Minimizing weak solutions for calabi's extremal metrics on toric   manifolds
  In this paper, we discuss a Donaldson's version of the modified $K$-energy associated to the Calabi's extremal metrics on toric manifolds and prove the existence of the weak solution for extremal metrics in the sense of convex functions which minimizes the modified $K$-energy. 
Bin Zhou and Xiaohua Zhu