Some new methods for Hamilton-Jacobi type nonlinear partial differential equations

I present two recent research directions in this dissertation. The first direction is on the study of the Nonlinear Adjoint Method for Hamilton-Jacobi equations, which was introduced recently by Evans [27]. The main feature of this new method consists of the introduction of an additional equation to derive new information about the solutions of the regularized Hamilton-Jacobi equations. More specifically, we linearize the regularizedHamilton-Jacobi equations first and then introduce the corresponding adjoint equations. Looking at the behavior of the solutions of the adjoint equations and using integration byparts techniques, we can prove new estimates, which could not be obtained by previous techniques.We use the Nonlinear Adjoint Method to study the eikonal-like Hamilton-Jacobi equations [79], and Aubry-Mather theory in the non convex settings [10]. The latter one is based on joint work with Filippo Cagnetti and Diogo Gomes. We are able to relax theconvexity conditions of the Hamiltonians in both situations and achieve some new results.The second direction, based on joint work with Hiroyoshi Mitake [69, 68], concerns the study of the properties of viscosity solutions of weakly coupled systems of Hamilton-Jacobi equations. In particular, we are interested in cell problems, large time behavior, and homogenization results of the solutions, which have not been studied much in the literature.We obtain homogenization results for weakly coupled systems of Hamilton-Jacobi equations with fast switching rates and analyze rigorously the initial layers appearing naturally in the systems. Moreover, some properties of the effective Hamiltonian are also derived.Finally, we study the large time behavior of the solutions of weakly coupled systems of Hamilton-Jacobi equations and prove the results under some specific conditions. The general case is still open up to now