Euler's tallest column revisited

Abstract

In 1757, Leonhard Euler started the study of the tallest column, i.e. the shape of a stable column with the symmetry of revolution, such that it attains the maximum height once the total mass is prescribed, buckling due to the effect of a load supported at its top. A more detailed analysis is due to Keller and Niordson in 1966 who formulated the problem in terms of an eigenvalue type problem under some coefficient constraints and also, by eliminating some of the unknowns, as a nonlocal boundary value problem for a p-Laplacian type operator with a negative exponent and with an infinite normal derivative in some of the boundaries. The main contribution of this work is the study of the existence and qualitative behavior of a weak solution completing the approach made by Keller and Niordson (developed merely by asymptotic analysis techniques). Under a suitable condition on the top load, we show that there exists a shape function a(x) for which the smallest eigenvalue is the largest one when a(x) is taken in a suitable class of shape functions (in contrast with the unload case according to a result due to Cox and McCarthy in 1998). We prove also that the nonlocal problem has a solution u such that u is an element of W(1,p)(0, 1) for any p is an element of [1, 3) but with u is not an element of W(1,3)(0, 1). We also give a sufficient condition for the uniqueness of the solution.