On A Class Of Degenerate Or Singular Monge-Ampere Equations

We introduce the concept of(a,?,?)type domain,and classify domains by convexity in more detail.By choosing proper auxiliary function,we get some estimates,then prove the existence and uniqueness of the Dirichlet problem of a class of degenerate and singular Monge-Ampere type equation on bounded convex domain.Through the boundary H ¨older estimates,we can easily found the relationship between the H ¨older exponent and the convexity of domains?The affine hyperbolic ball equation is a special case in our paper,which is related to some important equations in gemetry?In this cases,our boundary H ¨older estimates is optimal?And in some unbounded domains,we find some special solutions,then we can expand our results to unbounded domains?Our method also can be used to minimal graph equation in affine hyperbolic space?Similarly,we introduce(a,?,?)type domain and proper auxiliary function,and we get the global H ¨older estimate of concave solutions?...