IRIS Università degli Studi di Padovahttps://www.research.unipd.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Tue, 07 Apr 2020 11:33:29 GMT2020-04-07T11:33:29Z10151Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case.http://hdl.handle.net/11577/2469672Titolo: Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case.
Abstract: The aim of this paper is twofold. We construct an extension to a non-integrable case of Hopf’s formula, often used to produce viscosity solutions of Hamilton-Jacobi equations for p-convex integrable Hamiltonians. Further- more, for a general class of p-convex Hamiltonians, we present a proof of the equivalence of the minimax solution with the viscosity solution.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/11577/24696722006-01-01T00:00:00ZSOME GLOBAL FEATURES OF WAVE PROPAGATIONhttp://hdl.handle.net/11577/2467163Titolo: SOME GLOBAL FEATURES OF WAVE PROPAGATION
Abstract: After a discussion on two fundamental routes —weak discontinuity waves and high frequency asymptotic waves— both leading to Hamilton-Jacobi equation, we review two notions of weak solution for it, the minimax solution and the viscosity so- lution. We claim the coincidence of the two solutions for a general class of p-convex Hamiltonians of mechanical type and we sketch some technical details of the proof.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/11577/24671632006-01-01T00:00:00ZConvergence to the time average by stochastic regularizationhttp://hdl.handle.net/11577/2526385Titolo: Convergence to the time average by stochastic regularization
Abstract: In Ergodic Theory it is natural to consider the pointwise convergence of finite time averages of functions with respect to the flow of dynamical systems. Since the pointwise convergence is too weak for applications to Hamiltonian Perturbation Theory, requiring differentiability, we first introduce regularized averages obtained through a stochastic perturbation of an integrable Hamiltonian flow, and then we provide detailed estimates. In particular, for a special vanishing limit of the stochastic perturbation, we obtain convergence even in a Sobolev norm taking into account the derivatives.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/11577/25263852013-01-01T00:00:00ZNew estimates for Evans' Variational Approach to Weak KAM Theoryhttp://hdl.handle.net/11577/2523870Titolo: New estimates for Evans' Variational Approach to Weak KAM Theory
Abstract: We consider a recent approximate variational principle for weak KAM theory proposed by Evans. As in the case of classical integrability, for one dimensional mechanical Hamiltonian systems all the computations can be carried out explicitly. In this setting, we illustrate the geometric content of the theory and prove new lower bounds for the estimates related to its dynamic interpretation. These estimates also extend to the case of n degrees of freedom.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/11577/25238702013-01-01T00:00:00ZMather measures associated with a class of Bloch wave functionshttp://hdl.handle.net/11577/2658054Titolo: Mather measures associated with a class of Bloch wave functions
Abstract: We study the Wigner transform for a class of smooth Bloch wave functions on the
at torus. We select amplitudes and phase functions through a variational approach in the quantum
states space based on a semiclassical version of the classical effective Hamiltonian H(P) which is the central object of the weak KAM theory. Our main result is that the semiclassical limit of the Wigner transform admits subsequences converging in the weak* sense to Mather probability measures on the phase space. These measures are invariant for the classical dynamics and Action minimizing.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/11577/26580542012-01-01T00:00:00ZCauchy problems for stationary Hamilton-Jacobi equations under mild regularity assumptionshttp://hdl.handle.net/11577/2472936Titolo: Cauchy problems for stationary Hamilton-Jacobi equations under mild regularity assumptions
Abstract: For a Hamiltonian enjoying rather weak regularity assumptions, we provide necessary and sufficient conditions for the existence of a global viscosity solution to the corresponding stationary Hamilton–Jacobi equation at a fixed level a, taking a prescribed value on a given closed subset of the ground space. The analysis also includes the case where a is the Man ̃ ́e critical value. Our results are based on a metric method extending Maupertuis approach.
For general underlying spaces, compact or noncompact, we give a global ver- sion of the classical characteristic method based on the notion of a–characteristic. In the compact case, we propose an inf-sup formula producing the minimal so- lution of the problem, where the generalized Aubry set is involved.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/11577/24729362009-01-01T00:00:00ZA PDE approach to finite time indicators in Ergodic Theoryhttp://hdl.handle.net/11577/2438364Titolo: A PDE approach to finite time indicators in Ergodic Theory
Abstract: For dynamical systems defined by vector fields over a compact invariant set, we introduce a new class of approximated first integrals based on finite time averages and satisfying an explicit first order partial differential equation. These approximated first integrals can be used as finite time indicators of the dynamics. On the one hand, they provide the same results on applications than other popular indicators; on the other hand, their PDE based definition — that we show robust under suitable perturbations — allows one to study them using the traditional tools of PDE environment. In particular, we formulate this approximating device in the Lyapunov exponents framework and we compare the operative use of them to the common use of the Fast Lyapunov Indicators to detect the phase space structure of quasi-integrable systems.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/11577/24383642009-01-01T00:00:00ZOn $C^0$-variational solutions for Hamilton-Jacobi equationshttp://hdl.handle.net/11577/139416Titolo: On $C^0$-variational solutions for Hamilton-Jacobi equations
Abstract: Abstract. For evolutive Hamilton-Jacobi equations, we propose a refined def- inition of C0-variational solution, adapted to Cauchy problems for continuous initial data. This weaker framework enables us to investigate the semigroup property for these solutions. In the case of p-convex Hamiltonians, when vari- ational solutions are known to be identical to viscosity solutions, we verify directly the semigroup property by using minmax techniques. In the non- convex case, we construct a first explicit evolutive example where minmax and viscosity solutions are different. Provided the initial data allow for the sepa- ration of variables, we also detect the semigroup property for convex-concave Hamiltonians. In this case, and for general initial data, we finally give new upper and lower Hopf-type estimates for the variational solutions.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/11577/1394162011-01-01T00:00:00ZOn Poincaré-Birkhoff periodic orbits for mechanical Hamiltonian systems on $T^*{\mathbb T}^n$http://hdl.handle.net/11577/2456067Titolo: On Poincaré-Birkhoff periodic orbits for mechanical Hamiltonian systems on $T^*{\mathbb T}^n$
Abstract: Here, a version of the Arnol’d conjecture, first studied by Conley and Zehnder, giving a generalization of the Poincaré-Birkhoff last geometrical theorem, is proved inside Viterbo’s framework of the generating functions quadratic at infinity. We give brief overviews of some tools that are often utilized in symplectic topology.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/11577/24560672006-01-01T00:00:00ZChain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaceshttp://hdl.handle.net/11577/3217073Titolo: Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces
Abstract: The aim of this paper is twofold. On the one hand, we discuss the notions of strong chain recurrence and strong chain transitivity for flows on metric spaces, together with their characterizations in terms of rigidity properties of Lipschitz Lyapunov functions. This part extends to flows some recent results for homeomorphisms of Fathi and Pageault. On the other hand, we use these characterisations to revisit the proof of a theorem of Paternain, Polterovich and Siburg concerning the inner rigidity of a Lagrangian submanifold Λ contained in an optical hypersurface of a cotangent bundle, under the assumption that the dynamics on Λ is strongly chain recurrent. We also prove an outer rigidity result for such a Lagrangian submanifold Λ, under the stronger assumption that the dynamics on Λ.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/11577/32170732016-01-01T00:00:00Z