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Register a new userGlobal propagation of singularities for discounted Hamilton-Jacobi equations
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The main purpose of this paper is to study the global propagation of singularities of viscosity solution to discounted Hamilton-Jacobi equation begin{equation}label{eq:discount 1}tag{HJ$_lambda$} lambda v(x)+H( x, Dv(x) )=0 , quad xin mathbb{R}^n. end{equation} We reduce the problem for equation eqref{eq:discount 1} into that for a time-dependent evolutionary Hamilton-Jacobi equation. We proved that the singularities of the viscosity solution of eqref{eq:discount 1} propagate along locally Lipschitz singular characteristics which can extend to $+infty$. We also obtained the homotopy equivalence between the singular set and the complement of associated the Aubry set with respect to the viscosity solution of equation eqref{eq:discount 1}.
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For the discounted Hamilton-Jacobi equation,$$lambda u+H(x,d_x u)=0, x in M, $$we construct $C^{1,1}$ subsolutions which are indeed solutions on the projected Aubry set. The smoothness of such subsolutions can be improved under additional hyperbolicity assumptions. As applications, we can use such subsolutions to identify the maximal global attractor of the associated conformally symplectic flow and to control the convergent speed of the Lax-Oleinik semigroups
The well known phenomenon of exponential contraction for solutions to the viscous Hamilton-Jacobi equation in the space-periodic setting is based on the Markov mechanism. However, the corresponding Lyapunov exponent $lambda( u)$ characterizing the exponential rate of contraction depends on the viscosity $ u$. The Markov mechanism provides only a lower bound for $lambda( u)$ which vanishes in the limit $ u to 0$. At the same time, in the inviscid case $ u=0$ one also has exponential contraction based on a completely different dynamical mechanism. This mechanism is based on hyperbolicity of action-minimizing orbits for the related Lagrangian variational problem. In this paper we consider the discrete time case (kicked forcing), and establish a uniform lower bound for $lambda( u)$ which is valid for all $ ugeq 0$. The proof is based on a nontrivial interplay between the dynamical and Markov mechanisms for exponential contraction. We combine PDE methods with the ideas from the Weak KAM theory.
We prove that if a sequence of pairs of smooth commuting Hamiltonians converge in the $C^0$ topology to a pair of smooth Hamiltonians, these commute. This allows us define the notion of commuting continuous Hamiltonians. As an application we extend some results of Barles and Tourin on multi-time Hamilton-Jacobi equations to a more general setting.
For any compact connected manifold $M$, we consider the generalized contact Hamiltonian $H(x,p,u)$ defined on $T^*Mtimesmathbb R$ which is conex in $p$ and monotonically increasing in $u$. Let $u_epsilon^-:Mrightarrowmathbb R$ be the viscosity solution of the parametrized contact Hamilton-Jacobi equation [ H(x,partial_x u_epsilon^-(x),epsilon u_epsilon^-(x))=c(H) ] with $c(H)$ being the Ma~ne Critical Value. We prove that $u_epsilon^-$ converges uniformly, as $epsilonrightarrow 0_+$, to a specfic viscosity solution $u_0^-$ of the critical equation [ H(x,partial_x u_0^-(x),0)=c(H) ] which can be characterized as a minimal combination of associated Peierls barrier functions.
The long-time average behaviour of the value function in the calculus of variations, where both the Lagrangian and Hamiltonian are Tonelli, is known to be connected to the existence of the limit of the corresponding Abel means as the discount factor goes to zero. Still in the Tonelli case, such a limit is in turn related to the existence of solutions of the critical (or, ergodic) Hamilton-Jacobi equation. The goal of this paper is to address similar issues when the Hamiltonian fails to be Tonelli: in particular, for control systems that can be associated with a family of vector fields which satisfies the Lie Algebra rank condition. First, following a dynamical approach we characterise the unique constant for which the ergodic equation admits solutions. Then, we construct a critical solution which coincides with its Lax-Oleinik evolution.
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