Yv. Egorov et S. Karaa, ON THE OPTIMAL SHAPE OF A COLUMN AGAINST BUCKLING, Comptes rendus de l'Academie des sciences. Serie 1, Mathematique, 322(6), 1996, pp. 519-524
We give simple proofs for the existence and the uniqueness of the opti
mal shape of a column clamped at both ends (x = 0) and (x = 1) and hav
ing a unit volume.