To teach some classical methods of the Calculus of Variations, while giving also a flavor of some of the most recent developments of this important branch of Mathematical Analysis.
Measure Theory and some basic knowledge of Functional Analysis.
Course contents summary
The course will serve as an introduction to the Calculus of Variations in one and more dimensions.
The main topics that will be covered are he following.
Part I: Calculus of Variations in one dimension
- Motivations and presentation of some classical problems.
-Necessary minimality conditions: the Euler-Lagrange equations
-Sufficient minimality conditions: Weierstrass Fields
-Applications to some classical problems such as the brachistochrone problem.
Parte II: Calcolus of Variations in higher dimensions
- The Direct Method
-Convexity as necessary condition for the weak lower semiconinuity
-sufficient conditions for the strong-weak lower semicontinuity of integral functionals: Ioffe's Theorem.
- Existence of solutions to some boundary value problems for elliptic PDE's via a variational principle.
- The regularity problem and the theory of De Giorgi-Nash
Parte III: The Isoperimetric problem
-Some elementary proofs in 2 dimensions
- Introduction to the theory of sets of finite perimeter and De Giorgi's proof of the isoperimetry of the ball