Topological and variational methods in nonlinear partial equations 1000-1M25MTW
1. Sobolev spaces – intrudction. Weak solutions and a variational functional.
2. Mountain pass theorem:
a. deformation lemma and mountain pass lemma,
b. applications to nonlinear elliptic problems,
c. symmetries and compactness,
d. Sobolev inequality, critical nonlinearities.
3. Linking geometry:
a. Ekeland variational principle and general deformation lemma,
b. Rabinowitz saddle point theorem,
c. applications to nonlinear elliptic problems,
d. localization of critical points,
e. critical nonlinearities.
4. Fountain theorem:
a. equivariant deformations,
b. multiplicity of critical points,
c. convex and concave nonlinearities, critical nonlinearities.
Type of course
Mode
Prerequisites (description)
Course coordinators
Learning outcomes
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 applies the mountain pass theorem to show the existence of a critical point in the positive definite case.
3. Knows and applies the Rabinowitz linking theorem to show the existence of a critical point in the indefinite case.
4. Knows and can proof the mountain pass theorem and the linking geometry theorem.
5. Knows and understands the role of symmetries (group actions) of the space and its infuence to the compactness of minimizing sequences.
6. Applies the fountain theorem and appropriate versions of deformation lemma to show the multiplicity of critical points.
Assessment criteria
Written exam.
Additional information
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