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.