• Corpus ID: 1948527

A Short Course on Rearrangement Inequalities

@inproceedings{Burchard2009ASC,
  title={A Short Course on Rearrangement Inequalities},
  author={Almut Burchard},
  year={2009}
}

• A. Burchard
• Published 2009
• Mathematics

These notes grew out of introductory courses for graduate students that I gave at the First IMDEA Winter School in Madrid in January 2009, and at the Universita di Napoli “Federico II” in April 2009. The manuscript has been been slightly expanded to five sections, each providing material for one or two hours of lecture. The first section gives an overview of the classical rearrangement inequalities and their applications. The relationship between a function and its level sets is explored through the layer-cake decomposition and the co-area formula. The next couple of sections investigate sequences of rearrangements, such as iterated Steiner symmetrizations and polarizations. Both compactness arguments and the “competing symmetries” technique are described. The fourth section contains the proof of the three fundamental inequalities due to Riesz, Polya-Szegő and Talenti. Though only the simplest cases are stated, the proofs will more generally. As an afterthought, the fifth section gives a brief introduction to optimal transportation techniques for proving geometric inequalities. The text is accompanied by exercises, which could serve as starting points for discussions. References at the end of each section are not encyclopedic, but point to books, monographs, surveys, and original papers that the reader might find immediately related to each chapter.