Non compact variational problems

Period of duration of course
Course info
Number of course hours
Number of hours of lecturers of reference
Number of hours of supplementary teaching

Type of exam

Oral exam and seminars


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Functional Analysis, Elliptic Theory, Base notions in Calculus of Variations. Suggested to Ph.D. students and V-year Master students. 


Introduction and motivation

Strauss' Lemma

Existence and nonexistence for semilinear problems in R^n

Method of moving planes

Principle of concentration-compactness, I

Non-compact Palais-Smale sequences

Variational solutions for nonautonomous equations

Problems with critical exponent

Principle of concentration-compactness, I

Brezis-Nirenberg's theorem

Problems in conformal geometry

Educational aims

The purpose of the course is to illustrate a group of variational problems that exhibit lack of compactness, and to show general and more specific techniques that allow them to be addressed. The difficulties of these problems are due to the lack of compactness of embedding for appropriate function spaces, originating either from non-limiting domains (nonlinear Schroedinger equations) or from geometric properties of invariance by rescaling (conformal curvature prescription).
Techniques such as the use of symmetric spaces, the principle of compactness by concentration, and the use of asymptotic or perturbative estimates to make up for the problem of lack of compactness will be discussed.

Bibliographical references

Struwe: Variational Methods
Ambrosetti-Malchiodi: Nonlinear Analysis and Semilinear Elliptic Problems
Aubin: Some nonlinear problems in Riemannian Geometry