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Seminars 2003-2004

Quasi-periodic solutions for the completely resonant non-linear wave equation in 1D and 2D

We provide quasi-periodic solutions with two frequencies $\ome\in \R^2$, for a class of completely resonant non-linear wave equations in one and two spatial dimensions and with periodic boundary conditions. The chosen frequencies are close to that of the linear system in an uncountable zero measure Cantor set. The main idea is to work in an appropriate invariant subspace so that no small divisor problem arises.

Non-Linear Eigenvalues, Symplectic Geometry and Analytic regularity of sums of Squares

Periodic solutions of nonlinear wave equations with non-monotone forcing terms

Sub-Supersolutions and periodic boundary value problems

Lie-Algebraic Stability Criteria for Switched Systems

In this talk we discuss the connection between commutation relations among a family of asymptotically stable vector fields and stability properties of the corresponding switched system (or, more generally, a differential inclusion). We review some known results for the linear case and then present new results for the nonlinear case. The proof of the main new result involves considering an optimal control problem of finding the "most unstable" trajectory for an associated control system, and showing that there exists an optimal solution which is bang-bang with a bound on the total number of switches.

Mathematical Modeling in Linear Frictional Welding

Infinitely many bounded states for NLS

Improved Moser-Trudinger inequality.

Relaxation in BV of integral functionals with constraint on the norm

We study the relaxation of integral functionals with linear growth defined on Sobolev functions with values on the unit sphere. We prove an integral representation result for the relaxed functionals on the space of BV functions with values on the unit sphere. We give also an application to a generalized version of a variational model in micromagnetics.

Morse-Sard results and genericity for singular curves of distributions

We investigate singular curves of distributions. First, in the framework of sub-Riemannian geometry, we prove that they are scarce, giving several results of Morse-Sard type. The, taking the opposite point of view, we show that, generically (in a sense to be made precise), every nontrivial singular curve of a distribution is of minimal order and of corank one. We give some consequences of these results.

Nonlinear Maxwell equations

Dimension reduction in elasticity via Gamma-convergence

In this series of about 9 seminars we will discuss the derivation of elastic models for thin domains from three-dimensional elasticity by Gamma-convergence methods. Following seminars in April 27, May 4, June 8, 10, 11.

On geodesic equivalence of Riemannian and sub-Riemannian metrics

Our talk is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on generic corank 1 distributions and Engel distributions. Using Pontryagin Maximum Principle, we treat Riemannian and sub-Riemannian cases in an unified way and obtain some algebraic necessary conditions for geodesic equivalence of (sub-)Riemannian metrics, mentioned above, in terms of divisibility of some polynomials (on the fiber of the cotangent bundle of the ambient manifold) associated with these metrics. In this way first we obtain a new elementary proof of classical Levi-Civita's Theorem about the classification of all Riemannian geodesically equivalent metrics in a neighborhood of so-called regular (stable) point w.r.t. these metrics (in the case of a surface regularity means that the metrics are not proportional at the point). Secondly we prove that sub-Riemannian metrics on contact and Engel distributions are geodesically equivalent iff they are proportional. Then we describe all geodesically equivalent sub-Riemannian metrics on quasi-contact distributions. Finally we make the classification of all pairs of geodesically equivalent Riemannian metrics on a surface, which proportional in an isolated point. This is the simplest case, which was not covered by Levi-Civita's Theorem.

Chern-Simons vortex theory, 2

This second talk will deal with the mathematical aspects of vortex solutions.

Birkhof normal form for PDEs with Tame Modulus

On the minimal degree of common Lyapunov polynomial functions for planar switched systems

We consider planar switched systems of the type x'= u(t)Ax(t)+(1-u(t))Bx(t) , where x is in R^2 and A,B are 2x2 matrices, which are asymptotically stable at the origin for every switching function u:[0,+inf)->[0,1] . It is well-known that there are systems of this form such that a common quadratic Lyapunov function does not exist. Anyway the problem of finding an upper bound on the minimal degree of a common Lyapunov polynomial function was open up to now. We answer to this question proving that such a bound actually does not exist.

Chern-Simons vortex theory, 1

The first seminar is focused on the Chern-Simons theory applied to the study of the Quantum Hall effect, in order to explain existence of vortices (with factional charge and statistics) in some condensed-matter experiment.

The role of Liouville type equations in the study of vortices for the selfdual theory.

A nonlocal approach for the Perona-Malik equation and the Mumford-Shah functional

Periodic solutions of nonlinear wave equations and the Nash-Moser theorem

We prove existence of small amplitude $2\pi \slash \om$-periodic solutions of the completely resonant nonlinear wave equation u_{tt} - u_{xx} + f (x, u ) = 0, u ( t, 0 )= u( t, \pi ) = 0 where f(x,0)= f_u (x,0) =0, for any frequency $ \om $ belonging to a Cantor like set of positive measure. The proof is based on a Lyapunov-Schmidt reduction and on a variant of Nash-Moser Implicit Function Theorems to overcome the inherent "small divisor problem".

Reduction in the theory of Jacobi curves

Jacobi curves are the curves in the Lagrangian Grassmanian corresponding to a dynamical system. The differential geometry of these curves provides some basic symplectic invariants of the dynamical system such as the {\sl generalized curvature tensor } and the {\sl Ricci curvature}. In the present work we study the dependence of these invariants on the reduction in the phase space of the systems with symmetries. We also give some illustrating examples from classical mechanics.

On ergodic problems for Hamilton-Jacobi-Isaacs equations with state constraints.

We study the asymptotic behavior as $\lambda\rightarrow 0^+$ of $\lambda v_\lambda$, where $v_\lambda$ is the viscosity solution of the following Hamilton-Jacobi-Isaacs equation (infinite horizon case) $$ \lambda v_\lambda + H(x,Dv_\lambda)=0, $$ with $$ H(x,p):=\min_{b\in B}\max_{a \in A} \{-f(x,a,b)\cdot p -l(x,a,b)\}. $$ We discuss the case in which the state of the system is required to stay in the closure of a bounded domain $\Omega\subset{\Bbb R}^n$ with sufficiently smooth boundary (for example $\partial \Omega\in {\cal C}^2$). Under the uniform approximate controllability assumption of one player, we extend the convergence result of the term $\lambda v_\lambda (x)$ to a constant as $\lambda\rightarrow 0^+$ to Differential Games . We also show how to contruct nonanticipative strategies which satisfies some "good" estimates in order to obtain Holder regularity of the value function.

Some spectral properties of quasi-periodic Schroedinger operators

Homogenization

Water waves, III

Water waves, II

Differentiability problems in metric spaces

Water waves

Lindstedt series for periodic solutions on nonlinear wave equations

I describe a work, in collaboration with G.Gentile, in which we have defined a Linsdedt series expansion for the periodic solutions of nonlinear wave equations like the Klein-Gordon or the sine-Gordon equations. After suitable resummation this expansion is convergent for a large set of amplitudes and a large set of masses of full Lebesgue measure. We have recently extended this result, in collaboration also with M.Procesi, to the case of the string equation.

'Subgeneric' sets.

Regularity properties of subanalytic sets are often used in control theory to establish regularity of various characteristics of real analytic control systems. In some cases, the regularity of the corresponding characteristics of generic smooth control systems, while being of interest, is not known. We suggest an approach to these questions based on older observations due to Yu. Ilyashenko and S. Yakovenko, and S. Yakovenko and the speaker (obtained in a different context). Specifically, we intend to sketch a proof of a partial result in this direction. It is convenient to formulate this result as the logic-theoretic assertion that the extension of the first order theory of the real numbers by (a finite number of) generic smooth functions, restricted to the unit cube, is o-minimal. We will attempt to give a clear and concise explanation of the concepts involved, stressing the usefulness of the logic-theoretic point of view in the present context. In particular, no background in mathematical logic is assumed.

Pseudoholomorphic Maps and Dynamics in 3 Dimensions.

We shall describe tools useful in the study of 3-dimensional flows. The tools are based on partial differential equations of Cauchy-Riemann type. Their solutions allow the construction of global surfaces of sections and, as a generalization, of 2-dimensional foliations transversal to the flow in the complement of finitely many distinguished periodic orbits. Such a global system of transversal sections is established for Reeb-flows on the tight 3-sphere. The applications cover Hamiltonian systems on starlike energy surfaces.

Periodic solutions for the completely resonant wave equation with a general non-linearity.

We consider a non-linear wave equation for which all the solutions of the linearized problem at 0 are $2\pi$-periodic in time. Existence and multiplicity results for periodic solutions of fixed period are presented, under quite general assumptions on the non-linearity. These results are obtained by means of a variational principle.

Minimax solutions for Hamilton-Jacobi equations

Minimax solutions are weak solutions of Cauchy problems for Hamilton-Jacobi equations, constructed from generating families (quadratic at infinity) of the geometric solutions. We describe a new construction of the minimax in terms of Morse theory, and we show its stability by small perturbations of the generating family. We consider the wave front corresponding to the geometric solution as the graph of a multi-valued solution of the Cauchy problem, and we give a geometric criterion to find the graph of the minimax.

Dynamical Approach to Problem of Equivalence of Rank 2 Vector Distributions

Rank $k$ vector distribution on the $n$-dimensional manifold $M$ or $(k,n)$-distribution (where $k

On ergodic problems for Hamilton-Jacobi-Isaacs equations with state constraints

We study the asymptotic behavior as $\lambda\rightarrow 0^+$ of $\lambda v_\lambda$, where $v_\lambda$ is the viscosity solution of the following Hamilton-Jacobi-Isaacs (HJI) equation (infinite horizon case) $$\lambda v_\lambda + H(x,Dv_\lambda)=0, $$ with $$ H(x,p):=\min_{b\in B}\max_{a \in A} \{-f(x,a,b)\cdot p -l(x,a,b)\}. $$ We discuss the case in which the state of the system is required to stay in the closure of a bounded domain $\Omega\subset{\Bbb R}^n$ with sufficiently smooth boundary (for example $\partial \Omega\in {\cal C}^2$). Under the uniform approximate controllability assumption by the first player, we extend to Differential Games the convergence result of the term $\lambda v_\lambda (x)$ to a constant as $\lambda\rightarrow 0^+$. We also show how to contruct nonanticipative strategies which satisfies some "good" estimates in order to obtain H\"older regularity of the value function.

On parabolic and quasi-periodic solutions for some singular Hamiltonian systems.

We will describe some methods to get parabolic orbits, also called homoclinics to infinity, as well as quasi-periodic orbits for some singular Hamiltonian systems satisfying the strong force hypothesis. For parabolic orbits, we use some double approximatiom method, applying first classical variational methods to get heteroclinic orbits between two fixed points, then we let the points go to infinity to get parabolic orbits. As for quasi-periodic orbits, we get the solutions by applying variational methods to an associated second order elliptic P.D.E. on a torus using a generalized Poincare inequality, the we show that the obtained weak solutions correspond to regular quasi-periodic solutions of the Hamiltonian system. These are joint works with Pablo Padilla (UNAM, Mexico).

Aspects of a shallow water equation.

A recently derived nonlinear partial differential equation models the propagation of waves on shallow water. The equation is an integrable infinite dimensional Hamiltonian system and its solitary waves are peaked solitons. Moreover, the equation is a re-expression of geodesic flow on the diffeomorphism group of the circle. While certain solutions are global, others blow-up in finite time (the equation models wave breaking).

Ginzburg-Landau functionals and minimal surfaces.

Blow up for nonlinear differential equations.

Last modified: 18 December 2006

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