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

Abstract

In this paper we consider the following class of linear elliptic problems { − div ( A ( x ) ∇ u ) = x N k exp ( − | x | 2 2 ) f ( x ) in Ω , u = 0 on ∂ Ω ∖ { x N = 0 } , where k ⩾ 0 , Ω is a domain (possibly unbounded) of R + N = { x = ( x 1 , … , x N ) ∈ R N : x N > 0 } , f belongs to a suitable weighted Lebesgue space and A ( x ) = ( a i j ( x ) ) i j is a symmetric matrix with measurable coefficients satisfying x N k exp ( − | x | 2 2 ) | ζ | 2 ⩽ a i j ( x ) ζ i ζ j ⩽ C x N k exp ( − | x | 2 2 ) | ζ | 2 . We compare the solution to such a problem with the solution to a symmetric one-dimensional 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.