We study a 2-variational problem, in which the cost functional is an integral depending on the gradient through a convex but not strictly convex integrand, and the admissible functions have zero gradient on the complement of a given domain . We are interested in establishing whether solutions exist whose gradient “avoids” the region of non-strict convexity. Actually, the answer to this question is related to establishing whether homogenization phenomena occur in optimal thin torsion rods. We provide some existence results for different geometries of , and we study the nonstandard free boundary problem with a gradient obstacle, which is obtained through the optimality conditions.
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Jean Jacques Alibert, Guy Bouchitté, Ilaria Fragalà, Ilaria Lucardesi, A nonstandard free boundary problem arising in the shape optimization of thin torsion rods. Interfaces Free Bound. 15 (2013), no. 1, pp. 95–119DOI 10.4171/IFB/296