Convex approximations of an inhomogeneous anisotropic functional

The numerical minimization of the functional $F(u)=\int_\Omega \phi(x,\nu_u)|Du|+\int_{\partial\Omega\mu u -\int_\Omega\kappa u$, $u\in BV(\Omega,{-1,1})$, is addressed. The function $\phi$ is continuous, has linear growth, and is convex and positively homogeneous of degree one in the second variable. We prove that $F$ can be equivalently minimized on the convex set $BV(\Omega,[-1,1]$ and then regularized with a sequence $\{F_\epsilon(u)\}_\epsilon$ of strictly convex functionals. Then both $F$ and $F_\epsilon$ can be discretized by continuous linear finite elements. The convexity property of the functionals on $BV(\Omega,[-1,1])$ is useful in the numerical minimization of $F$. The $\Gamma$-convergence of the discrete functionals to F, as well as the compactness of any sequence of discrete absolute minimizers, are proven.