CM615: Nonlinear PDEs

Prof. Vicenţiu Rădulescu

 

1. The maximum principles of Hopf and Alexandrov-Bakelman. Maximum principles for nonlinear differential equations. Applications: Serrin’s theorem (overdetermined problems), positive solutions of boundary value problems. Blow-up boundary solutions: existence and asymptotic analysis of solutions.
2. Non-existence results for semilinear and quasilinear problems: the Pokhozaev identity for star-shaped domains.
3. Radial symmetry of solutions. The moving plane method and the Gidas-Ni-Nirenberg theorem.
4. Nonlinear operator theory methods: mappings on finite-dimensional spaces, Nemytskii operators, pseudo-monotone operators. Applications to PDEs.

Bibliography

 

1. V. Barbu, Partial Differential Equations and Boundary Value Problems, Mathematics and its Applications, vol. 441, Kluwer, 1998.
2. H. Brezis and L. Nirenberg, Nonlinear Functional Analysis and Applications to Partial Differential Equations, in preparation.
3. L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, Amer. Math. Soc., 1998.
4. J. Jost, Partial Differential Equations, Springer Verlag, 2002.
5. M. Renardy and R. Rogers, An Introduction to Partial Differential Equations, Springer Verlag, 1996.