ABSTRACT
The methods and techniques of symmetric decreasing rearrangements of functions and other related symmetrizations have become one of the important tools in analysis and its applications. This chapter discusses the compactness properties of symmetric decreasing rearrangements of minimising sequences arising in various problems in mathematical physics. It presents the complete proof of the Burchard-Guo theorem on compactness via symmetric decreasing rearrangements, which has interesting applications in the dynamical stability analysis of gaseous stars and stability of symmetric steady states in galactic dynamics. The chapter explores the compactness properties of minimising sequences of symmetric decreasing rearrangements. It proposes a number of applications of the technique of the symmetric decreasing rearrangements to several problems of mathematical physics. The chapter provides an application of the Burchard-Guo theorem to the dynamical stability of gaseous stars.
