Calculus of Variations and Geometric Measure Theory

G. P. Leonardi - G. Saracco

Rigidity and trace properties of divergence-measure vector fields

created by saracco on 04 Aug 2017
modified on 28 Feb 2022


Published Paper

Inserted: 4 aug 2017
Last Updated: 28 feb 2022

Journal: Adv. Calc. Var.
Volume: 15
Number: 1
Pages: 133--149
Year: 2022
Doi: 10.1515/acv-2019-0094

ArXiv: 1708.01393 PDF


We consider a $\varphi$-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 $\varphi$. In particular, we exhibit counterexamples to \emph{quadratic rigidity} (i.e., when $\varphi(t) = ct^2$) 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 $\xi$ on a $\mathcal{H}^{1}$-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.

Keywords: rigidity, weak normal trace, divergence-measure vector field