Weighted isoperimetric inequalities in cones and applications.

This paper deals with weighted isoperimetric inequalities relative to cones of R^N . We study the structure of measures that admit as an isoperimetric set the intersection of a cone with the ball centered at the vertex of the cone. For instance, in case that the cone is the half-space R^N_+={x ∈ R^N : x_N > 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.