A nonstandard free boundary problem arising in the shape optimization of thin torsion rods

Abstract

We study a 2$d$-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 $D$. 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 $D$, and we study the nonstandard free boundary problem with a gradient obstacle, which is obtained through the optimality conditions.