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Journal of Convex Analysis 26 (2019), No. 3, 855--876
Copyright Heldermann Verlag 2019

"On the Figure of Columns" of Lagrange Revisited

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

Francis Huot-Chantal
Dép. d'Informatique et de Recherche Opérationnelle, Université de Montréal, Montréal H3C 3J7, Canada


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 $p$-th power of the cross-sectional area for some $p>0$: $p=2$ (solid column) and $p=1$ (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.\par In this paper, we revisit some aspects of the \emph{unconstrained problem} with emphasis on the cases $01$. Additional results are given for \emph{degenerate profiles}, that is, profiles whose lower bound is zero at some points.

Keywords: Lagrange problem, column, buckling, eigenvalue, degenerate differential operator.

MSC: 34L10, 34L16, 74G60, 65K10, 49.

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