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 \textit{quadratic rigidity} (i.e., when $φ(t) = ct^2$) in dimension $n\ge 4$. 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 a $\mathcal{H}^{1}$-rectifiable set $S$, as soon as its weak normal trace $[ξ\cdot ν_S]$ 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.

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