Lipschitz minimizers for a class of integral functionals under the bounded slope condition

We consider the functional $\int_\Omega g(\nabla u+\textbf X^\ast)d\mathscr L^{2n}$ where $g$ is convex and $\textbf X^\ast(x,y)=2(-y,x)$ and we study the minimizers in $BV(\Omega)$ of the associated Dirichlet problem. We prove that, under the bounded slope condition on the boundary datum, and suitable conditions on $g$, there exists a unique minimizer which is also Lipschitz continuous.