Antonio Siconolfi

A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations

We consider the Hamilton--Jacobi equation \[ \partial_t u+H(x,Du)=0\qquad \hbox{in

On smooth time functions

(0,+\infty)\times\T^{N}

Metric character of Hamilton–Jacobi equations

}, \] where

Time-Dependent Measurable Hamilton–Jacobi Equations

\T^{N}

A metric approach to the converse Lyapunov theorem for continuous multivalued dynamics

is the flat N-dimensional torus, and the Hamiltonian

A Lagrangian Approach to Weakly Coupled Hamilton--Jacobi Systems

H(x,p)

A first order Hamilton-Jacobi equation with singularity and the evolution of level sets

is assumed continuous in x and strictly convex and coercive in p. We study the large time behavior of solutions, and we identify the limit through a Lax-type formula. Some convergence results are also given for H solely convex. Our qualitative method is based on the analysis of the dynamical properties of the Aubry set, performed in the spirit of [A. Fathi and A. Siconolfi, Calc. Var. Partial Differential Equations, 22 (2005), pp. 185-228]. This can be viewed as a generalization of the techniques used in [A. Fathi, C. R. Acad. Sci. Paris Ser. I Math., 327 (1998), pp. 267-270] and [J. M. Roquejoffre, J. Math. Pures Appl. (9), 80 (2001), pp. 85-104]. Analogous results have been obtained in [G. Barles and P. E. Souganidis, SIAM J. Math. Anal., 31 (2000), pp. 925-939] using PDE methods.

Randomly perturbed dynamical systems and Aubry-Mather theory

We are concerned with the existence of smooth time functions on connected time{oriented Lorentzian manifolds. The problem is tackled in a more general abstract setting, namely in a manifold M where is just dened a eld of tangent convex cones ( Cx)x2M enjoying mild continuity properties. Under some conditions on its integral curves, we will construct a time function. Our approach is based on the denition of an intrinsic length for curves indicating how a curve is far from being an integral trajectory ofCx. We nd connections with topics pertaining to Hamilton{Jacobi equations, and make use of tools and results issued from weak KAM theory.

Discontinuous solutions of a Hamilton-Jacobi equation with infinite speed of propagation

We deal with the metrics related to Hamilton-Jacobi equations of eikonal type. If no convexity conditions are assumed on the Hamiltonian, these metrics are expressed by an inf-sup formula involving certain level sets of the Hamiltonian. In the case where these level sets are star-shaped with respect to 0, we study the induced length metric and show that it coincides with the Finsler metric related to a suitable convexification of the equation.

Cycle characterization of the Aubry set for weakly coupled Hamilton–Jacobi systems

Abstract We study the equation with Hamiltonian H(x, p) measurable with respect to the state variable, and convex and coercive in p. We introduce a notion of solution based on viscosity test functions, appropriate averages in measure-theoretic sense of the Hamiltonian, and t-partial-sup-convolutions. We get existence results, comparison principles, and stability properties. We show that our solutions are uniform limits of viscosity solutions (in the sense of Crandall and Lions) of approximated continuous equations.