Minimal graphs in the hyperbolic space with singular asymptotic boundaries

We study asymptotic behaviors of solutions f to the Dirichlet problem for minimal graphs in the hyperbolic space with singular asymptotic boundaries under the assumption that the boundaries are piecewise regular with positive curvatures, a case which also arises in the study of a Chaplygin gas. We derive an estimate of such solutions by the corresponding solutions in the intersections of interior tangent balls. The positivity of curvatures plays an important role.