Isoperimetric inequalities for k-Hessian equations

We consider the homogeneous Dirichlet problem for a special k-Hessian equation of sub-linear type in a (k - 1)-convex domain Omega subset of R-n, 1 <= k <= n. We study the comparison between the solution of this problem and the ( radial) solution of the corresponding problem in a ball having the same (k - 1)-quermassintegral as Omega. Next, we consider the eigenvalue problem for the k-Hessian equation and study a comparison between its principal eigenfunction and the principal eigenfunction of the corresponding problem in a ball having the same (k - 1)-quermassintegral as Omega. Symmetrization techniques and comparison principles are the main tools used to get these inequalities.