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Abstract: We construct geometric barriers for minimal graphs in H^n xR. We prove theexistence and uniqueness of a solution of the vertical minimal equation in theinterior of a convex polyhedron in H^n extending continuously to the interiorof each face, taking infinite boundary data on one face and zero boundary valuedata on the other faces. In H^n xR, we solve the Dirichlet problem for thevertical minimal equation in a C^0 convex domain taking arbitrarily continuousfinite boundary and asymptotic boundary data. We prove the existence of anotherScherk type hypersurface, given by the solution of the vertical minimalequation in the interior of certain admissible polyhedron taking alternativelyinfinite values +\infty and -\infty on adjacent faces of this polyhedron. Thosepolyhedra may be chosen convex or non convex. We establish analogous resultsfor minimal graphs when the ambient is the Euclidean space R^ {n+1}.

Author: Ricardo Sá Earp, Eric Toubiana IMJ



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