Some unusual comparison properties of capillary surfaces. (Q701279)

Under typical physical conditions, the solution of the capillarity equation for a tube of circular section \(D\) will always exceed over \(D\) the solution obtained for a concentric tube of the same material and larger radius. In this paper the question raised by M. Miranda is examined. Namely, the conditions are described under which a solution over a general domain \(D_0\) will or will not exceed, over that section, the solution over any domain \(D_I\) strictly containing \(D_0\). The discussion includes some results of general geometric interest. All convex domains containing a disk, for which the ratio of perimeter to area is not less than for the disk are characterized.