We consider a φ-rigidity property for divergence-free vector fields in the Euclidean n-space, where φ(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 φ(t) = ct^2) in dimension n ≥ 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 a H1-rectifiable set S, as soon as its weak normal trace [ξ · νS ] 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.

Rigidity and trace properties of divergence-measure vector fields / Leonardi, Gian Paolo; Saracco, Giorgio. - In: ADVANCES IN CALCULUS OF VARIATIONS. - ISSN 1864-8266. - ELETTRONICO. - 15:1(2020), pp. 133-149. [10.1515/acv-2019-0094]

Rigidity and trace properties of divergence-measure vector fields

Gian Paolo Leonardi;Giorgio Saracco
2020-01-01

Abstract

We consider a φ-rigidity property for divergence-free vector fields in the Euclidean n-space, where φ(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 φ(t) = ct^2) in dimension n ≥ 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 a H1-rectifiable set S, as soon as its weak normal trace [ξ · νS ] 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.
2020
1
Leonardi, Gian Paolo; Saracco, Giorgio
Rigidity and trace properties of divergence-measure vector fields / Leonardi, Gian Paolo; Saracco, Giorgio. - In: ADVANCES IN CALCULUS OF VARIATIONS. - ISSN 1864-8266. - ELETTRONICO. - 15:1(2020), pp. 133-149. [10.1515/acv-2019-0094]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/279461
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