On the isoperimetric problem in Euclidean space with density

Abstract

ABSTRACT. We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density exp(|x| 2 ) by using symmetrization techniques.

Steiner and Schwarz symmetrization in warped products and fiber bundles with density

“…Proposition 5 provides more general symmetrization with uniqueness for regions (typically isoperimetric regions) which satisfy certain smoothness hypotheses, as in the proof by Rosales et al [17,Thm. 5.2] that in R n with density e r 2 , balls about the origin uniquely minimize perimeter for given volume.…”

“…4) as well as Steiner and Schwarz symmetrization. Proposition 5 treats the smooth case with an analysis of when equality holds after Rosales et al [17,Thm. 5.2].…”

Some calibrated surfaces in manifolds with density

“…Gauss space has many applications to probability and statistics. For more details about manifolds with density, we refer the reader to [13], [14], [15] and the entry "Manifolds with density" at Morgan's blog http://blogs.williams.edu/Morgan/.…”

“…The definition of the weighted mean curvature is fit for the first variation of weighted perimeter of a smooth region (see [5], [15]). For a stationary (i.e.…”

“…This condition means that u is orthogonal to e ψ in L 2 (Σ) and it is proved that any such u is the normal component of a vector field associated to a volume-preserving variation of Ω (see [7], [15]). …”