Some Methods for Studying Stability in Isoperimetric Type Problems

We review the method of quantitative symmetrization inequalities introduced in Fusco, Maggi and Pratelli, “The sharp quantitative isoperimetric inequality”, Ann. of Math. Introduction Our aim is to introduce certain ideas related to the quantitative study of isoperimetric type inequalities. The main efforts are devoted to the proof of a sharp quantitative version of the Euclidean isoperimetric inequality, first proved in [FMP1] improving on previous work by Hall, Hayman and Weitsman [HHW, Ha]. The general scheme of proof remains the same, but at different steps it has been possible to simplify the original arguments. Moreover, we have occasionally opted for a rather informal style of presentation, hoping to provide a broadly and easily accessible account of these techniques. There are many relevant variational problems [PS, Ka] intimately related to the same symmetrization inequalities studied here from a quantitative point of view, and these ideas have already been shown to be flexible enough to be employed in the study of quantitative forms of other variational problems, like sharp Sobolev, Faber-Krahn, isocapacitary, and Cheeger inequalities [FMP2, CFMP1, FMP3]. An even more surprising application is found in the quantitative study of the Gaussian isoperimetric inequality [CFMP2], where natural variants of these arguments still play a prominent role. Starting from the seminal paper by Brezis and Lieb [BL], the reader will find various other contributions on quantitative geometric-functional inequalities based on similar or related tools [BE, Ci1, Ci2, CEFT, EFT, AFN]. This work covers the material taught in the first part of a Ph.D. course held by the author at the Università di Napoli “Federico II” during May 2007. In the second part of that course we introduced a different approach to the quantitative isoperimetric inequality, where symmetrizations were dropped out in favor of a heavier use of transportation maps. This method allows us to attack problems where minimizers have no particular symmetry, such as in the case of the anisotropic isoperimetric inequality (4.4). This point of view is developed in the forthcoming paper [FiMP], and therefore it is not presented in here. Received by the editors August 29, 2007. 2000 Mathematics Subject Classification. Primary 49Q20. c ©2008 American Mathematical Society Reverts to public domain 28 years from publication