H\"older regularity for viscosity solutions of fully nonlinear, local or nonlocal, Hamilton-Jacobi equations with super-quadratic growth in the gradient

Viscosity solutions of fully nonlinear, local or non local, Hamilton-Jacobi equations with a super-quadratic growth in the gradient variable are proved to be H\"older continuous, with a modulus depending only on the growth of the Hamiltonian. The proof involves some representation formula for nonlocal Hamilton-Jacobi equations in terms of controlled jump processes and a weak reverse inequality

Weak KAM aspects of convex Hamilton-Jacobi equations with Neumann type boundary conditions

We establish the stability under the formations of infimum and of convex combinations of subsolutions of convex Hamilton-Jacobi equations, some comparison and existence results for convex and coercive Hamilton-Jacobi equations with the Neumann type boundary condition as well as existence results for the Skorokhod problem. We define the Aubry-Mather set associated with the Neumann type boundary problem and establish some properties of the Aubry-Mather set including the existence results for the ``calibrated'' extremals for the corresponding action functional (or variational problem).Comment: 39 pages, 1 figur

Invariant Lagrange submanifolds of dissipative systems

We study solutions of modified Hamilton-Jacobi equations H(du/dq,q) + cu(q) = 0, q \in M, on a compact manifold M

Discerned and Non-Discerned Particles in Classical Mechanics and Quantum Mechanics Interpretation

We introduce into classical mechanics the concept of non-discerned particles for particles that are identical, non-interacting and prepared in the same way. The non-discerned particles correspond to an action and a density which satisfy the statistical Hamilton-Jacobi equations and allow to explain the Gibbs paradox in a simple manner. On the other hand, a discerned particle corresponds to a particular action that satisfies the local Hamilton-Jacobi equations. We then study the convergence of quantum mechanics to classical mechanics when hbar -> 0 by considering the convergence for the two cases. These results provide an argument for a renewed interpretation of quantum mechanics