An instability of the Godunov scheme - Alberto Bressan; Helge Kristian Jenssen; Paolo Baiti (Comm. Pure Appl. Math. 59 (2006) 1604-1638, 2006)
We construct a solution to a $2\\\\times 2$ strictly hyperbolic system of conservation laws, showing that the Godunov scheme \\\\cite{Godunov59} can produce an arbitrarily large amount of oscillations. This happens when the speed of a shock is close to rational, inducing a resonance with the grid. Differently from the Glimm scheme or the vanishing viscosity method, for systems of conservation laws our counterexample indicates that no a priori BV bounds or $L^1$ stability estimates can in general be valid for finite difference schemes.

On Palais-Smale sequences for H-systems: some examples - Paolo Caldiroli; Roberta Musina (Adv. Differential Equations 11 (2006) 931-960, 2006)
We exhibit a series of examples of Palais-Smale sequences for the Dirichlet problem associated to the mean curvature equation with null boundary condition, and we show that in the case of nonconstant mean curvature functions different kinds of blow up phenomena appear and Palais-Smale sequences may have quite wild behaviour.

Q-curvature flow on S^4 - Andrea Malchiodi; Michael Struwe (J. Differential Geom. 73 (2006) 1-44, 2006)

Quasi-periodic solutions of completely resonant forced wave equations - Massimiliano Berti; Michela Procesi (Comm. Partial Differential Equations 31 (2006) 959 - 985, 2006)
We prove existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational number.

Quasistatic evolution problems for linearly elastic-perfectly plastic materials - Gianni Dal Maso; Antonio DeSimone; Maria Giovanna Mora (Arch. Ration. Mech. Anal. 180 (2006) 237-291, 2006)
The problem of quasistatic evolution in small strain associative elastoplasticity is studied in the framework of the variational theory for rate-independent processes. Existence of solutions is proved through the use of incremental variational problems in spaces of functions with bounded deformation. This provides a new approximation result for the solutions of the quasistatic evolution problem, which are shown to be absolutely continuous in time. Four equivalent formulations of the problem in rate form are derived. A strong formulation of the flow rule is obtained by introducing a precise definition of the stress on the singular set of the plastic strain.

Radial solutions concentrating on spheres of nonlinear Schrödinger equations with vanishing potentials - Antonio Ambrosetti; David Ruiz (Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 889-907, 2006)
We prove the existence of radial solutions of 1.2) concentrating at a sphere for potentials which might be zero and might decay to zero at\\r\\ninfinity. The proofs use a perturbation technique in a variational setting, through a Lyapunov-Schmidt reduction.

Recent analytical developments in micromagnetics - Antonio DeSimone; Robert V. Kohn; Stefan Müller; Felix Otto (The science of hysteresis / eds. Giorgio Bertotti, Isaak D. Mayergoyz. - Amsterdam: Elsevier, 2006. Vol.2, 269-381., 2006)

Stability of planar nonlinear switched systems - Ugo Boscain; Grégoire Charlot; Mario Sigalotti (Discrete Contin. Dyn. Syst. 15 (2006) 415-432, 2006)
We consider the time-dependent nonlinear system ˙ q(t) = u(t)X(q(t)) + (1 − u(t))Y (q(t)), where q ∈ R2, X and Y are two smooth vector fields, globally asymptotically stable at the origin and u : [0,∞) → {0, 1} is an arbitrary measurable function. Analysing the topology of the set where X and Y are parallel, we give some sufficient and some necessary conditions for global asymptotic stability, uniform with respect to u(.). Such conditions can be verified without any integration or construction of a Lyapunov function, and they are robust under small perturbations of the vector fields.

Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic field - Ugo Boscain; Paolo Mason (Journal of Mathematical Physics 47, 062101 (2006), 2006)
In this paper we consider the minimum time population transfer problem for the z-component\\nof the spin of a (spin 1/2) particle driven by a magnetic field, controlled along the x axis, with bounded amplitude. On the Bloch sphere (i.e. after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on 2-D manifolds. Let (-E,E) be the two energy levels, and |omega (t)| ≤ M the bound on the field amplitude. For each couple of values E and M, we determine the time optimal synthesis starting from the level -E and we provide the explicit expression of the time optimal trajectories steering the state one to the state two, in terms of a parameter that can be computed solving numerically a suitable equation. For M/E << 1, every time optimal trajectory is bang-bang and in particular the corresponding control is periodic with frequency of the order of the resonance frequency wR = 2E. On the other side, for M/E > 1, the time optimal trajectory steering the state one to the state two is bang-bang with exactly one switching. Fixed E we also prove that for M → ∞ the time needed to reach the state two tends to zero. In the case M/E > 1 there are time optimal trajectories containing a singular arc. Finally we compare these results with some known results of Khaneja, Brockett and Glaser and with those obtained by controlling the magnetic field both on the x and y directions (or with one external field, but in the rotating wave approximation). As byproduct we prove that the qualitative shape of the time optimal synthesis presents different patterns, that cyclically alternate as M/E → 0, giving a partial proof of a conjecture formulated in a previous paper.

On variational approach to differential invariants of rank two distributions - Igor Zelenko (Differential Geom. Appl. 24 (2006) 235-259, 2006)
n the present paper we construct differential invariants for generic rank 2 vector distributions on n-dimensional manifold. In the case n=5 (the first case containing functional parameters) E. Cartan found in 1910 the covariant fourth-order tensor invariant for such distributions, using his \\\"reduction-prolongation\\\" procedure. After Cartan\\\'s work the following questions remained open: first the geometric reason for existence of Cartan\\\'s tensor was not clear; secondly it was not clear how to generalize this tensor to other classes of distributions; finally there were no explicit formulas for computation of Cartan\\\'s tensor. Our paper is the first in the series of papers, where we develop an alternative approach, which gives the answers to the questions mentioned above. It is based on the investigation of dynamics of the field of so-called abnormal extremals (singular curves) of rank 2 distribution and on the general theory of unparametrized curves in the Lagrange Grassmannian, developed in our previous works with A. Agrachev . In this way we construct the fundamental form and the projective Ricci curvature of rank 2 vector distributions for arbitrary n greater than 4.\\nFor n=5 we give an explicit method for computation of these invariants and demonstrate it on several examples. In our next paper we show that in the case n=5 our fundamental form coincides with Cartan\\\'s tensor.

Variational problems in fracture mechanics - Gianni Dal Maso (2006)
We present some recent existence results for the variational model of crack growth in brittle materials proposed by Francfort and Marigo in 1998. These results, obtained in collaboration with Francfort and Toader, cover the case of arbitrary space dimension with a general quasiconvex bulk energy and with prescribed boundary deformations and applied loads.

Asymptotic Morse theory for the equation $\\\\Delta v=2v\\\\sb x\\\\wedge v\\\\sb y$ - Sagun Chanillo; Andrea Malchiodi (Comm. Anal. Geom. 13 (2005) 187-252, 2005)
Given a smooth bounded domain ${\\\\O}\\\\subseteq \\\\R^2$, we consider the equation $\\\\D v = 2 v_x \\\\wedge v_y$ in $\\\\O$, where $v: {\\\\O}\\\\to \\\\R^3$. We prescribe Dirichlet boundary datum, and consider the case in which this datum converges to zero. An asymptotic study of the corresponding Euler functional is performed, analyzing multiple-bubbling phenomena. This allows us to settle a particular case of a question raised by H. Brezis and J.M. Coron.

On the attainable set for Temple class systems with boundary controls - Fabio Ancona; Giuseppe Maria Coclite (SIAM J. Control Optim. 43 (2005) 2166-2190, 2005)
Consider the initial-boundary value problem for a strictly hyperbolic, genuinely nonlinear, Temple class system of conservation laws % $$ u_t+f(u)_x=0, \\\\qquad u(0,x)=\\\\ov u(x), \\\\qquad {{array}{ll} &u(t,a)=\\\\widetilde u_a(t), \\\\noalign{\\\\smallskip} &u(t,b)=\\\\widetilde u_b(t), {array}. \\\\eqno(1) $$ on the domain $\\\\Omega =\\\\{(t,x)\\\\in\\\\R^2 : t\\\\geq 0, a \\\\le x\\\\leq b\\\\}.$ We study the mixed problem (1) from the point of view of control theory, taking the initial data $\\\\bar u$ fixed, and regarding the boundary data $\\\\widetilde u_a, \\\\widetilde u_b$ as control functions that vary in prescribed sets $\\\\U_a, \\\\U_b$, of $\\\\li$ boundary controls. In particular, we consider the family of configurations $$ \\\\A(T) \\\\doteq \\\\big\\\\{u(T,\\\\cdot); ~ u {\\\\rm is a sol. to} (1), \\\\quad \\\\widetilde u_a\\\\in \\\\U_a, \\\\widetilde u_b \\\\in \\\\U_b \\\\big\\\\} $$ that can be attained by the system at a given time $T>0$, and we give a description of the attainable set $\\\\A(T)$ in terms of suitable Oleinik-type conditions. We also establish closure and compactness of the set $\\\\A(T)$ in the $lu$ topology.

On the Blow-up for a Discrete Boltzmann Equation in the Plane - Alberto Bressan; Massimo Fonte (Discrete Contin. Dyn. Syst. 13 (2005) 1-12, 2005)
We study the possibility of finite-time blow-up for a two dimensional Broadwell model. In a set of rescaled variables, we prove that no self-similar blow-up solution exists, and derive some a priori bounds on the blow-up rate. In the final section, a possible blow-up scenario is discussed.

Complete systems of invariants for rank 1 curves in Lagrange Grassmannians - Igor Zelenko (Differential geometry and its applications, 367-382, Matfyzpress, Prague, 2005, 2005)
Curves in Lagrange Grassmannians naturally appear when one studies intrinsically \\\"the Jacobi equations for extremals\\\", associated with control systems and geometric structures. In this way one reduces the problem of construction of the curvature-type invariants for these objects to the much more concrete problem of finding of invariants of curves in Lagrange Grassmannians w.r.t. the action of the linear Symplectic group. In the present paper we develop a new approach to differential geometry of so-called rank 1 curves in Lagrange Grassmannian, i.e., the curves with velocities being rank one linear mappings (under the standard identification of the tangent space to a point of the Lagrange Grassmannian with an appropriate space of linear mappings). The curves of this class are associated with \\\"the Jacobi equations for extremals\\\", corresponding to control systems with scalar control and to rank 2 vector distributions. In particular, we construct the tuple of m principal invariants, where m is equal to half of dimension of the ambient linear symplectic space, such that for a given tuple of arbitrary m smooth functions there exists the unique, up to a symplectic transformation, rank 1 curve having this tuple, as the tuple of the principal invariants. This approach extends and essentially simplifies some results of our previous paper (J. Dynamical and Control Systems, 8, 2002, No. 1, 93-140), where only the uniqueness part was proved and in rather cumbersome way. It is based on the construction of the new canonical moving frame with the most simple structural equation.

Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains - Andrea Malchiodi (Geometric and Functional Analysis 15 (6) 1162-1222 (2005), 2005)
We prove new concentration phenomena for the equation −ɛ2 Δu + u = up in a smooth bounded domain R3 and with Neumann boundary conditions. Here p > 1 and ɛ > 0 is small. We show that concentration of solutions occurs at some geodesics of ∂Ω when ɛ → 0.

Conservation laws with time dependent discontinuous coefficients - Giuseppe Maria Coclite; Nils Henrik Risebro (SIAM J. Math. Anal. 36 (2005) 1293-1309, 2005)
We consider scalar conservation laws where the flux function depends discontinuously on both the spatial and temporal location. Our main results are the existence and well-posedness of an entropy solution to the Cauchy problem. The existence is established by showing that a sequence of front tracking approximations is compact in L1, and that the limits are entropy solutions. Then, using the definition of an entropy solution taken form [11], we show that the solution operator is L1 contractive. These results generalize the corresponding results from [16] and [11].

On curvatures and focal points of distributions of dynamical Lagrangian distributions and their reductions by first integrals - Andrei A. Agrachev; Natalia N. Chtcherbakova; Igor Zelenko (J. Dyn. Control Syst. 11 (2005) 297-327, 2005)
Pairs (Hamiltonian system, Lagrangian distribution), called dynamical Lagrangian distributions, appear naturally in Differential Geometry, Calculus of Variations and Rational Mechanics. The basic differential invariants of a dynamical Lagrangian distribution w.r.t. the action of the group of symplectomorphisms of the ambient symplectic manifold are the curvature operator and the curvature form. These invariants can be seen as generalizations of the classical curvature tensor in Riemannian Geometry. In particular, in terms of these invariants one can localize the focal points along extremals of the corresponding variational problems. In the present paper we study the behavior of the curvature operator, the curvature form and the focal points of a dynamical Lagrangian distribution after its reduction by arbitrary first integrals in involution. The interesting phenomenon is that the curvature form of so-called monotone increasing Lagrangian dynamical distributions, which appear naturally in mechanical systems, does not decrease after reduction. It also turns out that the set of focal points to the given point w.r.t. the monotone increasing dynamical Lagrangian distribution and the corresponding set of focal points w.r.t. its reduction by one integral are alternating sets on the corresponding integral curve of the Hamiltonian system of the considered dynamical distributions. Moreover, the first focal point corresponding to the reduced Lagrangian distribution comes before any focal point related to the original dynamical distribution. We illustrate our results on the classical $N$-body problem.

A fourth order uniformization theorem on some four manifolds with large total Q-curvature - Zindine Djadli; Andrea Malchiodi (C. R. Acad. Sci. Paris, Ser. I 340 (2005) 341-346., 2005)
Given a four-dimensional manifold (M,g), we study the existence of a conformal metric for which the Q-curvature, associated to a conformally invariant fourth-order operator (the Paneitz operator), is constant. Using a topological argument, we obtain a new result in cases which were still open.

Global solutions of the Hunter-Saxton equation - Alberto Bressan; Adrian Constantin (SIAM J. Math. Anal. 37 (2005) 996-1026, 2005)
We construct a continuous semigroup of weak, dissipative solutions to a nonlinear partial differential equations modeling nematic liquid crystals. A new distance functional, determined by a problem of optimal transportation, yields sharp estimates on the continuity of solutions with respect to the initial data.

Ground states of nonlinear Schroedinger equations with potentials vanishing at infinity - Antonio Ambrosetti; Veronica Felli; Andrea Malchiodi (J. Eur. Math. Soc. 7 (2005) 117-144, 2005)
We deal with a class on nonlinear Schr\\\\\\\"odinger equations \\\\eqref{eq:1} with potentials $V(x)\\\\sim |x|^{-\\\\a}$, $0<\\\\a<2$, and $K(x)\\\\sim |x|^{-\\\\b}$, $\\\\b>0$. Working in weighted Sobolev spaces, the existence of ground states $v_{\\\\e}$ belonging to $W^{1,2}(\\\\Rn)$ is proved under the assumption that $p$ satisfies \\\\eqref{eq:p}. Furthermore, it is shown that $v_{\\\\e}$ are {\\\\em spikes} concentrating at a minimum of ${\\\\cal A}=V^{\\\\theta}K^{-2/(p-1)}$, where $\\\\theta= (p+1)/(p-1)-1/2$.

Hybrid necessary principle - Mauro Garavello; Benedetto Piccoli (SIAM J. Control Optim. 43 (2005) 1867-1887, 2005)
We consider a hybrid control system and general optimal control problems for this system. We suppose that the switching strategy imposes restrictions on control sets and we provide necessary conditions for an optimal hybrid trajectory, stating a hybrid necessary principle (HNP). Our result generalizes various necessary principles available in the literature.

Minimal surfaces in pseudohermitian geometry - Jih-Hsin Cheng; JennFang Hwang; Andrea Malchiodi; Paul Yang (Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5), 4 (2005) 129-177., 2005)
We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate some {\em extension} theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and to solve a Bernstein-type problem (for graphs over the $xy$-plane) in the Heisenberg group $H_1$. In $H_{1}$, identified with the Euclidean space $R^{3}$, the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set in two dimensions, and generalize to higher dimensions without any size control condition. We also show that there are no closed, connected, $C^{2}$ smoothly immersed constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard $S^{3}.$ This fact continues to hold when $S^{3}$ is replaced by a general spherical pseudohermitian 3-manifold.

On the Minimum Problem for Nonconvex Scalar Functionals - Sandro Zagatti (SIAM J. Math. Anal. 37 (2005) 982-995, 2005)
We study the minimum problem for scalar nonconvex functionals defined on Sobolev maps satisfying a Dirichlet boundary condition and refine well-known existence results under standard regularity assumptions.

Multiple clustered layer solutions for semilinear Neumann problems on a ball - Andrea Malchiodi; Wei-Ming Ni; Juncheng Wei (Ann. Inst. H. Poincare Anal. Non Lineaire 22 (2005) 143-163, 2005)

Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy - Ugo Boscain; Thomas Chambrion; Grégoire Charlot (Discrete Contin. Dyn. Syst. Ser. B 5 (2005) 957-990, 2005)
We apply techniques of subriemannian geometry on Lie groups and of optimal synthesis on 2-D manifolds to the population transfer problem in a three-level quantum system driven by two laser pulses, of arbitrary shape and frequency. In the rotating wave approximation, we consider a nonisotropic model i.e. a model in which the two coupling constants of the lasers are different. The aim is to induce transitions from the first to the third level, minimizing 1) the time of the transition (with bounded laser amplitudes),\\n2) the energy of lasers (with fixed final time). After reducing the problem to real variables, for the purpose 1) we develop a theory of time optimal syntheses for distributional problem on 2-D-manifolds, while for the purpose 2) we use techniques of subriemannian geometry on 3-D Lie groups. The complete optimal syntheses are computed.

Nonlinear Schrödinger Equations with vanishing and decaying potentials - Antonio Ambrosetti; Wang Zhi-Qiang (Differential Integral Equations 18 (2005), no. 12, 1321-1332, 2005)

An Optimal Transportation Metric for Solutions of the Camassa-Holm Equation - Alberto Bressan; Massimo Fonte (Methods Appl. Anal. 12 (2005) 191-219, 2005)
In this paper we construct a global, continuous flow of solutions to the Camassa-Holm equation on the entire space H1. Our solutions are conservative, in the sense that the total energy int[(u2 + u2x) dx] remains a.e. constant in time. Our new approach is based on a distance functional J(u, v), defined in terms of an optimal transportation problem, which satisfies d dtJ(u(t), v(t)) ≤ κ · J(u(t), v(t)) for every couple of solutions. Using this new distance functional, we can construct arbitrary solutions as the uniform limit of multi-peakon solutions, and prove a general uniqueness result.

Periodic solutions of nonlinear wave equations with non-monotone forcing terms - Massimiliano Berti; Luca Biasco (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 117-124, 2005)

Quasi-periodic oscillations for wave equations under periodic forcing - Massimiliano Berti; Michela Procesi (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 109-116, 2005)

Quasistatic Crack Growth in Nonlinear Elasticity - Gianni Dal Maso; Gilles A. Francfort; Rodica Toader (Arch. Ration. Mech. Anal. 176 (2005) 165-225, 2005)
In this paper, we prove a new existence result for a variational model of crack growth in brittle materials proposed in [15]. We consider the case of $n$-dimensional finite elasticity, for an arbitrary $n\\\\ge1$, with a quasiconvex bulk energy and with prescribed boundary deformations and applied loads, both depending on time.

Regularity properties of optimal trajectories of single-input control systems in dimension three - Mario Sigalotti (Journal of Mathematical Sciences 126 (2005) 1561-1573, 2005)
Let q=f(q)+ug(q) be a smooth control system on a three-dimensional manifold. Given a point q 0 of the manifold at which the iterated Lie brackets of f and g satisfy some prescribed independence condition, we analyze the structure of a control function u(t) corresponding to a time-optimal trajectory lying in a neighborhood of q 0. The control turns out to be the concatenation of some bang-bang and some singular arcs. More general optimality criteria than time-optimality are considered. The paper is a step toward to the analysis of generic single-input systems affine in the control in dimension 3. The main techniques used are second-order optimality conditions and, in particular, the index of the second variation of the switching times for bang-bang trajectories.

Self-similar folding patterns and energy scaling in compressed elastic sheets - Sergio Conti; Antonio DeSimone; Stefan Müller (Comput. Methods Appl. Mech. Engrg. 194 (2005) 2534-2549, 2005)
Thin elastic sheets under isotropic compression, such as for example blisters formed by thin films which debonded from the substrate, can exhibit remarkably complex folding patterns. We discuss the scaling of the elastic energy with respect to the film thickness, and show that in certain regimes the optimal energy scaling can be reached\\nby self-similar folding patterns that refine towards the boundary, in agreement with experimental observations. We then extend the analysis\\nto anisotropic compression, and discuss a simplified scalar model which suggests the presence of a transition between a regime where\\nthe deformation is governed by global properties of the domain and another one where the direction of maximal compression dominates and the scale of the folds is mainly determined by the distance to the boundary in the direction of the folds themselves.

A short introduction to optimal control - Ugo Boscain; Benedetto Piccoli (Contrôle non linéaire et applications: Cours donnés à l\\\'école d\\\'été du Cimpa de l\\\'Université de Tlemcen / Sari Tewfit [ed.]. - Paris: Hermann, 2005, 2005)

Solutions of Neumann problems in domains with cracks and applications to fracture mechanics - Gianni Dal Maso (2005)

Stability of solutions of quasilinear parabolic equations - Giuseppe Maria Coclite; Helge Holden (J. Math. Anal. Appl. 308 (2005) 221-239, 2005)
We bound the difference between solutions $u$ and $v$ of $u_t = a\\\\Delta u+\\\\Div_x f+h$ and $v_t = b\\\\Delta v+\\\\Div_x g+k$ with initial data $\\\\phi$ and $ \\\\psi$, respectively, by $\\\\Vert u(t,\\\\cdot)-v(t,\\\\cdot)\\\\Vert_{L^p(E)}\\\\le A_E(t)\\\\Vert \\\\phi-\\\\psi\\\\Vert_{L^\\\\infty(\\\\R^n)}^{2\\\\rho_p}+ B(t)(\\\\Vert a-b\\\\Vert_{\\\\infty}+ \\\\Vert \\\\nabla_x\\\\cdot f-\\\\nabla_x\\\\cdot g\\\\Vert_{\\\\infty}+ \\\\Vert f_u-g_u\\\\Vert_{\\\\infty} + \\\\Vert h-k\\\\Vert_{\\\\infty})^{\\\\rho_p} \\\\abs{E}^{\\\\eta_p}$. Here all functions $a$, $f$, and $h$ are smooth and bounded, and may depend on $u$, $x\\\\in\\\\R^n$, and $t$. The functions $a$ and $h$ may in addition depend on $\\\\nabla u$. Identical assumptions hold for the functions that determine the solutions $v$. Furthermore, $E\\\\subset\\\\R^n$ is assumed to be a bounded set, and $\\\\rho_p$ and $\\\\eta_p$ are fractions that depend on $n$ and $p$. The diffusion coefficients $a$ and $b$ are assumed to be strictly positive and the initial data are smooth.

Stress-dilatancy based modelling of granular materials and extensions to soils with crushable grains - Antonio DeSimone; Claudio Tamagnini (Int. J. Numer. Anal. Met. 29 (2005) 73-101, 2005)
Stress-dilatancy relations have played a crucial role in the understanding of the mechanical behaviour of soils and in the development of realistic constitutive models for their response. Recent investigations on the mechanical behaviour of materials with crushable grains have called into question the validity of classical relations such as those used in critical state soil mechanics.\\nIn this paper, a method to construct thermodynamically consistent (isotropic, three-invariant) elasto-plastic models based on a given stress-dilatancy relation is discussed. Extensions to cover the case of granular materials with crushable grains are also presented, based on the interpretation of some classical model parameters (e.g. the stress ratio at critical state) as internal variables that evolve according to suitable hardening laws.

Time minimal trajectories for two-level quantum systems with drift - Ugo Boscain; Paolo Mason (Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC \\\'05. 44th IEEE Conference on, 2005)
On a two-level quantum system driven by an external field, we consider the population transfer problem from the first to the second level, minimizing the time of transfer, with bounded field amplitude. On the Bloch sphere (i.e. after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on 2-D manifolds.

Time Optimal Synthesis for Left-Invariant Control Systems on SO(3) - Ugo Boscain; Yacine Chitour (SIAM J. Control Optim. 44 (2005) 111-139, 2005)
Consider the control system given by $\\\\dot x=x(f+ug)$, where $x\\\\in SO(3)$, $|u|\\\\leq 1$ and $f,g\\\\in so(3)$ define two perpendicular left-invariant vector fields normalized so that $\\\\|f\\\\|=\\\\cos(\\\\al)$ and $\\\\|g\\\\|=\\\\sin(\\\\al)$, $\\\\al\\\\in ]0,\\\\pi/4[$. In this paper, we provide an upper bound and a lower bound for $N(\\\\alpha)$, the maximum number of switchings for time-optimal trajectories. More precisely, we show that $N_S(\\\\al)\\\\leq N(\\\\al)\\\\leq N_S(\\\\al)+4$, where $N_S(\\\\al)$ is a suitable integer function of $\\\\al$ which for $\\\\al\\\\to 0$ is of order $\\\\pi/(4\\\\alpha).$ The result is obtained by studying the time optimal synthesis of a projected control problem on $R P^2$, where the projection is defined by an appropriate Hopf fibration. Finally, we study the projected control problem on the unit sphere $S^2$. It exhibits interesting features which will be partly rigorously derived and partially described by numerical simulations.

Traffic flow on a road network - Giuseppe Maria Coclite; Benedetto Piccoli; Mauro Garavello (SIAM J. Math. Anal. 36 (2005) 1862-1886, 2005)
This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from conservation of the number of cars,\\ndefined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions, hence we choose to have some fixed rules for the distribution of traffic plus an optimization criteria for the flux. We prove existence, uniqueness and stability of solutions to the Cauchy problem. Our method is based on wave front tracking approach, see [6], and works also for boundary data and time dependent coefficients of traffic distribution at junctions, so including traffic lights.

Vanishing viscosity solutions of nonlinear hyperbolic systems - Stefano Bianchini; Alberto Bressan (Ann. of Math. 161 (2005) 223-342, 2005)
We consider the Cauchy problem for a strictly hyperbolic, $n\\\\times n$ system in one space dimension: $u_t+A(u)u_x=0$, assuming that the initial data has small total variation.\\nWe show that the solutions of the viscous approximations $u_t+A(u)u_x=\\\\ve u_{xx}$ are defined globally in time and satisfy uniform BV estimates, independent of $\\\\ve$. Moreover, they depend continuously on the initial data in the $\\\\L^1$ distance, with a Lipschitz constant independent of $t,\\\\ve$. Letting $\\\\ve\\\\to 0$, these viscous solutions converge to a unique limit, depending Lipschitz continuously on the initial data. In the conservative case where $A=Df$ is the Jacobian of some flux function $f:\\\\R^n\\\\mapsto\\\\R^n$, the vanishing viscosity limits are precisely the unique entropy weak solutions to the system of conservation laws $u_t+f(u)_x=0$.

Wetting of rough surfaces: a homogenization approach - Antonio DeSimone; Giovanni Alberti (Proc. R. Soc. Lon. Ser. A 461 (2005) 79-97, 2005)
The contact angle of a drop in equilibrium on a solid is strongly affected by the roughness of the surface on which it rests. We study the roughness-induced enhancement of the hydrophobic or hydrophilic properties of a solid surface through homogenization theory. By relying on a variational formulation of the problem, we show that the macroscopic contact angle is associated with the solution of two cell problems, giving the minimal energy per unit macroscopic area for a transition layer between the rough solid surface and a liquid or vapor phase. Our results are valid for both chemically heterogeneous and homogeneous surfaces. In the latter case, a very transparent structure emerges from the variational\\napproach: the classical laws of Wenzel and Cassie-Baxter give bounds for the optimal energy, and configurations of minimal energy are those leading to the smallest macroscopic contact angle in the hydrophobic case, to the largest one in the hydrophilic case.

Asymptotic behaviour and correctors for linear Dirichlet problems with simultaneously varying operators and domains - Gianni Dal Maso; Francois Murat (Ann. Inst. H. Poincaré. Anal. Non Linéaire 21 (2004), (4), p. 445-486., 2004)
We consider a sequence of Dirichlet problems in varying domains (or, more generally, of relaxed Dirichlet problems involving measures in M_0) for second order linear elliptic operators in divergence form with varying matrices of coefficients. When the matrices H-converge to a matrix A^0, we prove that there exist a subsequence and a measure mu^0 in M_0 such that the limit problem is the relaxed Dirichlet problem corresponding to A^0 and mu^0. We also prove a corrector result which provides an explicit approximation of the solutions in the H^1-norm, and which is obtained by multiplying the corrector for the H-converging matrices by some special test function which depends both on the varying matrices and on the varying domains.

Bifurcation of free vibrations for completely resonant wave equations - Massimiliano Berti; Philippe Bolle (Boll. Unione Mat. Ital. Sez. B 7 (2004) 519-528, 2004)
We prove existence of small amplitude, 2 pi/omega -periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency omega belonging to a Cantor-like set of positive measure and for a generic set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser Implicit Function Theorem.

Calculation of impulsively started incompressible viscous flows - Marra, Andrea; Andrea Mola; Quartapelle, Luigi; Riviello, Luca (Int. J. Numer. Meth. Fluids, 2004)

Coarse-grained models of materials with non-convex free-energy: two case studies - Antonio DeSimone (Computer methods in applied mechanics and engineering , 193 (2004) 5129-5141, 2004)
Bridging across length scales is one of the fundamental challenges in the computational modelling of material systems whose mechanical response is driven by rough energy landscapes. The typical feature of such systems is that of exhibiting fine scale microstructures. Two case studies, namely, nematic elastomers and ferromagnetic shape memory alloys, are presented to illustrate the use of modern techniques from (non-convex) calculus of variations in developing coarse-grained models of microstructure-driven material response.

On the convergence rate of vanishing viscosity approximations - Alberto Bressan; Tong Yang (Comm. Pure Appl. Math. 57 (2004) 1075-1109, 2004)
Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound $\\\\big\\\\|u(t,\\\\cdot)-u^\\\\ve(t,\\\\cdot)\\\\big\\\\|_{\\\\L^1}= \\\\O(1)(1+t)\\\\cdot \\\\sqrt\\\\ve|\\\\ln\\\\ve|$ on the distance between an exact BV solution $u$ and a viscous approximation $u^\\\\ve$, letting the viscosity coefficient $\\\\ve\\\\to 0$. In the proof, starting from $u$ we construct an approximation of the viscous solution $u^\\\\ve$ by taking a mollification $u*\\\\phi_{\\\\strut \\\\sqrt\\\\ve}$ and inserting viscous shock profiles at the locations of finitely many large shocks, for each fixed $\\\\ve$. Error estimates are then obtained by introducing new Lyapunov functionals which control shock interactions, interactions between waves of different families and by using sharp decay estimates for positive nonlinear waves.

Energetics and switching of quasi-uniform states in small ferromagnetic particles - François Alouges; Sergio Conti; Antonio DeSimone; Ivo Pokern (M2AN Math. Model. Numer. Anal. 38 (2004) 235-248, 2004)
We present a numerical algorithm to solve the micromagnetic equations based on tangential-plane minimization for the magnetization update and a homothethic-layer decomposition of outer space for the computation of the demagnetization field. As a first application, detailed results on the flower-vortex transition in the cube of Micromagnetic Standard Problem number 3 are obtained, which confirm, with a different method, those already present in the literature, and validate our method and code. We then turn to switching of small cubic or almost-cubic particles, in the single-domain limit. Our data show systematic deviations from the Stoner-Wohlfarth model due to the non-ellipsoidal shape of the particle, and in particular a non-monotone dependence on the particle size.

Existence of H-bubbles in a perturbative setting - Paolo Caldiroli; Roberta Musina (Rev. Mat. Iberoamericana 20 (2004) 611-626, 2004)
Given a $C^{1}$ function $H: \\\\mathbb{R}^3 \\\\to \\\\mathbb{R}$, we look for $H$-bubbles, i.e., surfaces in $\\\\mathbb{R}^3$ parametrized by the sphere $\\\\mathbb{S}^2$ with mean curvature $H$ at every regular point. Here we study the case $H(u)=H_{0}(u)+\\\\epsilon H_{1}(u)$ where $H_{0}$ is some \\\"good\\\" curvature (for which there exist $H_{0}$-bubbles with minimal energy, uniformly bounded in $L^{\\\\infty}$), $\\\\epsilon$ is the smallness parameter, and $H_{1}$ is {\\\\em any} $C^{1}$ function.

H-bubbles in a perturbative setting: the finite-dimensional reduction\\\'s method - Paolo Caldiroli; Roberta Musina (Duke Math. J. 122 (2004), no. 3, 457--484, 2004)
Given a regular function $H\\\\colon\\\\mathbb{R}^{3}\\\\to\\\\mathbb{R}$, we look for $H$-bubbles, that is, regular surfaces in $\\\\mathbb{R}^{3}$ parametrized on the sphere $\\\\mathbb{S}+^{2}$ with mean curvature $H$ at every point. Here we study the case of $H(u)=H_{0}+\\\\varepsilon H_{1}(u)=:H_{\\\\varepsilon}(u)$, where $H_{0}$ is a nonzero constant, $\\\\varepsilon$ is the smallness parameter, and $H_{1}$ is any $C^{2}$-function. We prove that if $\\\\bar p\\\\in\\\\mathbb{R}^{3}$ is a ``good\\\'\\\' stationary point for the Melnikov-type function $\\\\Gamma(p)=-\\\\int_{|q-p|<|H_{0}|^{-1}}H_{1}(q)\\\\,dq$, then for $|\\\\varepsilon|$ small there exists an $H_{\\\\varepsilon}$-bubble $\\\\omega^{\\\\varepsilon}$ that converges to a sphere of radius $|H_{0}|^{-1}$ centered at $\\\\bar p$, as $\\\\varepsilon\\\\to 0$.

Higher order quasiconvexity reduces to quasiconvexity - Gianni Dal Maso; Irene Fonseca; Giovanni Leoni; Massimiliano Morini (Arch. Ration. Mech. Anal. 171 (2004) 55-81, 2004)
In this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions. Precisely, it is proved that a smooth strictly 2-quasiconvex function with p-growth at infinity, p>1, is the restriction to symmetric matrices of a 1-quasiconvex function with the same growth. As a consequence, lower semicontinuity results for second-order variational problems are deduced as corollaries of well-known first order theorems.

On the minimal degree of a common Lyapunov function for planar switched systems - Paolo Mason; Ugo Boscain; Yacine Chitour (43rd IEEE Conference on Decision and Control, 2004, 2786 - 2791 Vol.3, 2004)
In this paper, we consider linear switched systems x(t) = Au(t)x(t), x ε Rn, u ε U, and the problem of asymptotic stability for arbitrary switching functions, uniform with respect to switching (UAS for short). We first prove that, given a UAS system, it is always possible to build a polynomial common Lyapunov function. Then our main result is that the degree of that the common polynomial Lyapunov function is not uniformly bounded over all the UAS systems. This result answers a question raised by Dayawansa and Martin.

Multidimensional boundary layers for a singularly perturbed Neumann problem - Andrea Malchiodi; Marcelo Montenegro (Duke Math. J. 124 (2004) 105-143, 2004)

Multiplicity of periodic solutions of nonlinear wave equations - Massimiliano Berti; Philippe Bolle (Nonlinear Anal. 56 (2004) 1011-1046, 2004)

Periodic orbits close to elliptic tori and applications to the three-body problem - Massimiliano Berti; Luca Biasco; Enrico Valdinoci (Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004) 87-138, 2004)
We prove, under suitable non-resonance and non-degeneracy ``twist\\\'\\\' conditions, a Birkhoff-Lewis type result showing the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori (of Hamiltonian systems). We prove the applicability of this result to the spatial planetary three-body problem in the small eccentricity-inclination regime. Furthermore, we find other periodic orbits under some restrictions on the period and the masses of the ``planets\\\'\\\'. The proofs are based on averaging theory, KAM theory and variational methods. (Supported by M.U.R.S.T. Variational Methods and Nonlinear Differential Equations.)

Quasi-static evolution in brittle fracture: the case of bounded solutions - Gianni Dal Maso; Gilles A. Francfort; Rodica Toader (Quad. Mat. Dip. Mat. Seconda Univ. Napoli 14 (2004) 245-266, 2004)
The main steps of the proof of the existence result for the quasi-static evolution of cracks in brittle materials, obtained in [7] in the vector case and for a general quasiconvex elastic energy, are presented here under the simplifying assumption that the minimizing sequences involved in the problem are uniformly bounded in $L^\\\\infty$.

Resonance of minimizers for n-level quantum systems with an arbitrary cost - Ugo Boscain; Grégoire Charlot (ESAIM COCV 10 (2004) 593-614, 2004)
We consider an optimal control problem describing a laser-induced population transfer on a $n$-level quantum system.\\nFor a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for $n=2$ and $n=3$): instead of looking for minimizers on the sphere $S^{2n-1}\\\\subset\\\\C^n$ one is reduced to look just for minimizers on the sphere $S^{n-1}\\\\subset \\\\R^n$. Moreover, for the reduced problem, we investigate on the question of existence of strict abnormal minimizer.

The role of the spectrum of the Laplace operator on \\\\S2 in the H-bubble problem - Roberta Musina (J. Anal. Math. 94 (2004) 265-291, 2004)

Semi-cooperative strategies for differential games - Alberto Bressan; Wen Shen (Internat. J. Game Theory 32 (2004) 561-593, 2004)
The paper is concerned with a non-cooperative differential game for two players. We first consider Nash equilibrium solutions in feedback form. In this case, we show that the Cauchy problem for the value functions is generically ill-posed. Looking at vanishing viscosity approximations, one can construct special solutions in the form of chattering controls, but these also appear to be unstable. In the second part of the paper we propose an alternative \\\"semi-cooperative\\\" pair of strategies for the two players, seeking a Pareto optimum instead of a Nash equilibrium. In this case, we prove that the corresponding Hamiltonian system for the value functions is always weakly hyperbolic.

A sharp decay estimate for positive nonlinear waves - Alberto Bressan; Tong Yang (SIAM J. Math. Anal. 36 (2004) 659-677, 2004)
We consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension. A sharp decay estimate is proved for the positive waves in an entropy weak solution. The result is stated in terms of a partial ordering among positive measures, using symmetric rearrangements and a comparison with a solution of Burgers\\\' equation with impulsive sources.

Singularity perturbed elliptic equations with symmetry: existence of solutions concetrating on spheres, Part II - Antonio Ambrosetti; Andrea Malchiodi; Wei-Ming Ni (Indiana Univ. Math. J. 53 (2004) 297-392, 2004)

Small BV solutions of hyperbolic noncooperative differential games - Alberto Bressan; Wen Shen (SIAM J. Control Optim. 43 (2004) 194-215, 2004)
The paper is concerned with an n-persons differential game in one space dimension. We state conditions for which the system of Hamilton-Jacobi equations for the value functions is strictly hyperbolic. In the positive case, we show that the weak solution of a corresponding system of conservation laws determines an n-tuple of feedback strategies. These yield a Nash equilibrium solution to the non-cooperative differential game.

Solitary waves for Maxwell Schrodinger equations - Giuseppe Maria Coclite; Vladimir Georgiev (Electron. J. Differential Equations (2004) 94, 2004)
In this paper we study solitary waves for the coupled system of Schrodinger-Maxwell equations in the three-dimensional space. We prove the existence of a sequence of radial solitary waves for these equations with a fixed L^2 norm. We study the asymptotic behavior and the smoothness of these solutions. We show also that the eigenvalues are negative and the first one is isolated.

Solutions concentrating at curves for some singularly perturbed elliptic problems - Andrea Malchiodi (C. R. Acad. Sci. Paris, Ser. I 338 (2004) 775-780, 2004)

Soluzioni periodiche di PDEs Hamiltoniane - Massimiliano Berti (Bollettino dell\\\'Unione Matematica Italiana Serie 8 7-B (2004), p. 647-661, 2004)

Some remarks on multidimensional systems of conservation laws - Alberto Bressan (Rend. Mat. Acc. Lincei, s. 9, v. 15 (2004) 3-4, pp. 225 - 233, 2004)
This note is concerned with the Cauchy problem for hyperbolic systems of conservation\\nlaws in several space dimensions. We first discuss an example of ill-posedness, for a special system\\nhaving a radial symmetry property. Some conjectures are formulated, on the compactness of the set of\\nflow maps generated by vector fields with bounded variation.

Stability rates for patchy vector fields - Fabio Ancona; Alberto Bressan (ESAIM COCV 10 (2004) 168-200, 2004)
This paper is concerned with the stability of the set of trajectories of a patchy vector field, in the presence of impulsive perturbations. Patchy vector fields are discontinuous, piecewise smooth vector fields that were introduced in Ancona and Bressan (1999) to study feedback stabilization problems. For patchy vector fields in the plane, with polygonal patches in generic position, we show that the distance between a perturbed trajectory and an unperturbed one is of the same order of magnitude as the impulsive forcing term.

Well-posedness for general 2x2 systems of conservation laws - Fabio Ancona; Andrea Marson (Mem. Amer. Math. Soc. 169 (2004), no. 801, x+170 pp., 2004)

Autonomous integral functionals with discontinous nonconvex integrands: Lipschitz regularity of mimimizers, DuBois-Reymond necessary conditions and Hamilton-Jacobi equations - Gianni Dal Maso; Helene Frankowska (Applied Math.Optim. 48 (2003), no.1, p.39-66, 2003)

The calibration method for the Mumford-Shah functional and free-discontinuity problems - Giovanni Alberti; Guy Bouchitte; Gianni Dal Maso (Calc. Var. Partial Differential Equations 16 (2003) 299-333, 2003)
We present a minimality criterion for the Mumford-Shah functional, and more generally for non convex variational integrals on SBV which couple a surface and a bulk term. This method provides short and easy proofs for several minimality results.

Control Problems for Systems of Conservation Laws - Giuseppe Maria Coclite (SISSA, 2003)
Keywords: Asymptotic Stabilization

Drift in phase space: a new variational mechanism with optimal diffusion time - Massimiliano Berti; Luca Biasco; Philippe Bolle (J. Math. Pures Appl. 82 (2003) 613-664, 2003)
We consider non-isochronous, nearly integrable, a-priori unstable Hamiltonian systems with a (trigonometric polynomial) $O(\\\\mu)$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time $ T_d = O((1/ \\\\mu) \\\\log (1/ \\\\mu))$ by a variational method which does not require the existence of ``transition chains of tori\\\'\\\' provided by KAM theory. We also prove that our estimate of the diffusion time $T_d $ is optimal as a consequence of a general stability result derived from classical perturbation theory.

A finite element approximation of the Griffith\\\'s model in fracture mechanics - Matteo Negri (Numer. Math., 2003, 95, 653, 2003)
The Griffith model for the mechanics of fractures in brittle materials is consider in the weak formulation of SBD spaces. We suggest an approximation, in the sense of Gamma-convergence, by a sequence of discrete functionals defined on finite elements spaces over structured and adaptive triangulations. The quasi-static evolution for boundary value problems is also taken into account and some numerical results are shown.

Hybrid optimal control: case study of a car with gears - Ciro D'Apice; Mauro Garavello; Rosanna Manzo; Benedetto Piccoli (Int. J. Control 76 (2003) 1272-1284, 2003)
The purpose of this paper is to show the use of some analytical tools for hybrid optimal control. We illustrate both the hybrid maximum principle and the hybrid necessary principle at work on a simple example of a car with gears. The model is sufficiently rich to generate non-trivial optimization problems and the obtained results match with intuition. Finally, computer simulations confirm the theoretical analysis.

An ill posed Cauchy problem for a hyperbolic system in two space dimensions - Alberto Bressan (Università di Padova, 2003)
The theory of weak solutions for nonlinear conservation laws is now well developed in the case of scalar equations [3] and for one-dimensional hyperbolic systems [1, 2]. For systems in several space dimensions, however, even the global existence of solutions to the Cauchy problem remains a challenging open question. In this note we construct a conterexample showing that, even for a simple class of hyperbolic systems, in two space dimensions the Cauchy problem can be ill posed.

An interior estimate for a nonlinear parabolic equation - Giuseppe Maria Coclite (J.Math.Anal.Appl. 284 (2003) no.1, 49, 2003)

A lemma and a conjecture on the cost of rearrangements - Alberto Bressan (Rend. Sem. Mat. Univ. Padova 110 (2003) 97-102, 2003)
Consider a stack of books, containing both white and black books. Suppose that we want to sort them out, putting the white books on the right, and the black books on the left (fig.~1). This will be done by a finite sequence of elementary transpositions. In other words, if we have a stack of all black books of length $a$ followed by a stack of all white books of length $b$, we are allowed to reverse their order at the cost of $a+b$. We are interested in a lower bound on the total cost of the rearrangement.

On the local structure of optimal trajectories in R3 - Andrei A. Agrachev; Mario Sigalotti (SIAM J. Control Optim. 42 (2003) 513-531, 2003)
We analyze the structure of a control function u(t) corresponding to an optimal trajectory for the system $\\\\dot q =f(q)+u\\\\, g(q)$ in a three-dimensional manifold, near a point where some nondegeneracy conditions are satisfied. The kind of optimality which is studied includes time-optimality. The control turns out to be the concatenation of some bang and some singular arcs. Studying the index of the second variation of the switching times, the number of such arcs is bounded by four.

A note on singular limits to hyperbolic systems of conservation laws - Stefano Bianchini (Commun. Pure Appl. Ana., 2003, 2, 51-64, 2003)
In this note we consider two different singular limits to hyperbolic system of conservation laws, namely the standard backward schemes for non linear semigroups and the semidiscrete scheme. \\nUnder the assumption that the rarefaction curve of the corresponding hyperbolic system are straight lines, we prove the stability of the solution and the convergence to the perturbed system to the unique solution of the limit system for initial data with small total variation.

A note on the integral representation of functionals in the space SBD(O) - Francois Ebobisse; Rodica Toader (Rend. Mat. Appl. 23 (2003) 189-201, 2003)
In this paper we study the integral representation in the space SBD(O) of special functions with bounded deformation of some L^1-norm lower semicontinuous functionals invariant with respect to rigid motions.

Periodic solutions of nonlinear wave equations with general nonlinearities - Massimiliano Berti; Philippe Bolle (Comm.Math.Phys. 243 (2003) no.2, 315, 2003)

Positive solutions to a class of quasilinear elliptic equations on R - Antonio Ambrosetti; Wang Zhi-Qiang (Discrete Contin.Dyn.Syst. 9 (2003), no.1, 55-68, 2003)
We discuss the existence of positive solutions of perturbation to a class of quasilinear elliptic equations on R.

Prescribing scalar and boundary mean curvature on the three dimensional half sphere - Zindine Djadli; Andrea Malchiodi; Mohameden Ould Ahmedou (J. Geom. Anal. 13 (2003) 255-289, 2003)
We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove some existence and compactness results.

Sequences of Singularly Perturbed Functionals Generating Free-Discontinuity Problems - Massimiliano Morini (SIAM J. Math. Anal. 35 (2003) 759-805, 2003)
We prove that a wide class of singularly perturbed functionals generates as $\\\\Gamma$-limit a functional related to a free-discontinuity problem. Several applications of the result are shown.

Single-Input Control Affine Systems: Local Regularity of Optimal Trajectories and a Geometric Controllability Problem - Mario Sigalotti (SISSA, 2003)

Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I - Antonio Ambrosetti; Andrea Malchiodi; Wei-Ming Ni (Comm. Math. Phys. 235 (2003) no.3, 427-466, 2003)

Some results on the boundary control of systems of conservation laws - Alberto Bressan; Fabio Ancona; Giuseppe Maria Coclite (SIAM J.Control Optim. 41 (2003),no.2, 607, 2003)

A stability result for nonlinear Neumann problems under boundary variations - Gianni Dal Maso; Francois Ebobisse; Marcello Ponsiglione (J.Math. Pures Appl. (9) 82 (2003) no.5 , 503, 2003)
In this paper we study, in dimension two, the stability of the solutions of some nonlinear elliptic equations with Neumann boundary conditions, under perturbations of the domains in the Hausdorff complementary topology.

Admissible Riemann solvers for genuinely nonlinear P-systems of mixed type - Jean-Marc Mercier; Benedetto Piccoli (J. Differ. Equations, 2002, 180, 395, 2002)

Arnold diffusion: a functional analysis approach - Massimiliano Berti (Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 43, Part 1, 2, Natsīonal. Akad. Nauk Ukraïni, Īnst. Mat., Kiev, 2002, 2002)
We present, in the context of nearly integrable Hamiltonian systems, a functional analysis approach to study the “splitting of the whiskers” and the “shadowing problem” developed in collaboration with P. Bolle in the recent papers [1] and [2] . This method is applied to the problem of Arnold diffusion for nearly integrable partially isochronous systems improving known results.

On the Boundary Control of Systems of Conservation Laws - Alberto Bressan; Giuseppe Maria Coclite (SIAM J. Control Optim. 41 (2002) 607-622, 2002)
The paper is concerned with the boundary controllability of entropy weak solutions to hyperbolic systems of conservation laws. We prove a general result on the asymptotic stabilization of a system near a constant state. On the other hand, we give an example showing that exact controllability in finite time cannot be achieved, in general.

The Calibration Method for Free-Discontinuity Problems on Vector-Valued Maps - Maria Giovanna Mora (J. Convex Anal. 9 (2002) 1-29, 2002)
The calibration method is a classical minimality criterion, which has been recently adapted to functionals with free discontinuities by Alberti, Bouchitté, Dal Maso. In this paper we present a further generalization of this theory to functionals defined on vector-valued maps.

A center manifold technique for tracing viscous waves - Stefano Bianchini; Alberto Bressan (Commun. Pure Appl. Anal. 1 (2002) 161-190, 2002)
In this paper we introduce a new technique for tracing viscous travelling profiles. To illustrate the method, we consider a special 2 x 2 hyperbolic system of conservation laws with viscosity, and show that any solution can be locally decomposed as the sum of 2 viscous travelling profiles. This yields the global existence, stability and uniform BV bounds for every solution with suitably small BV data.

Chaotic dynamics for perturbations of infinite-dimensional Hamiltonian systems - Massimiliano Berti; Carlo Carminati (Nonlinear Anal. 48 (2002) 481-504, 2002)

Curvature theory of boundary phases: the two-dimensional case - Andrea Braides; Andrea Malchiodi (Interfaces Free Bound. 7 (2002) 345-370, 2002)
We describe the behaviour of minimum problems involving non-convex surface integrals in 2D, singularly perturbed by a curvature term. We show that their limit is described by functionals which take into account energies concentrated on vertices of polygons. Non-locality and non-compactness effects are highlighted.

Existence of minimal H-bubbles - Paolo Caldiroli; Roberta Musina (Commun. Contemp. Math. 4 (2002) 177-209, 2002)

Fast Arnold diffusion in systems with three time scales - Massimiliano Berti; Philippe Bolle (Discrete Contin. Dyn. Syst. 8 (2002) 795-811, 2002)
We consider the problem of Arnold Diffusion for nearly integrable partially isochronous Hamiltonian systems with three time scales. By means of a careful shadowing analysis, based on a variational technique, we prove that, along special directions, Arnold diffusion takes place with fast (polynomial) speed, even though the \\\"splitting determinant\\\" is exponentially small.

Flow Stability of Patchy Vector Fields and Robust Feedback Stabilization - Fabio Ancona; Alberto Bressan (SIAM J. Control Optim. 41 (2002) 1455-1476, 2002)
The paper is concerned with patchy vector fields, a class of discontinuous, piecewise smooth vector fields that were introduced in AB to study feedback stabilization problems. We prove the stability of the corresponding solution set w.r.t. a wide class of impulsive perturbations. These results yield the robusteness of patchy feedback controls in the presence of measurement errors and external disturbances.

A functional analysis approach to Arnold diffusion - Massimiliano Berti; Philippe Bolle (Ann. Inst. H. Poincare Anal. Non Lineaire 19 (2002) 395-450, 2002)
We discuss in the context of nearly integrable Hamiltonian systems a functional analysis approach to the \\\"splitting of separatrices\\\" and to the \\\"shadowing problem\\\". As an application we apply our method to the problem of Arnold Diffusion for nearly integrable partially isochronous systems improving known results.

Geometry of Jacobi Curves I - Andrei A. Agrachev; Igor Zelenko (J. Dynam. Control Systems 8 (2002) 93-140, 2002)
Jacobi curves are deep generalizations of the spaces of \\\"Jacobi fields\\\" along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. In our paper we develop differential geometry of these curves which provides basic feedback or gauge invariants for a wide class of smooth control systems and geometric structures. Two principal invariants are the generalized Ricci curvature, which is an invariant of the parametrized curve in the Lagrange Grassmannian endowing the curve with a natural projective structure, and a fundamental form, which is a fourth-order differential on the curve. The so-called rank 1 curves are studied in more detail. Jacobi curves of this class are associated with systems with scalar controls and with rank 2 vector distributions.\\nIn the forthcoming second part of the paper we will present the comparison theorems (i.e., the estimates for the conjugate points in terms of our invariants( for rank 1 curves an introduce an important class of \\\"flat curves\\\".

Geometry of Jacobi curves II - Andrei A. Agrachev; Igor Zelenko (J. Dynam. Control Systems 8 (2002), no. 2, 167--215, 2002)

Global calibrations for the non-homogeneous Mumford-Shah functional - Massimiliano Morini (Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002) 603-648, 2002)
Using a calibration method we prove that, if $\\\\Gamma\\\\subset \\\\Omega$ is a closed regular hypersurface and if the function $g$ is discontinuous along $\\\\Gamma$ and regular outside, then the function $u_{\\\\beta}$ which solves $$ \\\\begin{cases} \\\\Delta u_{\\\\beta}=\\\\beta(u_{\\\\beta}-g)& \\\\text{in $\\\\Omega\\\\setminus\\\\Gamma$} \\\\partial_{\\\\nu} u_{\\\\beta}=0 & \\\\text{on $\\\\partial\\\\Omega\\\\cup\\\\Gamma$} \\\\end{cases} $$ is in turn discontinuous along $\\\\Gamma$ and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional $$ \\\\int_{\\\\Omega\\\\setminus S_u}|\\\\nabla u|^2 dx +{\\\\cal H}^{n-1}(S_u)+\\\\beta\\\\int_{\\\\Omega\\\\setminus S_u}(u-g)^2 dx, $$ over $SBV(\\\\Omega)$, for $\\\\beta$ large enough. Applications of the result to the study of the gradient flow by the method of minimizing movements are shown.

On the K+P problem for a three-level quantum system: optimality implies resonance - Ugo Boscain; Thomas Chambrion; Jean-Paul Gauthier (J.Dynam. Control Systems 8 (2002),no.4, 547, 2002)

Linearized elasticity as gamma-limit of finite elasticity - Gianni Dal Maso; Matteo Negri; Danilo Percivale (Set-Valued Anal. 10 (2002), p.165-183, 2002)

Local calibrations for minimizers of the Mumford-Shah functional with a triple junction - Maria Giovanna Mora (Commun. Contemp. Math. 4 (2002) 297-326, 2002)
We prove that, if u is a function satisfying all Euler conditions for the Mumford-Shah functional and the discontinuity set of u is given by three line segments meeting at the origin with equal angles, then there exists a neighbourhood U of the origin such that u is a minimizer of the Mumford-Shah functional on U with respect to its own boundary conditions on the boundary of U. The proof is obtained by using the calibration method.

On a Lyapunov functional relating shortening curves and viscous conservation laws - Stefano Bianchini; Alberto Bressan (Nonlinear Anal. 51 (2002) 649-662, 2002)
We study a nonlinear functional which controls the area swept by a curve moving in the plane in the direction of curvature. In turn, this yields a priori estimates on solutions to a class of parabolic equations and of scalar viscous conservation laws. A further application provides an estimate on the \\\"change of shape\\\" of a BV solution to a scalar conservation law.

A model for the quasi-static growth of a brittle fracture: existence and approximation results - Gianni Dal Maso; Rodica Toader (Math. Models Methods Appl. Sci., 12 (2002), no. 12, 1773, 2002)
We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo.9 The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the L2-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Refs. 9 and 8, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith\\\'s criterion holds at the crack tips. We also prove that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough.

A model for the quasi-static growth of brittle fractures based on local minimization - Gianni Dal Maso; Rodica Toader (Math.Models Methods Appl. Sci., 12 (2002) , p.1773-1800., 2002)
We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo.9 The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the L2-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Refs. 9 and 8, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith\\\'s criterion holds at the crack tips. We also prove that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough.

A Model for the Quasi-Static Growth of Brittle Fractures: Existence and Approximation Results - Gianni Dal Maso; Rodica Toader (Arch. Ration. Mech. Anal. 162 (2002) 101-135, 2002)
We give a precise mathematical formulation of a variational model for the irreversible quasi-static evolution of brittle fractures proposed by G.A. Francfort and J.-J. Marigo, and based on Griffith\\\'s theory of crack growth. In the two-dimensional case we prove an existence result for the quasi-static evolution and show that the total energy is an absolutely continuous function of time, although we can not exclude that the bulk energy and the surface energy may present some jump discontinuities. This existence result is proved by a time discretization process, where at each step a global energy minimization is performed, with the constraint that the new crack contains all cracks formed at the previous time steps. This procedure provides an effective way to approximate the continuous time evolution.

A multiplicity result for the Schrodinger-Maxwell equations with negative potential - Giuseppe Maria Coclite (Ann. Pol. Math. 79 (2002) 21-30, 2002)
We prove the existence of a sequence of radial solutions with negative energy of the Schrödinger-Maxwell equations under the action of a negative potential.

Multiplicity results for the Yamabe problem on Sn - Antonio Ambrosetti (Proceedings of the National Academy of Sciences of the United States of America. 2002 Nov; 99(24):15252-6, 2002)
We discuss some results related to the existence of multiple solutions for the Yamabe problem.

An optimal fast-diffusion variational method for non isochronous system - Luca Biasco; Massimiliano Berti; Philippe Bolle (SISSA Library, 2002)

Optimal stability and instability results for a class of nearly integrable Hamiltonian systems - Massimiliano Berti; Luca Biasco; Philippe Bolle (Atti.Accad.Naz.Lincei Cl.Sci.Fis.Mat.Natur.Rend.Lincei (9) Mat.Appl.13(2002),no.2,77-84, 2002)

The passage from nonconvex discrete systems to variational problems in Sobolev spaces: the one-dimensional case - Andrea Braides; Maria Stella Gelli; Mario Sigalotti (Proc. Steklov Inst. Math. 236 (2002) 395-414, 2002)

Prescribing a fourth oder conformal invariant on the standard sphere - Part I: a perturbation result - Zindine Djadli; Mohameden Ould Ahmedou; Andrea Malchiodi (Commun. Contemp. Math., 2002, 4, 375, 2002)

Prescribing a fourth oder conformal invariant on the standard sphere - Part II: blow up analysis and applications - Zindine Djadli; Andrea Malchiodi; Mohameden Ould Ahmedou (Ann. Sc. Norm. Super. Pisa Cl. Sci., 2002, 1, 387, 2002)

On the reachability of quantized control systems - Antonio Bicchi; Alessia Marigo; Benedetto Piccoli (IEEE Trans. Automat. Contr. 47 (2002) 546-563, 2002)
In this paper, we study control systems whose input sets are quantized, i.e., finite or regularly distributed on a mesh. We specifically focus on problems relating to the structure of the reachable set of such systems, which may turn out to be either dense or discrete. We report results on the reachable set of linear quantized systems, and on a particular but interesting class of nonlinear systems, i.e., nonholonomic chained-form systems. For such systems, we provide a complete characterization of the reachable set, and, in case the set is discrete, a computable method to completely and succinctly describe its structure. Implications and open problems in the analysis and synthesis of quantized control systems are addressed.

The scalar curvature problem on $S\\\\sp n$: an approach via Morse theory - Andrea Malchiodi (Calc. Var. Partial Differential Equations 14 (2002), no. 4, 429-445, 2002)

Singular elliptic problems with critical growth - Paolo Caldiroli; Andrea Malchiodi (Comm. Partial Differential Equations 27 (2002), no. 5-6, 847-876, 2002)

Solutions concentrating on spheres to symmetric singularly perturbed problems - Antonio Ambrosetti; Andrea Malchiodi; Wei-Ming Ni (C.R.Math.Acad.Sci. Paris 335 (2002),no.2,145-150, 2002)
We discuss some existence results concerning problems (NLS) and (N), proving the existence of radial solutions concentrating on a sphere.

Stability of planar switched systems: the linear single input case - Ugo Boscain (SIAM J. Control Optim. 41 (2002), no. 1, 89-112, 2002)
We study the stability of the origin for the dynamical system $\\\\dot x(t)=u(t)Ax(t)+(1-u(t))Bx(t),$ where A and B are two 2 × 2 real matrices with eigenvalues having strictly negative real part, $x\\\\in {\\\\mbox{{\\\\bf R}}}^2$, and $u(.):[0,\\\\infty[\\\\to[0,1]$ is a completely random measurable function. More precisely, we find a (coordinates invariant) necessary and sufficient condition on A and B for the origin to be asymptotically stable for each function u(.). The result is obtained without looking for a common Lyapunov function but studying the locus in which the two vector fields Ax and Bx are collinear. There are only three relevant parameters: the first depends only on the eigenvalues of A, the second depends only on the eigenvalues of B, and the third contains the interrelation among the two systems, and it is the cross ratio of the four eigenvectors of A and B in the projective line CP1. In the space of these parameters, the shape and the convexity of the region in which there is stability are studied.

On the Stability of the Standard Riemann Semigroup - Stefano Bianchini; Rinaldo M. Colombo (P. Am. Math. Soc., 2002, 130, 1961, 2002)

On the Yamabe problem and the scalar curvature problems under boundary conditions - Antonio Ambrosetti; Li YanYan; Andrea Malchiodi (Math. Ann., 2002, 322, 667, 2002)

Adiabatic limits of closed orbits for some Newtonian systems in R-n - Andrea Malchiodi (Asymptotic Anal., 2001, 25, 149-181, 2001)
We deal with a Newtonian system like x\\\'\\\' + V\\\'(x) = 0. We suppose that V: \\\\R^n \\\\to \\\\R possesses an (n-1)-dimensional compact manifold M of critical points, and we prove the existence of arbitrarity slow periodic orbits. When the period tends to infinity these orbits, rescaled in time, converge to some closed geodesics on M.

A case study in vanishing viscosity - Stefano Bianchini; Alberto Bressan (Discrete Cont. Dyn. Syst. 7 (2001) 449-476, 2001)

Controllability for discrete systems with a finite control set - Yacine Chitour; Benedetto Piccoli (Math. Control Signals Systems 14 (2001) 173-193, 2001)

Dieletric breakdown: optimal bounds - Adriana Garroni; Vincenzo Nesi; Marcello Ponsiglione (Proc. of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences 457 (2001): p. 2317-2335, OCT. 8, 2001, 2001)

Existence and nonexistence results for a class of nonlinear, singular Sturm-Liouville equations - Paolo Caldiroli; Roberta Musina (Adv. Differential Equations 6 (2001), no. 3, 303-326, 2001)

Extremal synthesis for generic planar systems - Ugo Boscain; Benedetto Piccoli (J. Dynam. Control Systems, 2001, 7, 209, 2001)

Finite Difference Approximation of Free Discontinuity Problems - Massimo Gobbino; Maria Giovanna Mora (Proc. Royal Soc. Edinb. Ser. A 131 (2001), no. 3, 567-595, 2001)

Free-discontinuity problems: calibration and approximation of solutions - Massimiliano Morini (SISSA, 2001)
Keywords: Calibration of solutions

Gamma-limit of periodic obstacles - Gianni Dal Maso; Paola Trebeschi (Acta Appl. Math., 2001, 65, 207-215, 2001)
We compute the Gamma-limit of a sequence obstacle functionals in the case of periodic obstacles.

A Glimm type functional for a special Jin-Xin relaxation model - Stefano Bianchini (Ann. Inst. H. Poincare\\\' Anal. Non Lineaire 18 (2001), no. 1, 19-42, 2001)

Global continuous Riemann solver for nonlinear elasticity - Jean-Marc Mercier; Benedetto Piccoli (Arch. Ration. Mech. An., 2001, 156, 89, 2001)

Local calibrations for minimizers of the Mumford-Shah functional with a regular discontinuity set - Maria Giovanna Mora; Massimiliano Morini (Ann. I. H. Poincare - An., 2001, 18, 403, 2001)

A monotonicity approach to nonlinear Dirichlet problems in perforated domains - Gianni Dal Maso; Igor V. Skrypnik (Adv. Math. Sci. Appl. 11 (2001) 721-751, 2001)
We study the asymptotic behaviour of solutions to Dirichlet problems in perforated domains for nonlinear elliptic equations associated with monotone operators. The main difference with respect to the previous papers on this subject is that no uniformity is assumed in the monotonicity condition. Under a very general hypothesis on the holes of the domains, we construct a limit equation, which is satisfied by the weak limits of the solutions. The additional term in the limit problem depends only on the local behaviour of the holes, which can be expressed in terms of suitable nonlinear capacities associated with the monotone operator.

Morse properties for the minimum time function on 2-D manifolds - Ugo Boscain; Benedetto Piccoli (J. Dynam. Control Systems 7 (2001), no. 3, 385--423, 2001)

Multiple positive solutions of some elliptic equations in \\\\bold R\\\\sp N - Andrea Malchiodi (Nonlinear Anal. 43 (2001) 159-172, 2001)

Multiplicity results for some nonlinear Schrodinger equations with potentials - Antonio Ambrosetti; Andrea Malchiodi; Simone Secchi (Arch. Ration. Mech. An., 2001, 159, 253, 2001)

Non-compactness and multiplicity results for the Yamabe problem on Sn - Massimiliano Berti; Andrea Malchiodi (J. Funct. Anal. 180 (2001) 210-241, 2001)

Numerical Methods for Free-Discontinuity Problems Based on Approximations by Γ-Convergence - Matteo Negri (SISSA, 2001)
Keywords: Mumford-Shah functional

Numerical minimization of the Mumford-Shah functional - Matteo Negri; Maurizio Paolini (Calcolo, 2001, 38, 67, 2001)

S^2 type parametric surfaces with prescribed mean curvature and minimal energy - Paolo Caldiroli; Roberta Musina (Nonlinear equations : methods, models and applications (Bergamo, 2001) / Daniela Lupo, Carlo D. Pagani, Bernhard Ruf, editors. - Basel : Birkhäuser, 2003. - (Progress in nonlinear differential equations and their applications; 54). - p. 61-77, 2001)

On the spreading of characteristics for non-convex conservation laws - Helge Kristian Jenssen; Carlo Sinestrari (Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 909-925, 2001)
We study the spreading of characteristics for a class of one-dimensional scalar conservation laws for which the flux function has one point of inflection. It is well known that in the convex case the characteristic speed satisfies a one-sided Lipschitz estimate. Using Dafermos\\\' theory of generalized characteristics, we show that the characteristic speed in the non-convex case satisfies an Hölder estimate. In addition, we give a one-sided Lipschitz estimate with an error term given by the decrease of the total variation of the solution.

Stability of L-infinity solutions for hyperbolic systems with coinciding shocks and rarefactions - Stefano Bianchini (Siam J. Math. Anal., 2001, 33, 959, 2001)
We consider a hyperbolic system of conservation laws u_t + f(u)_x = 0 and u(0,\\\\cdot) = u_0, where each characteristic field is either linearly degenerate or genuinely nonlinear. Under the assumption of coinciding shock and rarefaction curves and the existence of a set of Riemann coordinates $w$, we prove that there exists a semigroup of solutions $u(t) = \\\\mathcal{S}_t u_0$, defined on initial data $u_0 \\\\in L^\\\\infty$. The semigroup $\\\\mathcal{S}$ is continuous w.r.t. time and the initial data $u_0$ in the $L^1_{\\\\text{loc}}$ topology. Moreover $\\\\mathcal{S}$ is unique and its trajectories are obtained as limits of wave front tracking approximations.

Stationary states for a two-dimensional singular Schrodinger equation - Paolo Caldiroli; Roberta Musina (Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 4 (2001), no. 3, 609-633., 2001)

On the subanalyticity of Carnot-Caratheodory distances - Andrei A. Agrachev; Jean-Paul Gauthier (Ann. I. H. Poincare - An., 2001, 18, 359, 2001)

On the symmetric scalar curvature problem on S\\\\sp n - Antonio Ambrosetti; Andrea Malchiodi (J. Differential Equations 170 (2001) 228-245, 2001)
We discuss some existence results dealing with the scalar curvature problem on S\\\\sp n in the presence of various symmetries.

Uniqueness of classical and nonclassical solutions for nonlinear hyperbolic systems - Paolo Baiti; Philippe G. LeFloch; Benedetto Piccoli (J. Differential Equations 172 (2001) 59-82, 2001)

Uniqueness of solutions to Hamilton-Jacobi equations arising in the Calculus of Variations - Gianni Dal Maso; Helene Frankowska (Optimal control and partial differential equations : in honour of professor Alain Bensoussan\\\'s 60th birthday / edited by José Luis Menaldi, Edmundo Rofman, and Agnès Sulem.,Amsterdam : IOS Press, 2001, p. 335-345, 2001)
We prove the uniqueness of the viscosity solution to the Hamilton-Jacobi equation associated with a Bolza problem of the Calculus of Variations, assuming that the Lagrangian is autonomous, continuous, superlinear, and satisfies the usual convexity hypothesis. Under the same assumptions we prove also the uniqueness, in a class of lower semicontinuous functions, of a slightly different notion of solution, where classical derivatives are replaced only by subdifferentials. These results follow from a new comparison theorem for lower semicontinuous viscosity supersolutions of the Hamilton-Jacobi equation, that is proved in the general case of lower semicontinuous Lagrangians.

Abnormal extremals for minimum time on the plane - Ugo Boscain; Benedetto Piccoli (SISSA Library, 2000)

Arnold's Diffusion in nearly integrable isochronous Hamiltonian systems - Massimiliano Berti; Philippe Bolle (SISSA Library, 2000)

BV estimates for multicomponent chromatography with relaxation - Alberto Bressan; Wen Shen (Discrete Contin. Dynam. Systems 6 (2000) 21-38, 2000)
We consider the Cauchy problem for a system of $2n$ balance laws which arises from the modelling of multi-component chromatography: $$\\\\left\\\\{ \\\\eqalign{u_t+u_x&=-{1\\\\over\\\\ve}\\\\big( F(u)-v\\\\big),\\\\cr v_t&={1\\\\over\\\\ve}\\\\big( F(u)-v\\\\big),\\\\cr}\\\\right. \\\\eqno(1)$$ This model describes a liquid flowing with unit speed over a solid bed. Several chemical substances are partly dissolved in the liquid, partly deposited on the solid bed. Their concentrations are represented respectively by the vectors $u=(u_1,\\\\ldots,u_n)$ and $v=(v_1,\\\\ldots,v_n)$. We show that, if the initial data have small total variation, then the solution of (1) remains with small variation for all times $t\\\\geq 0$. Moreover, using the $\\\\L^1$ distance, this solution depends Lipschitz continuously on the initial data, with a Lipschitz constant uniform w.r.t.~$\\\\ve$. Finally we prove that as $\\\\ve\\\\to 0$, the solutions of (1) converge to a limit described by the system $$\\\\big(u+F(u)\\\\big)_t+u_x=0,\\\\qquad\\\\qquad v=F(u).\\\\eqno(2)$$ The proof of the uniform BV estimates relies on the application of probabilistic techniques. It is shown that the components of the gradients $v_x,u_x$ can be interpreted as densities of random particles travelling with speed 0 or 1. The amount of coupling between different components is estimated in terms of the expected number of crossing of these random particles. This provides a first example where BV estimates are proved for general solutions to a class of $2n\\\\times 2n$ systems with relaxation.

BV solutions for a class of viscous hyperbolic systems - Stefano Bianchini; Alberto Bressan (Indiana Univ. Math. J. 49 (2000) 1673-1714, 2000)

The Calibration Method for Free Discontinuity Problems - Gianni Dal Maso (European Congress of Mathematics. Volume I : Barcelona, July 10-14, 2000 / Carles Casacuberta ... [et al.], editors. , Boston : Birkhauser, 2001, p. 317-326., 2000)
The calibration method is used to identify some minimizers of the Mumford-Shah functional. The method is then extended to more general free discontinuity problems.

On the convergence of Godunov scheme for nonlinear hyperbolic systems - Alberto Bressan; Helge Kristian Jenssen (Chinese Ann. Math. B, 2000, 21, 269, 2000)

Diffusion time and splitting of separatrices for nearly integrable - Massimiliano Berti; Philippe Bolle (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 2000, 11, 235, 2000)

Elliptic variational problems in $ R\\\\sp N$ with critical growth - Antonio Ambrosetti; Jesus Garcia Azorero; Ireneo Peral (J. Differential Equations 168 (2000), no. 1, 10--32, 2000)

Existence and multiplicity results for some nonlinear elliptic equations: a survey. - Antonio Ambrosetti; Jesus Garcia Azorero; Ireneo Peral (Rend. Mat. Appl., 2000, 20, 167, 2000)

Existence and multiplicity results for some problems in Riemannian geometry - Andrea Malchiodi (SISSA, 2000)
Keywords: Yamabe problem

Functionals depending on curvatures with constraints - Maria Giovanna Mora; Massimiliano Morini (Rend. Sem. Mat. Univ. Padova 104 (2000), 173--199, 2000)

High-order Averaging and Stability of Time-Varying Systems - Andrey Sarychev (SISSA Library, 2000)

Local calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity sets - Gianni Dal Maso; Maria Giovanna Mora; Massimiliano Morini (J. Math. Pures Appl. 79, 2 (2000) 141-162, 2000)

Minimization of functionals of the gradient by Baire's theorem - Sandro Zagatti (SIAM J. Control Optim. 38 (2000) 384-399, 2000)
We give sufficient conditions for the existence of solutions of the minimum problem $$ {\mathcal{P}}_{u_0}: \qquad \hbox{Minimize}\quad \int_Ω g(Du(x))dx, \quad u\in u_0 + W_0^{1,p}(Ω,{\mathbb{R}}), $$ based on the structure of the epigraph of the lower convex envelope of g, which is assumed be lower semicontinuous and to grow at infinity faster than the power p with p larger than the dimension of the space. No convexity conditions are required on g, and no assumptions are made on the boundary datum $u_0\in W_0^{1,p}(Ω,\mathbb{R})$.

A note on the scalar curvature problem in the presence of symmetries - Antonio Ambrosetti; Li YanYan; Andrea Malchiodi (Ricerche Mat. 49 (2000), suppl., 169-176, 2000)

Principal invariants of Jacobi curves - Andrei A. Agrachev; Igor Zelenko (Nonlinear control in the Year 2000 / Alberto Isidori, Francoise Lamnabhi-Lagarrigue, Witold Respondek (eds.) - Springer : Berlin, 2001. - (Lecture notes in control and information sciences ; 258). - ISBN 1-85233-363-4 (v. 1). - p. 9-22., 2000)
Jacobi curves are far going generalizations of the spaces of \\\"Jacobi fields\\\" along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. Differential geometry of these curves provides basic feedback or gauge invariants for a wide class of smooth control systems and geometric structures. In the present paper we mainly discuss two principal invariants: the generalized Ricci curvature, which is an invariant of the parametrized curve in the Lagrange Grassmanian providing the curve with a natural projective structure, and a fundamental form, which is a 4-oder differential on the curve.

Quantized control systems and discrete nonholonomy - Alessia Marigo; Benedetto Piccoli; Antonio Bicchi (Lagrangian and Hamiltonian Methods for Nonlinear Control : a proc. volume from the IFAC Workshop. Princeton, New Jersey, 16-18 March 2000 / ed. by N.E. Leonard, R. Ortega. - Oxford : Pergamon, 2000, 2000)

Reachability Analysis for a Class of Quantized Control Systems - Alessia Marigo; Benedetto Piccoli; Antonio Bicchi (Proc. 39th IEEE Int. Conf. on Decision and Control 4 (2000) 3963-3968, 2000)
We study control systems whose input sets are quantized. We specifically focus on problems relating to the structure of the reachable set of such systems, which may turn out to be either dense or discrete. We report on some results on the reachable set of linear quantized systems, and study in detail an interesting class of nonlinear systems, forming the discrete counterpart of driftless nonholonomic continuous systems. For such systems, we provide a complete characterization of the reachable set, and, in the case the set is discrete, a computable method to describe its lattice structure.

Regular Synthesis and Sufficiency Conditions for Optimality - Benedetto Piccoli; Hector J. Sussmann (SIAM J. Control Optim. 39 (2000) 359-410, 2000)
We propose a definition of \\\"regular synthesis\\\" that is more general than those suggested by other authors such as Boltyanskii and Brunovsky, and an even more general notion of \\\"regular presynthesis.\\\" We give a complete proof of the corresponding sufficiency theorem, a slightly weaker version of which had been stated in an earlier article, with only a rough outline of the proof. We illustrate the strength of our result by showing that the optimal synthesis for the famous Fuller problem satisfies our hypotheses. We also compare our concept of synthesis with the simpler notion of a \\\"family of solutions of the closed-loop equation arising from an optimal feedback law,\\\" and show by means of examples why the latter is inadequate, and why the difficulty cannot be resolved byusing other concepts of solution--such as Filippov solutions, or the limits of sample-and-hold solutions recently proposed as feedback solutions by Clarke, Ledyaev, Subbotin and Sontag -for equations with a non-Lipschitz and possibly discontinuous right-hand side.

Scalar curvature under boundary conditions - Antonio Ambrosetti; Li YanYan; Andrea Malchiodi (Cr. Acad. Sci. I-Math, 2000, 330, 1013, 2000)

The semigroup generated by a Temple class system with non-convex flux function - Stefano Bianchini (Differential Integral Equations 13 (2000) 1529-1550, 2000)
We consider the Cauchy problem for a nonlinear n × n system of conservation laws of Temple class, i.e. with coinciding shock and rarefaction curves and with a coordinate system made of Riemann invariants. Without any assumption on the convexity of the flux function, we prove the existence of a semigroup made of weak solutions of the equations and depending Lipschitz continuously on the initial data with bounded total variation.

On the shift differentiability of the flow generated by a hyperbolic system of conservation laws - Stefano Bianchini (Discrete Contin. Dynam. Systems 6 (2000), no. 2, 329-350, 2000)

Stability of L^infty Solutions of Temple Class Systems - Alberto Bressan; Paola Goatin (Differential Integral Equations 13 (2000) 1503-1528, 2000)
Let $u_t+f(u)_x=0$ be a strictly hyperbolic, genuinely nonlinear system of conservation laws of Temple class. In this paper, a continuous semigroup of solutions is constructed on a domain of $L^∞$ functions, with possibly unbounded variation. Trajectories depend Lipschitz continuously on the initial data, in the $L^1$ distance. Moreover, we show that a weak solution of the Cauchy problem coincides with the corresponding semigroup trajectory if and only if it satisfies an entropy condition of Oleinik type, concerning the decay of positive waves.

On a Steffen\\\'s result about parametric surfaces with prescribed mean curvature - Roberta Musina; Paolo Caldiroli (SISSA Library, 2000)

A Uniqueness Condition for Hyperbolic Systems of Conservation Laws - Alberto Bressan; Marta Lewicka (Discrete Contin. Dynam. Systems 6 (2000) 673-682, 2000)
Consider the Cauchy problem for a hyperbolic $n\\\\times n$ system of conservation laws in one space dimension: $$u_t+f(u)_x=0, u(0,x)=\\\\bar u(x).\\\\eqno(CP)$$ Relying on the existence of a continuous semigroup of solutions, we prove that the entropy admissible solution of (CP) is unique within the class of functions $u=u(t,x)$ which have bounded variation along a suitable family of space-like curves.

Value Functions for Bolza Problems with Discontinuous Lagrangians and Hamilton-Jacobi inequalities - Gianni Dal Maso; Helene Frankowska (ESAIM Control Optim. Calc. Var., 5 (2000), n. 5, p. 369-393., 2000)

Well-posedness of the Cauchy problem for n x n systems of conservation laws - Alberto Bressan; Graziano Crasta; Benedetto Piccoli (Mem. Amer. Math. Soc. 146 (2000), no. 694, 134 p., 2000)

The anisotropy introduced by the mesh in the finite element approximation of the Mumford-Shah functional - Matteo Negri (Numer. Funct. Anal. Optim. 20 (1999), no. 9-10, 957-982, 1999)
We compute explicitly the anisotropy effect in the H1 term, generated in the approximation of the Mumford-Shah functional by finite element spaces defined on structured triangulations.

Approximation, Stability and control for Conservation Laws - Andrea Marson (SISSA, 1999)

Asymptotic behaviour of nonlinear elliptic higher order equations in perforated domains - Gianni Dal Maso; Igor V. Skrypnik (Journal d\\\'Analyse Mathematique, Volume 79, 1999, Pages: 63-112, 1999)

Blowup asymptotics for scalar conservation laws with a source - Helge Kristian Jenssen; Carlo Sinestrari (Comm. in Partial Differential Equations 24 (1999) 2237-2261, 1999)

The calibration method for the Mumford-Shah functional - Giovanni Alberti; Guy Bouchitte; Gianni Dal Maso (C. R. Acad. Sci. Paris Ser. I Math. 329 (1999), no. 3, 249-254, 1999)
In this Note we adapt the calibration method to functionals of Mumford-Shah type, and provide a criterion (Theorem 1) to verify that a given function is energy minimizing. Among other applications, we use this criterion to show that certain triple-junction configurations are minimizing (Example 3).

Discrete approximation of the Mumford-Shah functional in dimension two - Antonin Chambolle; Gianni Dal Maso (M2AN 33 (1999) 651-672, 1999)

Evans-Vasilesco theorem in Dirichlet spaces - Gianni Dal Maso; Virginia De Cicco (Rendiconti di Matematica e delle sue Applicazioni. vol. 19, Issue 7, (1999), pages : 1-15, 1999)

Extremal faces of the range of a vector measure and a theorem of Lyapunov - Stefano Bianchini (J. Math. Anal. Appl. 231 (1999) 301-318, 1999)

Hyperbolic Systems of Conservation Laws - Alberto Bressan (Rev. Mat. Complut. 12 (1999) 135-200, 1999)
This is a survey paper, written in the occasion of an invited talk given by the author at the Universidad Complutense in Madrid, October 1998. Its purpose is to provide an account of some recent advances in the mathematical theory of hyperbolic systems of conservation laws in one space dimension. After a brief review of basic concepts, we describe in detail the method of wave-front tracking approximation and present some of the latest results on uniqueness and stability of entropy weak solutions.

L-1 stability estimates for n x n conservation laws - Alberto Bressan; Tai-Ping Liu; Tong Yang (Arch. Ration. Mech. Anal. 149 (1999), no. 1, 1--22, 1999)
Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\\\\times n$ system of conservation laws, each characteristic field being linearly degenerate or genuinely nonlinear. In this paper we explicitly define a functional $\\\\Phi=\\\\Phi(u,v)$, equivalent to the $L^1$ distance, which is `almost decreasing\\\', i.e., $\\\\Phi(u(t),v(t))-\\\\Phi(u(s),v(s))\\\\leq\\\\break O (\\\\epsilon)·(t-s)$ for all $t>s\\\\geq 0$, for every pair of $\\\\epsilon$-approximate solutions $u,v$ with small total variation, generated by a wave-front-tracking algorithm. The small parameter $\\\\epsilon$ here controls the errors in the wave speeds, the maximum size of rarefaction fronts and the total strength of all non-physical waves in $u$ and in $v$. From the above estimate, it follows that front-tracking approximations converge to a unique limit solution, depending Lipschitz continuously on the initial data, in the $L^1$ norm. This provides a new proof of the existence of the standard Riemann semigroup generated by an $n\\\\times n$ system of conservation laws.\\\'\\\'

A Lipschitz selection from the set of minimizers of a nonconvex functional of the gradient - Gianni Dal Maso; Vladimir V. Goncharov; Antonio Ornelas (Nonlinear Analysis, Theory, Methods and Applications. Volume 37, Issue 6, September 1999, Pages 707-717, 1999)
A constructive and improved version of the proof that there exist a continuous map that solves the convexified problem is presented. A Lipschitz continuous map is analyzed such that a map vector minimizes the functional at each vector satisfying Cellina\\\'s condition of existence of minimum. This map is explicitly given by a direct constructive algorithm.

A multiplicity result for the Yamabe problem on $S\\\\sp n$ - Antonio Ambrosetti; Andrea Malchiodi (J. Funct. Anal. 168 (1999), no. 2, 529-561, 1999)
We prove a multiplicity result for the Yamabe problem on the manifold (S, g), where g is a perturbation of the standard metric g0 of Sn. Solutions are found by variational methods via an abstract perturbation result.

Nonclassical Shocks and the Cauchy Problem for Nonconvex Conservation Laws - Debora Amadori; Paolo Baiti; Philippe G. LeFloch; Benedetto Piccoli (J. Differential Equations 151 (1999) 345-372, 1999)
The Riemann problem for a conservation law with a nonconvex (cubic) flux can be solved in a class of admissible nonclassical solutions that may violate the Oleinik entropy condition but satisfy a single entropy inequality and a kinetic relation. We use such a nonclassical Riemann solver in a front tracking algorithm, and prove that the approximate solutions remain bounded in the total variation norm. The nonclassical shocks induce an increase of the total variation and, therefore, the classical measure of total variation must be modified accordingly. We prove that the front tracking scheme converges strongly to a weak solution satisfying the entropy inequality.

Oleinik type estimates and uniqueness for n x n conservation laws - Alberto Bressan; Paola Goatin (J. Differential Equations 156 (1999), no. 1, 26--49, 1999)
Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\\\\times n$ system of conservation laws in one space dimension. Relying on the existence of a semigroup of solutions, we first establish the uniqueness of entropy admissible weak solutions to the Cauchy problem, under a mild assumption on the local oscillation of $u$ in a forward neighborhood of each point in the $t\\\\text{-}x$ plane. In turn, this yields the uniqueness of weak solutions which satisfy a decay estimate on positive waves of genuinely nonlinear families, thus extending a classical result proved by Oleĭnik in the scalar case.

Perturbation of $Δ u+u^(N+2)/(N-2)=0$, the scalar curvature problem in $R^N$, and related topics - Antonio Ambrosetti; Jesus Garcia Azorero; Ireneo Peral (J. Funct. Anal. 165 (1999) 117-149, 1999)
Some nonlinear elliptic equations on $R^N$ which arise perturbing the problem with the critical Sobolev exponent are studied. In particular, some results dealing with the scalar curvature problem in $R^N$ are given.

Projection singularities of extremals for planar systems - Ugo Boscain; Benedetto Piccoli (SISSA Library, 1999)

Renormalized solutions of elliptic equations with general measure data - Gianni Dal Maso; Francois Murat; Luigi Orsina; Alain Prignet (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), no. 4, 741-808, 1999)

On the scalar curvature problem under symmetry - Antonio Ambrosetti; Andrea Malchiodi (SISSA Library, 1999)

Some properties of the solutions of obstacle problems with measure data - Paolo Dall'Aglio; Gianni Dal Maso (Ricerche Matematiche., Supplemento dedicato a Ennio De Giorgi, vol. 48 (1999), page : 99-116, 1999)

Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws - Alberto Bressan; Philippe G. LeFloch (Indiana Univ. Math. J. 48 (1999), no. 1, 43--84, 1999)
The paper is concerned with the qualitative structure of entropy solutions to a strictly hyperbolic, genuinely nonlinear system of conservation laws. We first give an accurate description of the local and global wave-front structure of a BV solution, generated by a front tracking algorithm.

Vanishing viscosity solutions of hyperbolic systems on manifolds - Stefano Bianchini; Alberto Bressan (SISSA Library, 1999)

Variational formulation of softening phenomena in fracture mechanics. The one-dimensional case - Andrea Braides; Gianni Dal Maso; Adriana Garroni (Arch. Ration. Mech. Anal. 146 (1999), no. 1, 23--58, 1999)
Starting from experimental evidence, the authors justify a variational model for softening phenomena in fracture of one-dimensional bars where the energy is given by the contribution and interaction of two terms: a typical bulk energy term depending on elastic strain and a discrete part that depends upon the jump discontinuities that occur in fracture. A more formal, rigorous derivation of the model is presented by examining the $\\\\Gamma$-convergence of discrete energy functionals associated to an array of masses and springs. Close attention is paid to the softening and fracture regimes. \\nOnce the continuous model is derived, it is fully analyzed without losing sight of its discrete counterpart. In particular, the associated boundary value problem is studied and a detailed analysis of the stationary points under the presence of a dead load is performed. A final, interesting section on the scale effect on the model is included.

The vector measures whose range is strictly convex - Stefano Bianchini; Carlo Mariconda (J. Math. Anal. Appl. 232 (1999) 1-19, 1999)