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.

Keywords

• monge–ampère type equations
• rearrangements
• eigenvalues
• isoperimetric inequalities