Homogenization on arbitrary manifolds

We describe a setting for homogenization of convex hamiltonians on abelian covers of any compact manifold. In this context we also provide a simple variational proof of standard homogenization results.Comment: 17 pages, 1 figur

Convergence of the solutions of the discounted equation

We consider a continuous coercive Hamiltonian $H$ on the cotangent bundle of the compact connected manifold $M$ which is convex in the momentum. If $u_\lambda:M\to\mathbb R$ is the viscosity solution of the discounted equation $$\lambda u_\lambda(x)+H(x,d_x u_\lambda)=c(H),$$ where $c(H)$ is the critical value, we prove that $u_\lambda$ converges uniformly, as $\lambda\to 0$, to a specific solution $u_0:M\to\mathbb R$ of the critical equation $$H(x,d_x u)=c(H).$$ We characterize $u_0$ in terms of Peierls barrier and projected Mather measures.Comment: 35 page

Convergence of the solutions of the discounted equation: the discrete case

We derive a discrete version of the results of our previous work. If $M$ is a compact metric space, $c : M\times M \to \mathbb R$ a continuous cost function and $\lambda \in (0,1)$, the unique solution to the discrete $\lambda$-discounted equation is the only function $u_\lambda : M\to \mathbb R$ such that $$\forall x\in M, \quad u_\lambda(x) = \min_{y\in M} \lambda u_\lambda (y) + c(y,x).$$ We prove that there exists a unique constant $\alpha\in \mathbb R$ such that the family of $u_\lambda+\alpha/(1-\lambda)$ is bounded as $\lambda \to 1$ and that for this $\alpha$, the family uniformly converges to a function $u_0 : M\to \mathbb R$ which then verifies $$\forall x\in X, \quad u_0(x) = \min_{y\in X}u_0(y) + c(y,x)+\alpha.$$ The proofs make use of Discrete Weak KAM theory. We also characterize $u_0$ in terms of Peierls barrier and projected Mather measures.Comment: 15 page

Convergence of the solutions of the discounted equation: the discrete case

We derive a discrete version of the results of Davini et al. (Convergence of the solutions of the discounted Hamilton-Jacobi equation. Invent Math, 2016). If M is a compact metric space, a continuous cost function and , the unique solution to the discrete -discounted equation is the only function such that We prove that there exists a unique constant such that the family of is bounded as and that for this , the family uniformly converges to a function which then verifies The proofs make use of Discrete Weak KAM theory. We also characterize in terms of Peierls barrier and projected Mather measures

Hyperbolicity and exponential convergence of the Lax–Oleinik semigroup

AbstractFor a convex superlinear Lagrangian L:TM→R on a compact manifold M it is known that there is a unique number c such that the Lax–Oleinik semigroup Lt+ct:C(M,R)→C(M,R) has a fixed point. Moreover for any u∈C(M,R) the uniform limit u˜=limt→∞Ltu+ct exists.In this paper we assume that the Aubry set consists in a finite number of periodic orbits or critical points and study the relation of the hyperbolicity of the Aubry set to the exponential rate of convergence of the Lax–Oleinik semigroup