Rearrangements and the Monge-Ampère equations (Q6593021)

This paper is concerned with the so-called Talenti symmetrization estimate for the Laplacian. The author considers natural analogues for the real and complex Monge-Ampère equations, and shows that they do not hold. He shows that these estimates would imply certain isoperimetric inequalities, to which he then finds counterexamples. In addition, the author gives applications of the Talenti symmetrization estimate to obtain various optimal bounds for subharmonic functions on planar doamins, including improved estimates of \textit{H. Brézis} and \textit{F. Merle} [Commun. Partial Differ. Equations 16, No. 8--9, 1223--1253 (1991; Zbl 0746.35006)] and \textit{S. Benelkourchi} et al. [Ark. Mat. 43, No. 1, 85--112 (2005; Zbl 1092.31005)].