Relative Rearrangement

This book develops the properties of monotone rearrangement and relative rearrangement (sometimes called pseudo-rearrangement). It introduces applications to variational problems involving monotone rearrangements, a priori estimates for partial differential equations, and stationary or evolution problems associated with variable exponents. The properties of Sobolev embeddings for non-standard spaces such as BMO, VMO, Zygmung spaces and more general spaces invariant under rearrangement are also reviewed. The book is relatively self-contained - elementary details for non-specialists are covered in the first chapter, including, among other things, some punctual inequalities for the Sobolev embeddings and Pólya-Szegő type inequalities, which lead, for instance, to explicit and even precise estimates. The final chapter includes numerous exercises, with solutions. Based on the author’s Réarrangement relatif: un instrument d’estimations dans les problèmes aux limites (Springer, 2008), this edition contains additional recent results and new exercises concerning interpolation theory.

Jean Michel Rakotoson is retired from the Université de Poitiers, where he was a full professor in mathematics. He essentially taught analysis and topology, in Poitiers and a number of other universities in France, namely Nantes, Lyon and Paris XI (Orsay). He was a visiting professor at Indiana University at Bloomington, USA, the University of Naples in Italy, and the Universidad Complutense de Madrid in Spain, where he held short courses for graduate students. His book is partly based on a master’s course held around 1990 at the Université de Poitiers, but it also contains some of the author’s research.