Global propagation of singularities for discounted Hamilton-Jacobi equations

<p style='text-indent:20px;'>The main purpose of this paper is to study the global propagation of singularities of the viscosity solution to discounted Hamilton-Jacobi equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE333"> \begin{document}$ \begin{align} \lambda v(x)+H( x, Dv(x) ) = 0 , \quad x\in \mathbb{R}^n. \quad\quad\quad (\mathrm{HJ}_{\lambda})\end{align} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with fixed constant <inline-formula><tex-math id="M1">\begin{document}$ \lambda\in \mathbb{R}^+ $\end{document}</tex-math></inline-formula>. We reduce the problem for equation <inline-formula><tex-math id="M2">\begin{document}$(\mathrm{HJ}_{\lambda})$\end{document}</tex-math></inline-formula> into that for a time-dependent evolutionary Hamilton-Jacobi equation. We prove that the singularities of the viscosity solution of <inline-formula><tex-math id="M3">\begin{document}$(\mathrm{HJ}_{\lambda})$\end{document}</tex-math></inline-formula> propagate along locally Lipschitz singular characteristics <inline-formula><tex-math id="M4">\begin{document}$ {{\bf{x}}}(s):[0,t]\to \mathbb{R}^n $\end{document}</tex-math></inline-formula> and time <inline-formula><tex-math id="M5">\begin{document}$ t $\end{document}</tex-math></inline-formula> can extend to <inline-formula><tex-math id="M6">\begin{document}$ +\infty $\end{document}</tex-math></inline-formula>. Essentially, we use <inline-formula><tex-math id="M7">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-compactness of the Euclidean space which is different from the original construction in [<xref ref-type="bibr" rid="b4">4</xref>]. The local Lipschitz issue is a key technical difficulty to study the global result. As a application, we also obtain the homotopy equivalence between the singular locus of <inline-formula><tex-math id="M8">\begin{document}$ u $\end{document}</tex-math></inline-formula> and the complement of Aubry set using the basic idea from [<xref ref-type="bibr" rid="b9">9</xref>].</p>

Aubry-Mather theory for contact Hamiltonian systems II

<p style='text-indent:20px;'>In this paper, we continue to develop Aubry-Mather and weak KAM theories for contact Hamiltonian systems <inline-formula><tex-math id="M1">\begin{document}$ H(x,u,p) $\end{document}</tex-math></inline-formula> with certain dependence on the contact variable <inline-formula><tex-math id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula>. For the Lipschitz dependence case, we obtain some properties of the Mañé set. For the non-decreasing case, we provide some information on the Aubry set, such as the comparison property, graph property and a partially ordered relation for the collection of all projected Aubry sets with respect to backward weak KAM solutions. Moreover, we find a new flow-invariant set <inline-formula><tex-math id="M3">\begin{document}$ \tilde{\mathcal{S}}_s $\end{document}</tex-math></inline-formula> consists of <i>strongly</i> static orbits, which coincides with the Aubry set <inline-formula><tex-math id="M4">\begin{document}$ \tilde{\mathcal{A}} $\end{document}</tex-math></inline-formula> in classical Hamiltonian systems. Nevertheless, a class of examples are constructed to show <inline-formula><tex-math id="M5">\begin{document}$ \tilde{\mathcal{S}}_s\subsetneqq\tilde{\mathcal{A}} $\end{document}</tex-math></inline-formula> in the contact case. As their applications, we find some new phenomena appear even if the strictly increasing dependence of <inline-formula><tex-math id="M6">\begin{document}$ H $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M7">\begin{document}$ u $\end{document}</tex-math></inline-formula> fails at only one point, and we show that there is a difference for the vanishing discount problem from the negative direction between the <i>minimal</i> viscosity solution and <i>non-minimal</i> ones.</p>

Cohomology of Aubry-Mather type

This chapter defines cohomology of Aubry-Mather type and explains why it implies one of the diffusion mechanisms, after a generic perturbation. The definition of Aubry-Mather type includes a much simpler case, that is when the Aubry set is a hyperbolic periodic orbit, still contained in a normally hyperbolic invariant cylinder. This definition says that each of the two local components of the Aubry set is of Aubry-Mather type. There is another type of bifurcation in which one component of the Aubry set is of Aubry-Mather type with an invariant cylinder and another is a hyperbolic periodic orbit. This can be called the asymmetric bifurcation. This case appears at double resonance, when the shortest loop is simple non-critical.

Forcing Relation

This chapter describes forcing relations, different diffusion mechanisms, and Aubry-Mather types. The Aubry set can be decomposed into disjoint invariant sets called static classes, which gives important insight into the structure of the Aubry set. The chapter then formulates Theorem 2.2 and shows that it implies the book's main theorem. It utilizes the concept of forcing equivalence. The actual definition is not important for the current discussions, instead, the chapter states its main application to Arnold diffusion. The chapter also looks at symplectic coordinate changes. The definition of exact symplectic coordinate change for a time-periodic system is somewhat restrictive, and in particular, it does not apply directly to the linear coordinate change performed at the double resonance.

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

We study a class of weakly coupled systems of Hamilton–Jacobi equations using the random frame introduced in [H. Mitake, A. Siconolfi, H. V. Tran and N. Yamada, A Lagrangian approach to weakly coupled Hamilton–Jacobi systems, SIAM J. Math. Anal. 48(2) (2016) 821–846; doi: https://doi.org/10.1137/15M1010841 ]. We provide a cycle condition characterizing the points of Aubry set. This generalizes a property already known in the scalar case.

Aubry set for asymptotically sub-additive potentials

Given a topological dynamical systems [Formula: see text], consider a sequence of continuous potentials [Formula: see text] that is asymptotically approached by sub-additive families. In a generalized version of ergodic optimization theory, one is interested in describing the set [Formula: see text] of [Formula: see text]-invariant probabilities that attain the following maximum value [Formula: see text] For this purpose, we extend the notion of Aubry set, denoted by [Formula: see text]. Our central result provides sufficient conditions for the Aubry set to be a maximizing set, i.e. [Formula: see text] belongs to [Formula: see text] if, and only if, its support lies on [Formula: see text]. Furthermore, we apply this result to the study of the joint spectral radius in order to show the existence of periodic matrix configurations approaching this value.

Time functions revisited

In this paper we revisit our joint work with Antonio Siconolfi on time functions. We will give a brief introduction to the subject. We will then show how to construct a Lipschitz time function in a simplified setting. We will end with a new result showing that the Aubry set is not an artifact of our proof of existence of time functions for stably causal manifolds.

Regularization of Subsolutions in Discrete Weak KAM Theory

We expose different methods of regularizations of subsolutions in the context of discrete weak KAM theory that allow us to prove the existence and the density of C1,1 subsolutions. Moreover, these subsolutions can be made strict and smooth outside of the Aubry set.