Unit MODERN ANALYSIS
- Course
- Mathematics
- Study-unit Code
- 55A00086
- Curriculum
- Didattico-generale
- Teacher
- Roberta Filippucci
- Teachers
-
- Roberta Filippucci
- Hours
- 42 ore - Roberta Filippucci
- CFU
- 6
- Course Regulation
- Coorte 2020
- Offered
- 2021/22
- Learning activities
- Affine/integrativa
- Area
- Attività formative affini o integrative
- Academic discipline
- MAT/05
- Type of study-unit
- Opzionale (Optional)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Elements of Calculus of Variations. Minimization techniques: compact problems. Introduction to minimax methods. Deformation Lemma. Mountain Pass Theorem. Applications to partial differential equations. Minimization techniques: lack of compactness. Applications to some critical problems.
- Reference texts
- M. Badiale & E. Serra, Semilinear Elliptic Equations for beginners, Springer (2011)A. Ambrosetti & A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics 104 (2007).
M. Ghergu & V. Radulescu, Nonlinear PDEs. Mathematical models in biology, chemistry and population genetics. With a foreword by Viorel Barbu. Springer Monographs in Mathematics. Springer, Heidelberg, 2012. xviii+391 pp.
M. Struwe, Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems, Springer-Verlag, Berlin (2008). - Educational objectives
- The course is the natural completion of all the courses in Mathematical Analysis of the Degree in Mathematics, since all topics treated in those previous courses find here further applications and motivations. In particular, the main purpose is to provide students with the bases to recognize the nature of a variational problem in the applied sciences and to solve the easiest ones.
Main acquired knowledge:
- basic topics in the theory of distributions;
- properties of Nemitzskii operators in L^p spaces;
- minimum theorems and applications;
- fundamental minimax theorems and applications: mountain pass theorem.
Main competence:
- identification of the variational nature of a problem;
- determine the geometrical properties of the associated functional and choose the minimax theorem to apply;
- prove the existence of solutions for differential problems by a critical point theorem. - Prerequisites
- In order to be able to understand and apply the majority of the techniques described within the Course, students must have successfully passed the exam of Analisi Matematica V. Topics and techniques developed therein are indeed a mandatory prerequisite for students planning to follow this course with profit.
- Teaching methods
- Face to face lessons and applications
- Other information
- Optional but recommended attendance.
It is convenient to attend also lessons of Analisi Matematica VI. - Learning verification modality
- The exam consists of an oral interview of about 30 minutes, aiming at verifying the knowledge level and the understanding ability acquired by the student on the theoretical and methodological contents as indicated in the program. Moreover, the oral exam will test the student communication skills, her/his correct use of language and autonomy in the organization and exposure of the considered topics.
Upon request, students can take the exam in English. - Extended program
- Introducion adn basic results: function spaces, embeddings, Poincarè e Sobolev inequalities, Riesz's theorem, Banach-Alaoglu theorem, Gateaux and Frechet differentiation, Examples in abstract spaces and then in concrete spaces, functional of Eulero Lagrange, critical points and weak solutions. Funzionali convessi. Some spectral properties of elliptic operators.
Minimization techniques: compact problems. Coercive problems, A min-max problem. Superlinear problems and constrained minimization (on spheres and on Nehari manifold) Nonhomeneous nonlinearities. The p-Laplacian operator: basic theory and two applications.
Introduction to minimax methods: deformation lemma, Palais Smale sequences, the minimax principle, mountain pass theorem. Some applications.
Minimization techniques: lack of compactness. Problems with critical exponent. Pohozaev identity and a nonexistence result.