  Utilizing a weight matrix we study surfaces of prescribed weighted mean
curvature which yield a natural generalisation to critical points of
anisotropic surface energies. We first derive a differential equation for the
normal of immersions with prescribed weighted mean curvature, generalising a
result of Clarenz and von der Mosel. Next we study graphs of prescribed
weighted mean curvature, for which a quasilinear elliptic equation is proved.
Using this equation, we can show height and boundary gradient estimates.
Finally, we solve the Dirichlet problem for graphs of prescribed weighted mean
curvature.
