Calculus of Variations and Geometric Measure Theory

I. Lucardesi

Concentration phenomena in the optimal design of thin rods

created by lucardesi on 10 Aug 2018


Published Paper

Inserted: 10 aug 2018
Last Updated: 10 aug 2018

Journal: Journal of Convex Analysis
Volume: 22
Number: 2
Pages: 303-338
Year: 2015


In this paper we analyze the concentration phenomena which occur in thin rods, solving the following optimization problem: a given fraction of elastic material must be distributed into a cylindrical design region with infinitesimal cross section in an optimal way, so that it maximizes the resistance to a given external load. For small volume fractions, the optimal configuration of material is described by a measure which concentrates on 2-rectifiable sets. For some choices of the external charging, the concentration phenomena turn out to be related to some new variants of the Cheeger problem of the cross section of the rod. The same study has already been carried out in the particular case of pure torsion regime in G. Bouchitté, I. Fragalà, I. Lucardesi, P. Seppecher: Optimal thin torsion rods and Cheeger sets, SIAM J. Math. Anal. 44 (2012). Here we extend those results by enlarging the class of admissible loads.