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

We consider the functional ∫Ωg(∇u+X∗)dL2n where g is convex and X∗(x,y)=2(−y,x) and we study the minimizers in BV(Ω) 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. The assumptions on g allow to consider both the case with superlinear growth and the one with linear growth. Moreover neither uniform ellipticity nor smoothness of g are assumed.