Some isoperimetric inequalities on $\mathbb{R}^{N}$ with respect to weights $\left\vert x\right\vert ^{\alpha }$

We solve a class of isoperimetric problems on RNwith respect to weights that are powers of the distance to the origin. For instance we show that, if k∈[0, 1], then among all smooth sets Ωin RNwith fixed Lebesgue measure, ∂Ω|x|kHN−1(dx)achieves its minimum for a ball centered at the origin. Our results also imply a weighted Pólya–Szegö principle. In turn, we establish radiality of optimizers in some Caffarelli–Kohn–Nirenberg inequalities, and we obtain sharp bounds for eigenvalues of some nonlinear problems.