Journal of Convex Analysis, Vol. 4, No. 2, pp. 353-361 (1997)

Are Some Optimal Shape Problems Convex?

Bernd Kawohl and Jan Lang

Mathematisches Institut, Universität zu Köln, D-50923 Köln, Germany,, and Mathematical Institute, Czech Academy of Sciences, Zitna 25, 115 67 Prague 1, Czech Republic,

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 threshhold. The functions $u$ and $y$ are related to each other via the differential equation $\Delta(b u^p \Delta y)=f$, 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.

Keywords: optimal shape, plate equation, convexity, maximal deformation, beam

Classification (MSC2000): 49J20; 73K40, 35Q72

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