Radially symmetric boundary value problems for real and complex elliptic Monge-Ampére equations

Radially symmetric Dirichlet and Neumann problems for real and complex Monge-Ampére equations are considered. Existence of radially symmetric solutions is proved by transforming the differential equations into integral ones, solvable by means of fixed point arguments. Then, taking advantage of integral formulae, regularity and convexity of the radial solutions are checked. Fairly weak assumptions are required in that process. In the real case, a priori radial symmetry is also discussed.