Existence of Global Smooth Solutions to Dirichlet Problem for Degenerate Elliptic Monge-Ampere Equations

摘要

In the present paper the existence of global smooth solutions to the homogeneous Dirichlet problem for degenerate elliptic Monge-Ampere equation: det(_(ij)) = K(x) f(x, u, Du) in a smooth, strictly convex domain Ω ? R~2 is proved provided that 0 < f ε C~∞(Ω? × R~1 × R~2) and K ε C~∞(Ω?) positive in Ω and degenerate at finite degree on ?Ω. Some applications to the regularity of solutions for eigenvalue problem for Monge-Ampere equations is discussed and in two dimensions an affirmative answer to a related problem raised by Trudinger is given.