Weighted isoperimetric inequalities in cones and applications

This paper deals with weighted isoperimetric inequalities relative to cones of RNRN. We study the structure of measures that admit as isoperimetric sets the intersection of a cone with balls centered at the vertex of the cone. For instance, in case that the cone is the half-space R+N={x∈RN:xN>0} and the measure is factorized, we prove that this phenomenon occurs if and only if the measure has the form dμ=axNkexp(c|x|2)dx, for somea>0a>0, k,c≥0k,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.