  We study a class of second-order boundary-degenerate elliptic equations in
two dimensions with minimal regularity assumptions. We prove a maximum
principle and a Harnack inequality at the degenerate boundary, and assuming
local boundedness, we prove continuity. On globally defined non-negative
solutions we provide strong constraints on behavior at infinity, and prove a
Liouville-type theorem for entire solutions on the closed half-plane. The class
of PDE in question includes many from mathematical finance, Keldysh- and
Tricomi-type PDE, and the 2nd order reduction of the fully non-linear 4th order
Abreu equation from K\"ahler geometry.
