Abstract
The connection between the minimal elastic compliance problem and Monge transport involving the euclidean metric cost has been evidenced in the year 1997. The aim of this paper is to renew this connection and adapt it to some variants in optimal design, focusing in particular on the optimal pre-stressed 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 PDE optimality conditions, we present a geometrical approach arising from a two-point scheme in which geodesics with respect to the optimal metric play a central role. It turns out from examples that optimal structures are very often 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 that is of independent interest.
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