Rigidity and trace properties of divergence-measure vector fields

Abstract

We consider a $\phi$-rigidity property for divergence-free vector fields in the Euclidean $n$-space, where $\phi (t)$ 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 $\phi$. In particular, we exhibit counterexamples to \textit{quadratic rigidity} (i.e., when $\phi (t)=c{t}^2$) 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.

19 pages, 3 figures

References in corpus (6)

• Anzellotti's pairing theory and the Gauss--Green theorem
• The Cheeger constant of a Jordan domain without necks
• An extension of the pairing theory between divergence-measure fields and BV functions
• Two examples of minimal Cheeger sets in the plane
• Liouville type theorems on the steady Navier-Stokes equations in R3
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Cited by in corpus (5)

• Anzellotti's pairing theory and the Gauss--Green theorem
• An extension of the pairing theory between divergence-measure fields and BV functions
• Fractional divergence-measure fields, Leibniz rule and Gauss-Green formula
• Geometric criteria for the existence of capillary surfaces in tubes
• On the structure of divergence-free measures on ${\mathbb{R}}^2$