Optimal thin torsion rods and Cheeger sets

We carry out an asymptotic analysis of the following shape optimization problem: a given volume fraction of elastic material must be distributed in a cylindrical design region of infinitesimal cross section in order to maximize resistance to a twisting load. We derive a limit rod model written in different equivalent formulations and for which we are able to give necessary and sufficient conditions characterizing optimal configurations. Eventually we show that for a convex design region and for very small volume fractions, the optimal shape tends to concentrate section by section near the boundary of the Cheeger set of the design. These results were announced in [G. Bouchitté, I. Fragalà, and P. Seppecher, C. R. Math., 348 (2010), pp. 467-471]. © 2012 Society for Industrial and Applied Mathematics.