstein distance to an approximation when the number of points goes to infinity. The main result gives an equivalent when the space is a Riemannian manifold and the approximated measure is absolutely continuous and compactly supported.

A control system is said to be finite if the Lie algebra generated by its vector fieldsis finite dimensional. Sufficient conditions for such a system on a compact manifold to becontrollable are stated in terms of its Lie algebra. The proofs make use of theequivalence theorem of [Ph. Jouan, ESAIM: COCV 16 (2010)956–973]. and of the existence of an invariant measure on certain compact homogeneousspaces.

We consider chained systems that model various systems of mechanical or biological origin. It is known according to Brockett that this class of systems, which are controllable, is not stabilizable by continuous stationary feedback (i.e. independent of time). Various approaches have been proposed to remedy this problem, especially instationary or discontinuous feedbacks. Here, we look at another stabilization strategy (by continuous stationary or discontinuous feedbacks) to ensure the asymptotic stability even in finite time for some variables, while other variables do converge, and not necessarily toward equilibrium. Furthermore, we build feedbacks that permit to vanish the two first components of the Brockett integrator in finite time, while ensuring the convergence of the last one. The considering feedbacks are continuous and discontinuous and regular outside zero.

We consider a two-player zero-sum-game in a bounded open domain Ω described as follows: at a point x ∈ Ω, Players I and II play an ε-step tug-of-war game with probability α, and with probability β (α + β = 1), a random point in the ball of radius ε centered at x is chosen. Once the game position reaches the boundary, Player II pays Player I the amount given by a fixed payoff function F. We give a detailed proof of the fact that the value functions of this game satisfy the Dynamic Programming Principle

for x ∈ Ω with u(y) = F(y) when y ∉ Ω. This principle implies the existence of quasioptimal Markovian strategies.

This paper concerns continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacsoperators. We establish such an estimate for the parabolic Cauchy problem in the wholespace  [0, +∞) × ℝn and, under some periodicity and eitherellipticity or controllability assumptions, we deduce a similar estimate for the ergodicconstant associated to the operator. An interesting byproduct of the latter result will bethe local uniform convergence for some classes of singular perturbation problems.

We analyze the stability and stabilizability properties of mixed retarded-neutral typesystems when the neutral term may be singular. We consider an operator differentialequation model of the system in a Hilbert space, and we are interested in the criticalcase when there is a sequence of eigenvalues with real parts converging to zero. In thiscase, the system cannot be exponentially stable, and we study conditions under which itwill be strongly stable. The behavior of spectra of mixed retarded-neutral type systemsprevents the direct application of retarded system methods and the approach of pureneutral type systems for the analysis of stability. In this paper, two techniques arecombined to obtain the conditions of asymptotic non-exponential stability: the existenceof a Riesz basis of invariant finite-dimensional subspaces and the boundedness of theresolvent in some subspaces of a special decomposition of the state space. For unstablesystems, the techniques introduced enable the concept of regular strong stabilizabilityfor mixed retarded-neutral type systems to be analyzed.

Traffic flow is modeled by a conservation law describing the density of cars. It isassumed that each driver chooses his own departure time in order to minimize the sum of adeparture and an arrival cost. There are N groups of drivers, Thei-th group consists of κidrivers, sharing the same departure and arrival costsϕi(t),ψi(t).For any given population sizesκ1,...,κn,we prove the existence of a Nash equilibrium solution, where no driver can lower his owntotal cost by choosing a different departure time. The possible non-uniqueness, and acharacterization of this Nash equilibrium solution, are also discussed.

We give a first contribution to the homogenization of many-body structures that are exposed to large deformations and obey the noninterpenetration constraint. The many-body structures considered here resemble cord-belts like they are used to reinforce pneumatic tires. We establish and analyze an idealized model for such many-body structures in which the subbodies are assumed to be hyperelastic with a polyconvex energy density and shall exhibit an initial brittle bond with their neighbors. Noninterpenetration of matter is taken into account by the Ciarlet-Nečas condition and we demand deformations to preserve the local orientation. By studying Γ-convergence of the corresponding total energies as the subbodies become smaller and smaller, we find that the homogenization limits allow for deformations of class special functions of bounded variation while the aforementioned kinematic constraints are conserved. Depending on the many-body structures’ geometries, the homogenization limits feature new mechanical effects ranging from anisotropy to additional kinematic constraints. Furthermore, we introduce the concept of predeformations in order to provide approximations for special functions of bounded variation while preserving the natural kinematic constraints of geometrically nonlinear solid mechanics.

The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain Ω ⊂ ℝ2 the functional is \hbox{$I_{\ep}(u)=\frac{1}{2}\int_{\Omega} \ep^{-1}\lt|1-\lt|Du\rt|^2\rt|^2+\ep\lt|D^2 u\rt|^2 {\rm d}z$}${\mathit{I}}_{\mathit{ϵ}}\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}{{}^{\mathrm{\int }}}_{\mathit{\Omega }}{\mathit{ϵ}}^{-1}{\left|\mathrm{1}\mathrm{-}{\left|\mathit{Du}\right|}^{\mathrm{2}}\right|}^{\mathrm{2}}\mathrm{+}\mathit{ϵ}{\left|{\mathit{D}}^{\mathrm{2}}\mathit{u}\right|}^{\mathrm{2}}\mathrm{d}\mathit{z}$ where u belongs to the subset of functions in \hbox{$W^{2,2}_{0}(\Omega)$}${\mathit{W}}_{\mathrm{0}}^{\mathrm{2}\mathit{,}\mathrm{2}}\mathrm{\left(}\mathit{\Omega }\mathrm{\right)}$ whose gradient (in the sense of trace) satisfies Du(x)·ηx = 1 where ηx is the inward pointing unit normal to ∂Ω at x. In [Ann. Sc. Norm. Super. Pisa Cl. Sci. 1 (2002) 187–202] Jabin et al. characterized a class of functions which includes all limits of sequences \hbox{$u_n\in W^{2,2}_0(\Omega)$}${\mathit{u}}_{\mathit{n}}\mathrm{\in }{\mathit{W}}_{\mathrm{0}}^{\mathrm{2}\mathit{,}\mathrm{2}}\mathrm{\left(}\mathit{\Omega }\mathrm{\right)}$ with Iϵn(un) → 0 as ϵn → 0. A corollary to their work is that if there exists such a sequence (un) for a bounded domain Ω, then Ω must be a ball and (up to change of sign) u: = limn → ∞un = dist(·,∂Ω). Recently [Lorent, Ann. Sc. Norm. Super. Pisa Cl. Sci. (submitted), http://arxiv.org/abs/0902.0154v1] we provided a quantitative generalization of this corollary over the space of convex domains using ‘compensated compactness’ inspired calculations of DeSimone et al. [Proc. Soc. Edinb. Sect. A 131 (2001) 833–844]. In this note we use methods of regularity theory and ODE to provide a sharper estimate and a much simpler proof for the case where Ω = B1(0) without the requiring the trace condition on Du.

Motivated by a long-standing conjecture of Pólya and Szegö about the Newtonian capacityof convex bodies, we discuss the role of concavity inequalities in shape optimization, andwe provide several counterexamples to the Blaschke-concavity of variational functionals,including capacity. We then introduce a new algebraic structure on convex bodies, whichallows to obtain global concavity and indecomposability results, and we discuss theirapplication to isoperimetric-like inequalities. As a byproduct of this approach we alsoobtain a quantitative version of the Kneser-Süss inequality. Finally, for a large class offunctionals involving Dirichlet energies and the surface measure, we perform a localanalysis of strictly convex portions of the boundary via second ordershape derivatives. This allows in particular to exclude the presence of smooth regionswith positive Gauss curvature in an optimal shape for Pólya-Szegö problem.