Adv. Nonlinear Anal. 1 (2012), 181 – 203 DOI 10.1515 / anona-2011-0006 © de Gruyter 2012 Isoperimetric inequalities for k-Hessian equations Ahmed Mohammed, Giovanni Porru and Abdessalam Safoui Abstract. We consider the homogeneous Dirichlet problem for a special k-Hessian equa- tion of sub-linear type in a .k  1/-convex domain   R n , 1  k  n. We study the comparison between the solution of this problem and the (radial) solution of the corre- sponding problem in a ball having the same .k  1/-quermassintegral as . Next, we con- sider 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 . Symmetrization techniques and comparison principles are the main tools used to get these inequalities. Keywords. Monge–Ampère type equations, rearrangements, eigenvalues, isoperimetric inequalities. 2010 Mathematics Subject Classification. 35A23, 35B51, 35J96, 35P30, 47J20, 52A40. 1 Introduction In the seminal paper [23], G. Talenti pioneered an important method for establish- ing sharp a priori estimates of quantities involving solutions to boundary value problems of second order elliptic linear PDE’s. In the subsequent papers [24, 25], he obtained optimal estimates of a priori solutions to Dirichlet problems with ho- mogeneous boundary conditions by comparing them to corresponding quantities of solutions of problems that have better symmetry. These papers have inspired the use of similar methods in numerous investigations involving both linear and non- linear elliptic problems. In the paper [28], K. Tso applies the ideas in the paper [24] to develop sym- metrization schemes for obtaining isoperimetric inequalities of quantities involv- ing solutions to k-Hessian equations. More specifically, let  be a bounded, open convex set in R n , and  be a positive, smooth real-valued function defined on . Consider the Dirichlet problem F k .D 2 u/ D .x/ in ; u D 0 on @: (1.1) Here 1  k  n, F k .A/ is the sum of the principal minors of order k of the n  n real matrix A, and D 2 u is the Hessian of u 2 C 2 ./. In [28], K. Tso derives iso- perimetric inequalities of quantities involving strictly convex solutions u to (1.1)