114 research outputs found
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 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 is missing in the hypothesis of
Theorem A and related Lemmas in the published version. This version contains
the correct statement
On the generalized Cheeger problem and an application to 2d strips
In this paper we consider the generalization of the Cheeger problem which
comes by considering the ratio between the perimeter and a certain power of the
volume. This generalization has been already sometimes treated, but some of the
main properties were still not studied, and our main aim is to fill this gap.
We will show that most of the first important properties of the classical
Cheeger problem are still valid, but others fail; more precisely, long and thin
rectangles will give a counterexample to the property of Cheeger sets of being
the union of all the balls of a certain radius, as well as to the uniqueness.
The shape of Cheeger set for rectangles and strips is then studied as well as
their Cheeger constant.Comment: 15 pages, 1 figur
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