The Dirichlet problem for a prescribed anisotropic mean curvature equation: existence, uniqueness and regularity of solutions

We discuss existence, uniqueness, regularity and boundary behaviour of solutions of the Dirichlet problem for the prescribed anisotropic mean curvature equation \begin{equation*} {\rm -div}\left({\nabla u}/{\sqrt{1 + |\nabla u|^2}}\right) = -au + {b}/{\sqrt{1 + |\nabla u|^2}}, \end{equation*} where $a,b>0$ are given parameters and $\Omega$ is a bounded Lipschitz domain in $\RR^N$. This equation appears in the modeling theory of capillarity phenomena for compressible fluids and in the description of the geometry of the human cornea.