Abstract
This paper deals with various questions related to the isoperimetric problem for a smooth positive measure dμ=φ(x)dx, with x∈Ω⊂RN. Firstly we find some necessary conditions on the density of the measure φ(x) that render the intersection of half spaces with Ω a minimum in the isoperimetric problem. We then identify the unique isoperimetric set for a wide class of factorized finite measures. These results are finally used in order to get sharp inequalities in weighted Sobolev spaces and a comparison result for solutions to boundary value problems for degenerate elliptic equations.
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