Non compact variational problems
Prerequisites
Functional Analysis, Elliptic Theory, Base notions in Calculus of Variations. Suggested to Ph.D. students and V-year Master students.
Programme
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