A.A. 2014/2015 Corso di Laurea Magistrale in Matematica Advanced Analysis B Codice SCC0540 Daniele Cassani CF U SSD Lezioni Esercitazioni Laboratorio 8 MAT/0 5 64 - - (ore) (ore) (ore) [inserire voce: es. attività di campo; seminari; uscite;…] (ore) Ann o Lingu a - I english Overview This course aims to give “basic” tools within modern theory of nonlinear differential equations and critical point theory. Topics Euler –Lagrange equations and solutions of partial differential equations via the Dirichlet principle of minimal energy: brachistochrona, isoperimetric problem, catenaria. Weak solutions. A few facts from Functional Analysis: L^p spaces and Hilbert spaces, Hölder’s inequality and duality, mollifiers and density of smooth functions, pseudo-orthogonality and lack of compactness in infinite dimensional spaces, weak topologies and compactness, the Riesz lemma and Stampacchia’s lemma, the Lax-Milgram theorem. Weak derivatives and Sobolev spaces: embedding inequalities, the Rellich-Kondrachov theorem, extensions and traces. A direct method in the Calculus of Variations, minima of weakly lower semicontinuous functionals: applications to nonlinear Schroedinger’s equation and the nonlinear pendulum equation. Introduction to topological methods in Nonlinear Analysis for indefinite functionals: deformation lemma and the mountain-pass theorem by AmbrosettiRabinowitz, applications to semilinear elliptic equations. Critical growth problems, lack of compactness and Pohozaev identity. Quantization of energy and the Brezis-Nirenberg theorem. Teaching methods Classical chalk-lectures. References H. Brezis, Functional Analysis, Springer; M. Struwe, Variational Methods, Springer; M. Willem, Minimax Theorems, Birkauser. Final exam Oral examination possibly combined with a presentation of a seminar on a subject arranged with the instructor. Office hours By appointment. Teaching schedule Link to orari e sedi del CdS Exam sessions Link to bacheca appelli