Rigidity and trace properties of divergence-measure vector fields
AbstractWe consider a φ-rigidity property for divergence-free vector fields in the Euclidean n-space, where {\varphi(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 {\varphi(t)=ct^{2}}) in dimension {n\geq 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 {\mathcal{H}^{1}}-rectifiable set S, as soon as its weak normal trace {[\xi\cdot\nu_{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.
