Course Objective To provide a gentle introduction to the direct methods in Calculus of Variations concerning minimizations problems, excluding minmax methods. The focus will be on illustrating the main methods using important prototype examples and not on proving the most general or the sharpest results.
Target audience This course is primarily intended for students of Mathematics with interests in Analysis, PDE and/or differential Geometry and geometric analysis, especially minimal surfaces. However, students of physics and different branches of engineering ( especially mechanical engineering ) and economics would still probably find a portion of the course useful for them.
Course contents and outline Our goal is to cover the following topics:
Classical Methods: Euler-Lagrange equations, Lagrangian and Hamiltonian formulations, Hamilton-Jacobi equations, constrained problems and Lagrange multipliers, An illustration of the methods: Geodesic curves.
Direct Methods: Dirichlet integral and $p$-Dirichlet Integral: Existence of minimizers: Existence theorem for convex functional with lower order terms, examples and counterexamples, weak form of the Euler-Lagrange equations, Dirichlet Principle, weak continuity of determinants. Regularity questions.
Plateauâ€™s problem and minimal surfaces: Parametric Plateauâ€™s problem: Douglas-Courant-Tonelli method, Regularity, uniqueness and nonuniqueness, Nonparametric minimal surfaces, Isoperimetric inequality.