Variational methods in partial differential equations

elective monographs

Functional Analysis 1000-135AF

The aim of the course is to familiarize students with selected methods of the calculus of variations applied to partial differential equations. In particular, using direct method of calculus of variations, mountain pass and Nehari manifold method we will show existence of solutions to some elliptic problems.

1. Introduction to Sobolev spaces. H^1 space and its properties.

2. Weak solutions to the Dirichlet problem. The energy functional.

3. Direct method of calculus of variations.

4. Palais-Smale sequences and boundedness (Ambrosetti-Rabinowitz condition). Ekeland’s variational principle.

5. Mountain pass theorem with applications.

6. Nehari manifold method in the smooth case.

7. Homeomorphism between Nehari manifold and the sphere in a Hilbert space. Appliactions to equations without smoothness of the Nehari manifold.

M. Willem: Minimax theorems, Birkhäuser 1997

M. Struwe: Variational methods, Springer-Verlag 2008

M. Badiale, E. Serra: Semilinear Elliptic Equations for Beginners, Springer-Verlag 2011

A. Szulkin, T. Weth: Ground state solutions for some indefinite variational problems, Journal of Functional Analysis, Volume 257, Issue 12 (2009)

1. Knows the concept of Sobolev space H^1, weak derivatives with basic properties, the concept of variational energy functional and weak solutions.

2. Knows and can apply direct method of the calculus of variations.

3. Knows the concept of a Palais-Smale sequence, knows and understands conditions that imply its boundedness (in particular, the Ambrosetti-Rabinowitz condition).

4. Knows and can proof the mountain pass theorem.

5. Applies the mountain pass theorem to show the existence of a nontrivial solution.

6. Knows the concept of the Nehari manifold and its basic properties.

7. Can apply the Nehari manifold method to show the existence of the least energy solutions (ground state solution).

Written exam.