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Optimal design versus maximal Monge-Kantorovich metrics

A remarkable connection between optimal design and Monge transport was initiated in the years 1997 in the context of the minimal elastic compliance problem and where the euclidean metric cost was naturally involved. In this paper we present different variants in optimal design of mechanical structures, in particular focusing on the optimal pre-stressed elastic membrane problem. We show that the underlying metric cost is associated with an unknown maximal monotone map which maximizes the Monge-Kantorovich distance between two measures. In parallel with the classical duality theory leading to existence and (in a smooth case) to PDE optimality conditions, we present a general geometrical approach arising from a two-point scheme in which geodesics with respect to the optimal metric play a central role. These two aspects are enlightened by several explicit examples and also by numerical solutions in which optimal structures very often turn out to be truss-like i.e supported by piecewise affine geodesics. In case of a discrete load, we are able to relate the existence of such truss-like solutions to an extension property of maximal monotone maps which is of independent interest and that we propose here as a conjecture.