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We investigate the differences and similarities of the Dirichlet problem ofthe mean curvature equation in the Euclidean space and in the Lorentz-Minkowskispace. Although the solvability of the Dirichlet problem follows standardstechniques of elliptic equations, we focus in showing how the spacelikecondition in the Lorentz-Minkowski space allows to drop the hypothesis on themean convexity which is required in the Euclidean case.

The Dirichlet problem of the constant mean curvature equation in Lorentz-Minkowski space and in Euclidean space