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

In this paper we consider the following class of linear elliptic problems ⎧⎪ ⎨⎪ ⎩ −div A(x)∇u = xk N exp −|x|2 2 f (x) in Ω, u =0 on ∂Ω \ {xN = 0}, where k 0, Ω is a domain (possibly unbounded) of RN+ = {x = (x1, . . . , xN) ∈ RN : xN > 0}, f belongs to a suitable weighted Lebesgue space and A(x) = (ai j (x))i j is a symmetric matrix with measurable coefficients satisfying xk N exp −|x|2 2 |ζ |2 ai j (x)ζiζ j Cxk N exp −|x|2 2 |ζ |2. We compare the solution to such a problem with the solution to a symmetric onedimensional problem belonging to the same class. Our approach use classical symmetrization methods adapted to a relative isoperimetric inequality with respect to a measure related to the structure of the equation.