Abstract

In the design of clamped-clamped circular columns that can accommodate the largest vertical load before buckling, the moment of inertia of the horizontal section is assumed to be proportional to the pp-th power of the cross-sectional area for some p>0p>0: p=2p=2 (solid column) and p=1p=1 (hollow column). Existence of maximizing profiles has been established by Cox and Overton [SIAM J. Math. Anal. 23 (1992) 287–325] under the assumption of strictly positive lower and upper bounds on the profiles. Yet, their numerical computations indicate that the bounds are not necessary to get maximizing profiles. In this paper, we revisit some aspects of the unconstrained problem with emphasis on the cases 0<p≤10<p\le1. The existence of a maximizing profile in L1(0,1)L^1(0,1) is established for 0<p<10<p<1 without bounds. The existence theorem of Cox and Overton is extended to continuous profiles in C[0,1]C[0,1]. In addition, their strictly positive lower bound assumption is relaxed for p≤1p\le1 in both L∞(0,1)L^\infty(0,1) and C[0,1]C[0,1]. The dual inf sup problem is shown to be bounded above by 4848 for p≤1p\le1, but is +∞+\infty for p>1p>1. Additional results are given for degenerate profiles, that is, profiles whose lower bound is zero at some points.

Author information

Contact details are reproduced from the original publication and may be historical.

Michel C. Delfour

Centre de Recherches Mathématiques, and: Dép. de Mathématiques et de Statistique, Université de Montréal, Montréal H3C 3J7, Canada

delfour@crm.umontreal.ca

Keywords

  • Lagrange problem
  • column
  • buckling
  • eigenvalue
  • degenerate differential operator

Mathematics Subject Classification

34L10, 34L16, 74G60, 65K10, 49

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Suggested citation

M. C. Delfour, F. Huot-Chantal. “"On the Figure of Columns" of Lagrange Revisited.” Journal of Convex Analysis 26 (2019), No. 3, 855–876. https://doi.org/10.68381/jca26045

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.