(in Polish) Metody wariacyjne w nieliniowych równaniach różniczkowych cząstkowych
7404-MAT-WAR
1. Introduction to Sobolev spaces. Space H^1 and its properties.
2. Weak solution of the Dirichlet problem. Energy functional.
3. Direct method of the calculus of variations.
4. Palais-Smale sequences and their boundedness (Ambrosetti-Rabinowitz condition). Ekeland's principle.
5. Mountain pass theorem with applications.
6. Nehari manifold method in the smooth case.
7. Homeomorphism of the Nehari manifold with the sphere in Hilbert space. Application to equations with non-regular Nehari manifold.
Total student workload
1. Hours conducted with teacher participation
a) lecture – 30 hours.
2. Time dedicated to individual student work necessary to successfully complete the course
a) studying literature – 30 hours.
3. Time required to prepare for assessment (e.g., exams):
a) exam preparation – 30 hours.
TOTAL: 90 hours
3 ECTS credits
Learning outcomes - knowledge
1. Understands the concept of Sobolev space H^1, weak derivatives, and their basic properties, as well as the concept of variational energy functional and weak solutions. (P8S_WG)
2. Understands the direct method of the calculus of variations. (P8S_WG)
3. Understands the concept of a Palais-Smale sequence and knows and understands the conditions that guarantee its boundedness (particularly the Ambrosetti-Rabinowitz condition). (P8S_WG)
4. Understands the mountain pass theorem. (P8S_WG)
5. Understands the concept of the Nehari manifold and its basic properties. (P8S_WG, P8S_WG)
Learning outcomes - skills
1. Is able to apply the direct method of the calculus of variations. (P8S_UW, P8S_UO)
2. Applies the mountain pass theorem to demonstrate the existence of a non-trivial solution. (P8S_UW, P8S_UO)
3. Is able to prove the mountain pass theorem and apply the deformation lemma. (P8S_UW, P8S_UO)
4. Is able to apply the Nehari manifold technique to obtain solutions with the lowest energy (ground state solutions). (P8S_UW, P8S_UO)
Learning outcomes - social competencies
1. Shares their knowledge and insights with others while maintaining intellectual integrity. (P8S_KR, P8S_KK)
2. Is aware of the limitations of their knowledge, has the ability to critically assess the topic under consideration, and is skilled in seeking solutions based on principles of logic and various sources of information. (P8S_KK)
Observation/demonstration teaching methods
- display
Expository teaching methods
- informative (conventional) lecture
- problem-based lecture
Exploratory teaching methods
- classic problem-solving
Type of course
elective course
Prerequisites
Functional analysis
Course coordinators
Assessment criteria
Bibliography
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)
Additional information
Additional information (registration calendar, class conductors,
localization and schedules of classes), might be available in the USOSweb system: