The paper is concerned with a class of optimal blocking problems in the plane. We consider a time dependent set R(t) ⊂ ℝ2, described as the reachable set for a differential inclusion. To restrict its growth, a barrier Γ can be constructed, in real time. This is a one-dimensional rectifiable set which blocks the trajectories of the differential inclusion. In this paper we introduce a definition of “regular strategy”, based on a careful classification of blocking arcs. Moreover, we derive local and global necessary conditions for an optimal strategy, which minimizes the total value of the burned region plus the cost of constructing the barrier. We show that a Lagrange multiplier, corresponding to the constraint on the construction speed, can be interpreted as the “instantaneous value of time”. This value, which we compute by two separate formulas, remains constant when free arcs are constructed and is monotone decreasing otherwise.

We prove an observability estimate for awave equation with rapidly oscillating density,in a bounded domain with Dirichlet boundary condition.

We prove by giving an example that when n ≥ 3 the asymptotic behavior of functionals $\int_\Omega \varepsilon|\nabla^2 u|^2+(1-|\nabla u|^2)^2/\varepsilon$ is quite different with respect to the planar case. In particular we show that the one-dimensional ansatz due to Aviles and Giga in the planar case (see [2]) is no longer true in higher dimensions.

Let $u\in\phi+ W_0^{1,1}(\Omega)$ be a minimum for

$\[I(v)=\int_{\Omega}g(x,v(x))+f(\nabla v(x))\,{\rm d}x\]$

where f is convex, $v\mapsto g(x,v)$ is convex for a.e. x. We prove that u shares the same modulus of continuity of ϕ whenever Ω is sufficiently regular, the right derivative of g satisfies a suitable monotonicity assumption and the following inequality holds

$\forall \gamma\in\partial\Omega\qquad |u(x)-\phi(\gamma)|\le\omega(|x-\gamma|) \quad\text{a.e. }x\in\Omega.$

This result generalizes the classical Haar-Rado theorem forLipschitz functions.

This paper describes a compact perceptual image model intended for morphological representation of the visual information contained in natural images. We explain why the total variation can be a criterion to split the information between the two main visual structures, which are the sketch and the microtextures. We deduce a morphological decomposition scheme, based on a segmentation where the borders of the regions correspond to the location of the topological singularities of a topographic map. This leads to propose a new and morphological definition of edges. The sketch is computed by approximating the image with a piecewise smooth non-oscillating function, using a Lipshitz interpolant given as the solution of a PDE. The data needed to reconstruct the sketch image are very compact, so that an immediate outcome of this image model is the design of a progressive, and artifact-free, image compression scheme.

We study the Bellman equation for undiscounted exit time optimal control problems with fully nonlinear Lagrangians and fully nonlinear dynamics using the dynamic programming approach. We allow problems whose non-Lipschitz dynamics admit more than one solution trajectory for some choices of open loop controls and initial positions. We prove a uniqueness theorem which characterizes the value functions of these problems as the unique viscosity solutions of the corresponding Bellman equations that satisfy appropriate boundary conditions. We deduce that the value function for Sussmann's Reflected Brachystochrone Problem for an arbitrary singleton target is the unique viscosity solution of the corresponding Bellman equation in the class of functions which are continuous in the plane, null at the target, and bounded below. Our results also apply to degenerate eikonal equations, and to problems whose targets can be unbounded and whose Lagrangians vanish for some points in the state space which are outside the target, including Fuller's Example.

We study the homogenization of parabolic or hyperbolic equations like \[\rho_\varepsilon{\partial^n u_\varepsilon\over \partial t^n}- {\rm div}(a_\varepsilon\nabla u_\varepsilon) =f \ \ \hbox{ in } \quad {\O\times(0,T)}+\ \ \hbox{\rm boundary conditions}, \quad n \in \{1,2\},\] when the coefficients $\rho_\varepsilon$ , $a_\varepsilon$ (defined in Ω) take possibly high values on a ε-periodic set of grain-like inclusions of vanishing measure.Memory effects arise in the limit problem.

We study the rank–2 distributions satisfying so-called Goursat condition (GC); that is to say, codimension–2 differential systems forming with their derived systems a flag. Firstly, we restate in a clear way the main result of[7] giving preliminary local forms of such systems. Secondly – and this is the main part of the paper – in dimension 7 and 8 we explain which constants in those local forms can be made 0, normalizing the remaining ones to 1. All constructed equivalences are explicit. The complete list of local models in dimension 7 contains 13 items, and not 14, as written in[7], while the list in dimension 8 consists of 34 models (and not 41, as could be concluded from some statements in[7]). In these dimensions (and in lower dimensions, too) the models are eventually discerned just by their small growth vector at the origin.

To find a zero of a maximal monotone operator, an extension of the Auxiliary Problem Principle to nonsymmetric auxiliary operators is proposed. The main convergence result supposes a relationship between the main operator and the nonsymmetric component of the auxiliary operator. When applied to the particular case of convex-concave functions, this result implies the convergence of the parallel version of the Arrow-Hurwicz algorithm under the assumptions of Lipschitz and partial Dunn properties of the main operator. The latter is systematically enforced by partial regularization. In the strongly monotone case, it is shown that the convergence is linear in the average. Moreover, if the symmetric part of the auxiliary operator is linear, the Lipschitz property of the inverse suffices to ensure a linear convergence rate in the average.

Rate-independent evolution for material models with nonconvexelastic energies is studied without any spatial regularization ofthe inner variable; due to lack of convexity, the model is developedin the framework of Young measures. An existence result for thequasistatic evolution is obtained in terms of compatible systems ofYoung measures. We also show as this result can be equivalentlyreformulated with probabilistic language and leads to thedescription of the quasistatic evolution in terms of stochasticprocesses on a suitable probability space.

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.

The aim of this paper is to study a class of domains whose geometry strongly depends on time namely. More precisely, we consider parabolic equations in perforated domains with rapidly pulsing (in time) periodic perforations, with a homogeneous Neumann condition on the boundary of the holes. We study the asymptotic behavior of the solutions as the period ε of the holes goes to zero. Since standard conservation laws do not hold in this model, a first difficulty is to get a priori estimates of the solutions. We obtain them in a weighted space where the weight is the principal eigenfunction of an “adjoint” periodic time-dependent eigenvalue problem. This problem is not a classical one, and its investigation is an important part of this work. Then, by using the multiple scale method, we construct the leading terms of a formal expansion (with respect to ε) of the solution and give the limit “homogenized” problem. An interesting peculiarity of the model is that, depending on the geometry of the holes, a large convection term may appear in the limit equation.

In Carnot groups of step  ≤ 3, all subriemannian geodesics are proved to be normal. Theproof is based on a reduction argument and the Goh condition for minimality of singularcurves. The Goh condition is deduced from a reformulation and a calculus of the end-pointmapping which boils down to the graded structures of Carnot groups.

We study the homogenization process of ferromagnetic multilayers in the presence of surface energies:super-exchange, also called interlayer exchange coupling,and surface anisotropy. The two main difficulties are the non-linearity of the Landau-Lifshitz equation and the absence of a good sequenceof extension operators for the multilayer geometry.First, we consider the case when surface anisotropy is the dominant term, then the case when the magnitude of the super-exchange interaction is inversely proportional to the interlayer distance. We establishthe homogenized equation in these two situations.

This paper deals with a model describing damage processes in a (nonlinear) elastic body which is in contact with adhesion with a rigid support. On the basis of phase transitions theory, we detailthe derivation of the model written in terms of a PDE system, combined with suitable initial and boundary conditions. Some internal constraints on the variables are introduced in the equations and on the boundary, to get physical consistency. We prove theexistence of global in time solutions (to a suitable variational formulation) of therelated Cauchy problem by means of a Schauder fixed point argument, combinedwith monotonicity and compactness tools. We also perform an asymptotic analysis of the solutions as the interfacial damage energy (between the body and the contact surface) goes to +∞.

We consider an optimal control problem for a class of non-linearelliptic equations. A result of existence and uniqueness of the state equation is proven under weaker hypotheses than in theliterature. We also prove the existence of an optimalcontrol. Applications to some lubrication problems and numericalresults are given.

We are extending to linear recurrent codes, i.e., to time-varying convolutional codes, most of the classic structural properties of fixed convolutional codes. We are also proposing a new connection between fixed convolutional codes and linear block codes. These results are obtained thanks to a module-theoretic framework which has been previously developed for linear control.

Motivated by a classical work of Erdős we give rather precise necessary and sufficient growth conditions on the nonlinearity in a semilinear wave equation in order to have global existence for all initial data. Then we improve some former exact controllability theorems of Imanuvilov and Zuazua.

Let W be a function from the real m×n-matrices to the real numbers. If W is quasiconvex in the sense of the calculus of variations, then we show that W can be approximated locally uniformly by quasiconvex polynomials.

In his 1957 paper [1] L. Dubins considered the problem of finding shortest differentiable arcs in the plane with curvature bounded by a constantand prescribed initial and terminal positions and tangents. One can generalize this problem to non-euclidean manifolds as well as to higherdimensions (cf. [15]). 
Considering that the boundary data - initial and terminal position and tangents - are genuinely three-dimensional, it seems natural to ask if then-dimensional problem always reduces to the three-dimensional case. In this paper we will prove that this is true in the euclidean as well as inthe noneuclidean case. At first glance one might consider this a trivial problem, but we will also give an example showing that this is not thecase.