The isoperimetric problem in the plane with the sum of two Gaussian densities. (English) Zbl 1391.49078

Berry, John; Dannenberg, Matthew; Liang, Jason; Zeng, Yingyi

Involve 11, No. 4, 549-567 (2018).

A positive continuous function \(e^\psi\) on \(\mathbb R^n\) used to weight volume and hypersurface area is called a density. The weighted volume of a region \(R\) is given by \(\int_Re^\psi dV_0\) and the weighted hypersurface area of its boundary \(\partial R\) is given by \(\int_{\partial R}e^\psi dA_0\). The region \(R\) is called isoperimetric if no other region of the same weighted volume has a boundary with smaller hypersurface area. In [Riemannian geometry. A beginner’s guide. 2nd ed., revised paperback printing. Natick, MA: A K Peters (2009; Zbl 1234.53001)], F. Morgan considered the existence and regularity of isoperimetric regions for densities of finite total volume and showed that if \(e^\psi\) is a density in the line or plane such that the line or plane has finite measure \(A_0\), then for any \(0<A<A_0\), an isoperimetric region \(R\) of weighted volume \(A\) exists and is a finite union of intervals bounded by finitely many points in the line or a finite union of regions with smooth boundaries in the plane.
In this paper, the authors consider the isoperimetric problem for the sum of two Gaussian densities in the line and the plane. They prove that the double Gaussian isoperimetric regions in the line are rays and that if the double Gaussian isoperimetric regions in the plane are half-spaces, then they must be bounded by vertical lines.

Reviewer: Andrew Bucki (Edmond)

MSC:

49Q10	Optimization of shapes other than minimal surfaces
53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Keywords:

isoperimetric region; Gaussian densities

Citations:

Zbl 1234.53001

References:

[1]	10.1007/s12220-009-9109-4 · Zbl 1193.49050
[2]	10.1007/s00526-007-0104-y · Zbl 1126.49038