The interplay between variational functionals and the Brunn-Minkowski Theory is currently a well-established phenomenon that has been widely investigated in the last thirty years. In this talk, we present a result on the existence of solutions to the even logarithmic Minkowski problems arising from torsional rigidity. The blueprint we lay down works as well for more generic functionals by adapting the volume case from Böröczky, Lutwak, Yang, and Zhang. We show how these results imply self-similar solutions to variational flow problems a la Firey's worn stone problem.