Steven J. Cox, C. Maeve McCarthy, The shape of the tallest column, SIAM Journal on Mathematical Analysis, 29 no. 3 (1998). pp. 547-554.
The height at which an unloaded column will buckle under its own weight is the fourth root of the least eigenvalue of a certain Sturm-Liouville operator. We show that the operator associated with the column proposed by Keller and Niordson [J. Math. Mech., 16 (1966), pp. 433-446] does not possess a discrete spectrum. This calls into question their formal use of perturbation theory, so we consider a class of designs that permits a tapered summit yet still guarantees a discrete spectrum. Within this class we prove that the least eigenvalue increases when one replaces a design with its decreasing rearrangement. This leads to a very simple proof of the existence of a tallest column.
Steven J. Cox, C. Maeve McCarthy, The shape of the tallest column: corrected, SIAM Journal on Mathematical Analysis, 31 no. 4 (1998). pp. 940-940.
Our summary of the work of Keller and Niordson (J. Math. Mech. 16, pp. 433–446, 1966) was inaccurate. We offer a correction.