Regularity of Degenerate Monge–Ampère and Prescribed Gaussian Curvature Equations in Two Dimensions

Abstract We use a priori inequalities for quasilinear equations to obtain a regularity theorem for the Dirichlet problem for the Monge–Ampère equation, $$u_{xx}u_{yy}-(u_{xy})^{2}=k(x,y),$$ and the prescribed Gaussian curvature equation, $$u_{xx}u_{yy}-(u_{xy})^{2}=k(x,y)(1+u_{x}^{2}+u_{y}^{2})^{2},$$ where k(x,y) is close to a function of one variable alone when k is small, but permitted to vanish to infinite order.