  In this paper we consider the isoperimetric problem with double density in an
Euclidean space, that is, we study the minimisation of the perimeter among
subsets of $\mathbb{R}^n$ with fixed volume, where volume and perimeter are
relative to two different densities. The case of a single density, or
equivalently, when the two densities coincide, has been well studied in the
last years; nonetheless, the problem with two different densities is an
important generalisation, also in view of applications. We will prove the
existence of isoperimetric sets in this context, extending the known results
for the case of single density.
