Some isoperimetric inequalities on $\mathbb{R} ^N$ with respect to   weights $|x|^\alpha $
  We solve a class of isoperimetric problems on $\mathbb{R}^N $ with respect to weights that are powers of the distance to the origin. For instance we show that if $k\in [0,1]$, then among all smooth sets $\Omega$ in $\mathbb{R} ^N$ with fixed Lebesgue measure, $\int_{\partial \Omega } |x|^k \, \mathscr{H}_{N-1} (dx)$ achieves its minimum for a ball centered at the origin. Our results also imply a weighted Polya-Sz\"ego 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. 
A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo, M.R. Posteraro
math.FA