Frank Morgan

Rodrigo Banuelos suggested studying the isoperimetric problem for the radial density 1/(1+r²) corresponding to the square root of the Laplacian just as the most important Gaussian density corresponds to the Laplacian itself.

Proposition. Consider Rⁿ with density 1/(1+r²). For n > 1 minimizers of perimeter for given volume do not exist: the perimeter can go to zero as the region goes off to infinity. On the line, for more than half the volume the minimizer is a ball about the origin, for less than half, the complement, for exactly half, the ball, its complement, or a half-line. In particular, balls about the origin are minimizing while stable, up to radius 1, with (log density)” = 2(x²-1)/(x²+1)².

Proof. For n=2, the projected density on the line is π(1+x²)^-1/2, for which perimeter goes to 0 as an interval of arbitrary fixed volume goes off to infinity. For n > 2, balls going off to infinity have perimeter approaching 0. (Consider balls about (0,R) of radius R^α with α between 2/n and 2/(n-1).)

For n=1, since the total volume is finite, minimizers exist and must be half-lines, intervals, or complements [RCBM, Thm. 4.3]. The only stable interval is the ball about the origin of radius at least 1. (There is an unstable equilibrium from -1/a to a.) At half the total volume, the half-lines tie the ball; as volume increases, the ball does better for a while because curvature, 2r/(1+r²), starts out greater for the ball. That situation eventually reverses, but the half-line never catches the ball, as you can see by considering volume near the maximum (volume π, perimeter 0), where perimeter of the half-line comes from integrating a larger curvature.

[RCBM] César Rosales, Antonio Cañete, Vincent Bayle, and Frank Morgan, On the isoperimetric problem in Euclidean space with density, Calc. Var. PDE 31 (2008), 27–46; arXiv.org (2006).