On weighted isoperimetric inequalities with non-radial densities

Abstract

We consider a class of isoperimetric problems on R N + where the volume and the area element carry two different weights of the type |x| l x α N . We solve them in a special case while a more detailed study is contained in [2]. Our results imply a weighted Polya-Szëgo principle and a priori estimates for weak solutions to a class of boundary value problems for degenerate elliptic equations

The isoperimetric problem for a class of non-radial weights and applications

“…Proofs of Theorem 6.1 are given in various subsections, each of which addresses one of the cases ofTheorem 1.1. First let us recall that the proof of case (i) of Theorem 1.1 has been given in [2].…”

“…So that we did not try to adapt the techniques contained in [25], and, depending on the regions where the three parameters lie, we use different methods. The proof in the case (i) is given in [2]. It is based on Gauss's Divergence Theorem.…”

“…The claim is achieved using the fact that such a variation at a minimizing set must be nonnegative, together with a nontrivial weighted Poincaré inequality on the sphere derived in [10]. Part of these results have been announced in [2].…”

On existence and nonexistence of isoperimetric inequality with differents monomial weights

“…A great attention has been given recently to the isoperimetric inequalities with weights, see for instance [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [12], [14], [15], [16], [17], [18], [19] and the references therein. However, in the wide literature, most works approach volume functional and perimeter functional carrying the same weight.…”

Some Weighted Isoperimetric Problems on $${\mathbb {R}}^N _+ $$ with Stable Half Balls Have No Solutions

“…The theorem proved in [2] states that all spheres about the origin are isoperimetric for a certain range of the powers. One can modify this problem by inserting a further homogeneous perturbation term, namely x α N , both in the volume and in the perimeter, see [1] and [3]:…”