|Feb 23, 2017
We will review the work by K. Astala, T. Iwaniec, and G. Martin of the Alexandrov-Bakelman-Pucci maximum principle for Elliptic PDEs in the plane. The ABP maximum principle is a basic tool in the theory of linear Elliptic PDEs in nondivergence form. In two dimensions, Astala-Iwaniec-Martin proved the sharp form of the maximum principle as conjectured by Pucci in 1996. These results are obtained by using the advances in the planar theory of quasiconformal mappings.