Nonlinear Partial Differential Equations

I intend to investigate two closely related classes of nonlinear artial differential equations: Hessian and curvature equations. These have a wide range of applications in differential geometry. Monge-Ampere equations are a particularly important subclass having close connections to mass transfer problems, which have a wide range of applications in mathematics and in other fields. I will develop a comprehensive theory of these equations and their applications, encompassing boundary value problems, extremal solutions, mass transfer problems and curvature flow problems.