Some Isoperimetric Problems in Planes with Density

Abstract

We study the isoperimetric problem in Euclidean space endowed with a density. We first consider piecewise constant densities and examine particular cases related to the characteristic functions of half-planes, strips and balls. We also consider continuous modification of Gauss density in R 2 . Finally, we give a list of related open questions.

Some calibrated surfaces in manifolds with density

“…. , n} (see also [4], Section 2.1). (3) The 3-covector Φ = (e * 1 ∧e * 2 + e * 3 ∧e * 4 )∧e * 5 is a calibration in R 5 with density 1.…”

The ε - εβProperty in the Isoperimetric Problem with Double Density, and the Regularity of Isoperimetric Sets

“…| · | Eucl and P Eucl (·). The isoperimetric problem with single density is a wide generalisation of the classical Euclidean isoperimetric problem, and it has been deeply studied in the last decades, we refer the interested reader to [8,10,11,16,22,25,28,29,32] and the references therein. The case of double density is yet a further important generalisation, since many of the possible applications correspond to two different densities.…”

Some Weighted Isoperimetric Problems on $${\mathbb {R}}^N _+ $$ with Stable Half Balls Have No Solutions

“…Hence we remind the interested reader to [20] and [22] and the references therein. One natural issue in this setting consists of finding families of densities for which one can determine the explicit form of the isoperimetric set, see for instance [24], [5], [17], [8], [11], [25], [7], [6], [9].…”