Rearrangements and the Monge–Ampère equations

Abstract

We show that the direct counterpart of the Talenti symmetrization estimate for the Laplacian does not hold neither for the complex nor real Monge–Ampère equations. We also use this Talenti result to improve some known estimates for subharmonic functions in C,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}},$$\\end{document} where the constant depends on the area of the domain, instead of the diameter.