Ilmanen's Lemma on Insertion of C1,1^{1,1} Functions

We give a proof of Ilmanen's lemma, which asserts that between a locally semi-convex and a locally semi-concave function it is possible to find a C$^{1,1}$ function.Comment: 17 pages, 1 figure, accepted for publication in Rend. Semin. Mat. Univ. Padov

On the Hausdorff Dimension of the Mather Quotient

Under appropriate assumptions on the dimension of the ambient manifold and the regularity of the Hamiltonian, we show that the Mather quotient is small in term of Hausdorff dimension. Then, we present applications in dynamics

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

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

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

Uniqueness of Invariant Lagrangian Graphs in a Homology or a Cohomology Class

Given a smooth compact Riemannian manifold $M$ and a Hamiltonian $H$ on the cotangent space $T^*M$, strictly convex and superlinear in the momentum variables, we prove uniqueness of certain ergodic invariant Lagrangian graphs within a given homology or cohomology class. In particular, in the context of quasi-integrable Hamiltonian systems, our result implies global uniqueness of Lagrangian KAM tori with rotation vector $\rho$. This result extends generically to the $C^0$-closure of KAM tori.Comment: 20 pages. Version published on Ann. Sc. Norm. Super. Pisa Cl. Sci.(5) Vol. 8, no. 4, 659-680, 200