Introduction.- Convex Analysis and the Scalar Case.- Convex Sets and Convex Functions.- Lower Semicontinuity and Existence Theorems.- The one Dimensional Case.- Quasiconvex Analysis and the Vectorial Case.- Polyconvex, Quasiconvex and Rank one Convex Functions.- Polyconvex, Quasiconvex and Rank one Convex Envelopes.- Polyconvex, Quasiconvex and Rank one Convex Sets.- Lower Semi Continuity and Existence Theorems in the Vectorial Case.- Relaxation and Non Convex Problems.- Relaxation Theorems.- Implicit Partial Differential Equations.- Existence of Minima for Non Quasiconvex Integrands.- Miscellaneous.- Function Spaces.- Singular Values.- Some Underdetermined Partial Differential Equations.- Extension of Lipschitz Functions on Banach Spaces.- Bibliography.- Index.- Notations.
This book is developed for the study of vectorial problems in the calculus of variations. The subject is a very active one and almost half of the book consists of new material. This is a new edition of the earlier book published in 1989 and it is suitable for graduate students. The book has been updated with some new material and examples added. Applications are included.
This second edition is the successor to "Direct methods in the calculus of variations" which was published in the Applied Mathematical Sciences series and is currently out of print. Although the core and the structure of the present book is similar to the first edition, it is much more than a revised version. Fifteen years have passed since the publication of the "Direct methods in the calculus of variations" book and since the subject is a very active one, almost half of the book presently consists of new material. The perspective has also slightly changed, indeed, a new subject, "quasiconvex analysis" has now been developed. The present edition, which is essentially a reference book on the subject of quasiconvex analysis can be used, as was the earlier book, for an advanced course on the calculus of variations.