Some isoperimetric inequalities on RN with respect to weights |x|α

We solve a class of isoperimetric problems on RN with 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 Ω in RN with 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.