A beam that can only bend on the Cantor set

Abstract

In this work we address the following question: is it possible for a one-dimensional, linearly elastic beam to only bend on the Cantor set and, if so, what would the bending energy of such a beam look like? We answer this question by considering a sequence of beams, indexed by $n$, each one only able to bend on the set associated with the $n$-th step in the construction of the Cantor set and compute the $\Gamma$-limit of the bending energies. The resulting energy in the limit has a structure similar to the traditional bending energy, a key difference being that the measure used for the integration is the Hausdorff measure of dimension $\ln 2/\ln 3$, which is the dimension of the Cantor set.