Variational methods in nonlinear analysis (Q2776617)

The paper under review consists of the lecture notes of a course given by the author at a Summer School in Cuernavaca. The notes cover the very broad domain of variational problems, and are addressed to nonspecialist readers; in particular, no background on function spaces such as Sobolev spaces or \(BV\) spaces are required. The topics are presented in a very enjoyable way, in the spirit of motivating the reader to study more deeply what is here only sketched.NEWLINENEWLINENEWLINEIn some few pages the author concentrates some of the main topics of the calculus of variations and nonlinear analysis: direct methods, Euler-Lagrange equation, second variation, degree theory, Morse theory, minimax methods, as well as some of their most classical applications. NEWLINENEWLINENEWLINEThe paper is enriched by several classical examples such as Fermat's principle on reflecting and refracting light rays, the brachystochrone and the tautochrone problems, the characterization of shapes of heavy chains, the existence of geodesic curves on a manifold, and many others. Also, several exercises are proposed to the reader.NEWLINENEWLINEFor the entire collection see [Zbl 0977.00018].