Weighted isoperimetric inequalities in cones and applications

Abstract

This paper deals with weighted isoperimetric inequalities relative to cones of R N . We study the structure of measures that admit as isoperimetric sets the intersection of a cone with balls centered at the vertex of the cone. For instance, in case that the cone is the half-space R N + = x ∈ R N : xN > 0 and the measure is factorized, we prove that this phenomenon occurs if and only if the measure has the form dµ = ax k N exp c |x| 2 dx, for some a > 0, k, c ≥ 0. Our results are then used to obtain isoperimetric estimates for Neumann eigenvalues of a weighted Laplace-Beltrami operator on the sphere, sharp Hardy-type inequalities for functions defined in a quarter space and, finally, via symmetrization arguments, a comparison result for a class of degenerate PDE's.

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

“…and (3.6) holds if and only if C k,l,N,α = C rad k,l,N,α . Finally, we recall the following weighted isoperimetric inequality proved, for example, in [10] (see also [13] and [37]).…”

“…We recall that the isoperimetric constant C rad 0,0,N,α is explicitly computed in [10], see also [37] for the case N = 2.…”

“…In the proof we firstly evaluate the second variation of the perimeter functional. 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 weighted isoperimetric inequalities with non-radial densities

“…Hence we have that C k,l,N,α = C rad k,l,N,α . Finally, we recall the following weighted isoperimetric inequality proved, for example, in [7] (see also [8,10,11,19])…”

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

“…volume-preserving smooth perturbations at the half ball is nonnegative for α ∈ (−1, +∞). Note that in [7], see Proposition 2.1, the case of nonnegative α is addressed.…”