Abstract

The optimal shape problem in this paper is to construct plates or beams of minimal weight. The thickness u(x) is variable, but the vertical deformation y(x) should not exceed a certain threshold. The functions u and y are related to each other via the differential equation Δ(buᵖΔy) = f, see (1.2) below. We investigate under which boundary conditions on y the class of admissible thickness functions u is convex. In two out of three cases we give a positive answer, contrary to the common belief that these optimal shape problems are nonconvex. Moreover, under one type of boundary condition, the answer is different for beam and plate. Nonconvexity is shown by means of counterexamples which were found using MAPLE.

Author information

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

Bernd Kawohl

Mathematisches Institut, Universität zu Köln, 50923 Köln, Germany.

Jan Lang

Mathematical Institute, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic

Keywords

  • optimal shape
  • plate equation
  • convexity
  • maximal deformation
  • beam

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Suggested citation

B. Kawohl, J. Lang. “Are Some Optimal Shape Problems Convex?.” Journal of Convex Analysis 4 (1997), No. 2, 353–361. https://doi.org/10.68381/jca04020

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Published by Heldermann Verlag, 1997. Rights now held by Banach Press LLC.