Frank Morgan

An interesting class of densities for the isoperimetric problem on Rⁿ is exp(±r^a). The Log-convex Density Conjecture says that spheres about the origin are isoperimetric if and only if 0≤a≤1 with the minus sign or a ≥ 1 with the plus sign, known for a=2 [Borell; see Rosales et al.] and in R² for a≥2 [Maurmann-Morgan]. Henceforth we’ll focus on R².

Density exp(r^a) on R². Circles about the origin are stable unless 0<a<1. They are conjectured isoperimetric for  a≥1, proven for a≥2 [Maurmann-Morgan] and for a>1 for large balls (Kolesnikov-Zhdanov, Rmk. 6.9, generalized by Howe [H]), and supported by some numerical evidence for a=1 [L, Prop. 6.8]. For a<0, the density is decreasing to 1 at infinity with severe nonconvex blow-up at the origin, and minimizers probably disappear at infinity; yet a circle S about the origin is probably perimeter minimizing in competition with any other curve C such that C–S bounds net area 0, as follows for small circles by Howe [H].

For 0<a<1 circles about the origin are unstable, but minimizers exist [Rosales et al., Thm. 3.5]. They are probably transcendental ovals containing the origin.

Density exp(–r^a) on R². When 0<a<1, the density has severe nonconvexity at the origin but decreases rapidly to 0 at infinity, yielding finite total area and hence existence of isoperimetric curves; a circle about the origin is probably isoperimetric. In the famous case of Gaussian density (a=2), straight lines are uniquely isoperimetric. As a decreases from 2 to 1, numerical studies [Kolesnikov-Zhdanov, Sect. 5] show a nice transition from a straight line to a curved line to a large closed curve to a circle; as a increases from 2, the line curves in the other direction.

In the interesting case of a < 0, the density increases to 1 at infinity. Isoperimetric curves exist [MP]. They probably pass through the origin, where the density vanishes.

[H]  Sean Howe, The Log-Convex Density Conjecture and vertical surface area in warped products,arXiv.org (2011).

[L]  Yifei Li, Michael Mara, Isamar Rosa Plata, and Elena Wikner, Tiling with penalties and isoperimetry with density, Rose-Hulman Und. Math. J. 13 (1) (2012).
[MP]  Frank Morgan and Aldo Pratelli, Existence of isoperimetric regions in Rⁿ with density, arXiv.org (2011).