Syllabus
1. differential calculus in Banach spaces directional derivative, differential, mean value theorems, chain rule, Taylor formula
2. Inverse mapping theorem, Implicit function theorem
3. Extrema local extrema, Fermat's condition, Euler-Lagrange equation, Lagrange sufficient and necessary conditions for local extrema, Lagrange multiplier theorem
4. Application on Nemitskii operator and integral functionals Caratheodory function, measurability of composed function, continuity and boundedness of Nemitskii operator from $L^p$ to $L^q$
5. Direct methods in the Calculus of Variations convexity and weak lower semicontinuity, basic theorem of Calculus of Variations
6. Counterexamples to the existence of a minimizer
7. Classical problems of the Calculus of Variations (briefly)
8. Degree of a mapping uniqueness and existence of the degree in finite dimension, Sard's theorem, Brouwer theorem, Borsuk theorem
9. Leray-Schauder degree definition, Schauder fixed point theorem, Leray-Schauder index of an isolated solution
10. Monotone operators in Hilbert space continuous, monotone and weakly coercive operators, monotone operators in reflexive spaces (briefly)
Annotation
Recommended for master students of mathematical analysis.
Content: differential calculus in Banach spaces, implicit function theorem, calculus of variations.