A beam that can only bend on the Cantor set

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 (Formula presented.), each one only able to bend on the set associated with the (Formula presented.) -th step in the construction of the Cantor set and compute the (Formula presented.) -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 (Formula presented.), which is the dimension of the Cantor set.