Rigidity and trace properties of divergence-measure vector fields

TL;DR

This paper investigates the rigidity properties of divergence-free vector fields in Euclidean space, revealing dimension-dependent behaviors and implications for trace existence and geometric properties of solutions to mean curvature equations.

Contribution

It establishes dimension-specific rigidity results for divergence-measure vector fields and links these to trace properties and geometric behavior of solutions.

Findings

Rigidity always holds in 2D for convex functions.

Counterexamples show rigidity fails in dimensions ≥4 for quadratic functions.

Trace existence is linked to maximal weak normal traces on rectifiable sets.

Abstract

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 \textit{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.