Some isoperimetric inequalities onRNwith respect to weights |x|

Abstract

We solve a class of isoperimetric problems on R 2 + := (x, y) ∈ R 2 : y > 0 with respect to monomial weights. Let α and β be real numbers such that 0 ≤ α < β +1, β ≤ 2α. We show that, among all smooth sets Ω in R 2 + with fixed weighted measure Ω y β dxdy, the weighted perimeter ∂Ω y α ds achieves its minimum for a smooth set which is symmetric w.r.t. to the y-axis, and is explicitly given. Our results also imply an estimate of a weighted Cheeger constant and a lower bound for the first eigenvalue of a class of nonlinear problems.

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

“…We conclude with a result that has been obtained for the cases α = 0 and α > 0 in the papers [2] and [1], respectively. We proceed similarly as in [1], proof of Theorem 4.1.…”

“…If M is any measurable subset of R N + , with 0 < µ ℓ,α (M) < +∞, we set The conditions (4.1), (4.3) and (4.2) have been made to ensure that the integrals (4.6) and (4.7) converge. The cases α = 0 and α > 0 were analysed in the articles [2] and [1], respectively. Here we are only interested in the case α ∈ (−1, 0), that is, our weight functions are singular on the hyperplane {x N = 0}.…”

“…The problem becomes more challenging when perimeter and volume carry two different weights. One important example is when the manifold is R N , (N ≥ 2), and the two weight functions are powers of the distance from the origin, see [2], and the references cited therein. The theorem proved in [2] states that all spheres about the origin are isoperimetric for a certain range of the powers.…”

“…One important example is when the manifold is R N , (N ≥ 2), and the two weight functions are powers of the distance from the origin, see [2], and the references cited therein. 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]:…”

Sobolev embeddings, rearrangement-invariant spaces and Frostman measures

“…For instance, when X(Ω) = L p (Ω), Y (Ω, µ) = L q (Ω, µ 1 ) and m = 1, the weighted Sobolev inequality in question coincides with a special instance of the Caffarelli-Kohn-Nirenberg inequality, that has attracted the interest of researchers over the years in connection with the existence and description of its extremals if Ω = R n -see e.g. [4,9,24,25,28].…”