Mathematics

The course aims to introduce students to the theory of viscosity solutions and to the metric and variational aspects of first-order Hamilton-Jacobi equations (weak KAM Theory) and to present some applications to asymptotic problems. 1. Knowledge and understanding. At the end of the lectures the student will be familiar with the basic notions and results of the theory of viscosity solution and with the metric and variational aspects of first-order HJ equations (weak KAM Theory). 2. Applied knowledge and understanding. Students who have passed the exam will be able to derive explicit expressions for solutions of first-order HJ equations in some simple examples and to derive qualitative information in more general cases. 3. Making judgments. The students will acquire a satisfactory knowledge of the main tools and results of weak KAM Theory, which will provide them of a valuable insight on the geometric and dynamical phenomena taking place in the study of first-order HJ equations. 4. Communication skills Ability to present the content during the oral exam. 5. Learning skills Students will acquire the necessary tools to face the study of first-order Hamilton-Jacobi equations and to possibly approach research topics.