An isoperimetric inequality for Gauss--like product measures

This paper deals with various questions related to the isoperimetic problem for smooth positive measure $dμ= φ(x)dx$, with $x \in Ω\subset \mathbb{R}^N$. 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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