Polar factorization and pseudo-rearrangements: applications to Pólya–Szegö type inequalities

✍ Scribed by Adele Ferone; Roberta Volpicelli

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 244 KB
• Volume: 53
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(03)00025-7

✦ Synopsis

We are interested in the polar factorization of a function f deÿned in an open bounded set ⊆ R N . It is well known that there exists a measure preserving map such that f=f * • where f * is the decreasing rearrangement of f. We prove that, under suitable assumptions, besides the classical polar factorization of f we have f = f u • where f u is a pseudo-rearrangement of f with respect to the measurable function u and is the measure preserving map such that u = u * • . As an application, we characterize those functions that realize equality in the PÃ olya-Szeg o inequality.