CM512: Elliptic Equations and Variational Problems

Prof. Vicenţiu Rădulescu

 

1. Basic notions on Sobolev spaces: main properties and embedding theorems. The Lax-Milgram lemma and Garding’s inequality.
2. Lower and upper solutions. The linear case. Stable solutions, uniqueness results (the Kransoselskii and the Brezis-Oswald theorems).
3. The variational characterization of the method of lower and upper solutions.
4. Bifurcation problems. The Implicit Function Theorem and applications to nonlinear bifurcation problems. The Amman and the Crandall-Rabinowitz theorems.

Bibliography

 

1. V. Barbu, Partial Differential Equations and Boundary Value Problems, Mathematics and its Applications, vol. 441, Kluwer, 1998.
2. H. Brezis, Analyse fonctionnelle, Masson, Paris, 1992.
3. H. Brezis and L. Nirenberg, Nonlinear Functional Analysis and Applications to Partial Differential Equations, in preparation.
4. D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, 1983.
5. V. Radulescu, http://www.inf.ucv.ro/~radulescu/articles/nonlpde.pdf