Variational solutions to extremal metrics on toric surfaces

In this paper, we prove the interior regularity of minimizers of Donaldson’s version of K-energy associated to Calabi’s extremal metrics on toric surfaces. A new idea is to use solutions to the first boundary value problem of Abreu’s equation to approximate a minimizer of K-energy. By the regularity, we also prove the uniqueness of minimizers in dimension 2.