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Prescribed mean curvature problems on the torus have been considered in one dimension. In this paper, we prove the existence of a graph on the  n -dimensional torus  n{\mathbb{T}^{n}}   , the mean curvature vector of which equals the normal component of a given vector field satisfying suitable conditions for a Sobolev norm, the integrated value, and monotonicity.

Prescribed mean curvature equation on torus