Weak KAM Theorie : Existenz und Eindeutigkeit von Viskositätslösungen der Hamilton-Jacobi Gleichung

Abstract

The goal of this master thesis is to give an introduction to weak KAM theory and prove the existence of weak KAM solutions, which solve the Hamilton-Jacobi equation in the sense of viscosity solutions. Furthermore, we prove an important determination result for weak KAM solutions.In the preliminaries, we recall the notion of smooth manifolds, the tangent bundle and vector fields. Moreover, we define the Legendre Transformation as link between the Lagrangian and the Hamiltonian.The next chapter relates to calculus of variations, which first will be discussed on open subsets of Rn and then more generally on arbitrary smooth manifolds. We prove the Euler-Lagrange equation as necessary condition for a curve to be minimizing and define the Euler-Lagrange flow.In the fourth chapter, we change the perspective to the Hamiltonian point of view and therefore define the Hamiltonian and its vector fields, where it comes natural to discuss symplectic aspects.We continue with the definition of Tonelli Lagrangians and their properties. With the help of absolutely continuous curves, we can prove the existence of minimizing curves and examine their regularity.With the work done so far, we arrive at the main section of this manuscript: Weak KAM theory. We define dominated functions, calibrated curves and weak KAM solutions. Existence is proved by means of the Lax-Oleinik semi-group in two different ways and we show that weak KAM solutions indeed solve the Hamilton-Jacobi equation in the sense of viscosity solutions. Finally, we define minimizing measures and prove that weak KAM solutions are equal if they coincide on the Mather set.