The Geometric Measure Theoretical Characterization of Viscosity Solutions to Parabolic Monge-Ampere Type Equation

Abstract

By an involved approach a geometric measure theory is established for parabolic Monge-Ampère operator acting on convex-monotone functions, the theory bears complete analogy with Aleksandrov's classical ones for elliptic Monge-Ampère operator acting on convex functions. The identity of solutions in weak and viscosity sense to parabolic Monge-Ampère equation is proved. A general result on existence and uniqueness of weak solution to BVP for this equation is also obtained.