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Isoperimetric inequalities for -Hessian equations

Abstract. We consider the homogeneous Dirichlet problem for a special -Hessian equation of sub-linear type in a -convex domain , . We study the comparison between the solution of this problem and the (radial) solution of the corresponding problem in a ball having the same -quermassintegral as . Next, we consider the eigenvalue problem for the -Hessian equation and study a comparison between its principal eigenfunction and the principal eigenfunction of the corresponding problem in a ball having the same -quermassintegral as . Symmetrization techniques and comparison principles are the main tools used to get these inequalities.