We study a prescribed mean curvature problem where we seek a surface whose mean curvature vector coincides with the normal component of a given vector field. We prove that the problem has a solution near a graphical minimal surface if the prescribed vector field is sufficiently small in a dimensionally sharp Sobolev norm.
"The Dirichlet problem for a prescribed mean curvature equation." Hiroshima Math. J. 50 (3) 325 - 337, November 2020. https://doi.org/10.32917/hmj/1607396492