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. Continue reading ‘Density 1/(1+r^2)’ »