Are Some Optimal Shape Problems Convex?

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.