Rigidity and trace properties of divergence-measure vector fields

Abstract

Abstract We consider a φ-rigidity property for divergence-free vector fields in the Euclidean n-space, where φ ⁢ ( t ) {\varphi(t)} is a non-negative convex function vanishing only at t = 0 {t=0} . We show that this property is always satisfied in dimension n = 2 {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 ) = c ⁢ t 2 {\varphi(t)=ct^{2}} ) in dimension n ≥ 4 {n\geq 4} . The validity of the quadratic rigidity, which we prove in dimension n = 2 {n=2} , implies the existence of the trace of a divergence-measure vector field ξ on an ℋ 1 {\mathcal{H}^{1}} -rectifiable set S, as soon as its weak normal trace [ ξ ⋅ ν S ] {[\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.