A symmetrization result for Monge–Ampère type equations

Abstract

Key words Monge-Ampère equations, eigenvalue problems MSC (2000) 35J25, 35J65In this paper we prove some comparison results for Monge-Ampère type equations in dimension two. We also consider the case of eigenfunctions and we derive a kind of "reverse" inequalities.

Estimates for the first eigenfunction of linear eigenvalue problems via Steiner symmetrization

“…Chiti's results have been generalized to nonlinear equations (see, for instance, [1] and [8]) and to the eigenvalue problem for the Hermite equation via the Gaussian symmetrization (see [7]). We finally recall that Chiti's type estimates were also used by Ashbaugh and Benguria in order to solve the well-known Payne-Pólya-Weinberger conjecture (see [3]) and its generalization on the n-dimensional sphere S n (see [4]).…”

Isoperimetric estimates for eigenfunctions of Hessian operators

“…Remark 2 Inequality (4.1) for k = 1 and k = n becomes respectively the results of Talenti [25] and of Brandolini and Trombetti [7].…”

“…the Monge-Ampère operator, in dimension n = 2, in [7] it has been proven a similar inequality, where in the right-hand side of (1.8) does not appear u q , but u q , where u is the rearrangement of u with respect to the perimeter of its level lines.…”

“…Proof We argue as in [2,7]. First we prove the existence of µ 1 and of the corresponding eigenfunction.…”

“…In the proof of this result we argue as in [1,2,7]. It is hence based on a comparison result between u and a suitable eigenfunction v of k-Hessian operator with replaced by a ball B, centered at the origin and such that the corresponding eigenvalue is equal to λ k .…”