The isoperimetric problem in the plane with the sum of two Gaussian   densities
  We consider the isoperimetric problem for the sum of two Gaussian densities in the line and the plane. We 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. 
John Berry, Matthew Dannenberg, Jason Liang, Yingyi Zeng
math.GM