Rigidity and trace properties of divergence-measure vector fields

We consider a φ-rigidity property for divergence-free vector fields in the Euclidean n-space, where φ(t) is a non-negative convex function vanishing only at t = 0. We show that this property is always satisfied in dimension n = 2, while in higher dimension it requires some further restriction on φ. In particular, we exhibit counterexamples to quadratic rigidity (i.e. when φ(t) = ct^2) in dimension n ≥ 4. The validity of the quadratic rigidity, which we prove in dimension n = 2, implies the existence of the trace of a divergence-measure vector field ξ on an H^1-rectifiable set S, as soon as its weak normal trace [ξ . νS] is maximal on S. As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.