Study-unit Code
Roberta Filippucci
  • Roberta Filippucci
  • 42 ore - Roberta Filippucci
Course Regulation
Coorte 2023
Learning activities
Attività formative affini o integrative
Academic discipline
Type of study-unit
Opzionale (Optional)
Type of learning activities
Attività formativa monodisciplinare
Language of instruction
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.
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 Functional Analysis. 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
Other information
Optional but recommended attendance.¿ It could be convenient to attend also lessons of Applied Functional Analysis.
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.¿
Extended program
Introduction and 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. Convex Functionals. Some spectral properties of elliptic operators.¿¿ Minimization techniques for 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 to superlunar problems as well as to problems asymptotically linear.¿¿ Minimization techniques for problems with lack of compactness. Problems with critical exponent. Pohozaev identity and a nonexistence result.
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