A class of degenerate elliptic equations and a Dido's problem with respect to a measure

In the present paper we study a class of linear elliptic equations in dimension N, whose coefficients have a type of singularity which appears, for instance, when one looks for axially symmetric solutions to elliptic equations in dimension greater then N, or, more in general, solutions which are symmetric with respect to a group of variables. We are interested in sharp apriori bounds for the solution to such a type of equations. Such type of problems can be studied by classical symmetrization methods. However the degeneracy of the operator does not allow to use the classical approach via Schwarz symmetrization . This leads us to consider an appropriate weighted symmetrization based on the structure of the problem and an suitable relative isoperimetric inequality.