Anisotropic minimal surface equation with Dirichlet boundary condition

This paper investigates the Dirichlet problem for the anisotropic minimal surface equation in a bounded domain.

Under the natural assumption of non-negative boundary anisotropic mean curvature, we establish the unique solvability of the Dirichlet problem for continuous boundary data.

To achieve this, an essential a priori gradient estimate is established, which also allows us to prove a weak version of Bernstein's theorem for entire solutions with linear growth.

Moreover, using the direct method in the calculus of variations, we prove the existence and local Lipschitz regularity of generalized minimizers in $BV(\Omega)$ with $L^1(\partial\Omega)$ boundary data.

We also find that this variational formulation naturally yields a Neumann boundary condition, geometrically explaining why the Neumann problem requires no boundary curvature constraints.