Prescribed Mean Curvature Equations for Functions of Bounded Variation

Let L(u) = L(u,∇u) be a functional on W1,1(Ω) whose formal Euler-Lagrange equation at the critical point u of L is the prescribed mean curvature equation:

−div(∇u /√1 + |∇u|2)= g(x, u).

Suppose L(u) = L(u,Du) is a relaxed functional of L(u), the weakly lower semicontinuous extension of L on the space of functions of bounded variation. How dose the relaxation affect the prescribed mean curvature equation? Instead of an Euler-Lagrange equation, we obtain here the so-called Euler-Lagrange system of equations which the critical points u of L and their derivatives Du necessarily satisfy.