Calculus of variations is a field that studies optimization problems. On the one hand, optimization problems can be solved by solving the corresponding partial differential equation. On the other hand, we can use variational methods to study and solve partial differential equations. We define the basic variational problem and derive the Euler-Lagrange equation. We study some typical variational problems such as the shortest path problem, the brachistochrone problem and the Plateau problem. We also discuss the direct method which is used to prove the existence of the minimizer. For this purpose we introduce Hilbert spaces and derive some of their properties, such as weak compactness. We use this in an example to illustrate the direct method and to introduce the concept of the weak formulation and weak solution. Lax-Milgram lemma, which we prove in the end, guarantees existence and uniqueness of the weak solution.
|
