We discuss the global regularity of solutions f to the Dirichlet problem for minimal graphs in the hyperbolic space when the boundary of the domain ⊂ R n has a nonnegative mean curvature and prove an optimal regularity f ∈ C 1 n+1 (¯ ). We can improve the Hölder exponent for f if certain combinations of principal curvatures of the boundary do not vanish, a phenomenon observed by F.-H. Lin.

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In this paper we discuss some issues related to Poincaré's inequality for a special class of weighted Sobolev spaces. A common feature of these spaces is that they can be naturally associated with differential operators with variable diffusion coefficients that are not uniformly elliptic. We give a classification of these spaces in the 1-D case bases on a measure of degeneracy of the corresponding weight coefficient and study their key properties.