On the isoperimetric problem with double density

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.Comment: 19 pages, 1 figure. A subscript $r$ is missing in the hypothesis of Theorem A and related Lemmas in the published version. This version contains the correct statement

The Cheeger constant of curved strips

We study the Cheeger constant and Cheeger set for domains obtained as strip-like neighbourhoods of curves in the plane. If the reference curve is complete and finite (a "curved annulus"), then the strip itself is a Cheeger set and the Cheeger constant equals the inverse of the half-width of the strip. The latter holds true for unbounded strips as well, but there is no Cheeger set. Finally, for strips about non-complete finite curves, we derive lower and upper bounds to the Cheeger set, which become sharp for infinite curves. The paper is concluded by numerical results for circular sectors.Comment: 18 pages, 22 figures; typos and a gap in the proof of Lemma 6 correcte