Strong solutions for the Levi curvature equation

Abstract

We consider the prescribed Levi curvature equation, a second order quasilinear equation whose associated operator can be represented as a sum of squares of nonlinear vector fields. For this equation we introduce a notion of derivatives modeled on the geometry of the associated operator and prove an a priori $L^2$ estimate for these second order intrinsic derivatives of a viscosity solution. We then show that viscosity solutions are strong solutions in a natural sense and satisfy the equation almost everywhere.