Rigidity and trace properties of divergence-measure vector fields

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. - STAMPA. - 2020(2020).



Titolo: Rigidity and trace properties of divergence-measure vector fields
Autori: Leonardi, Gian Paolo; Saracco, Giorgio
Autori Unitn: 
Leonardi, Gian Paolo
mostra contributor esterni 
Titolo del periodico: ADVANCES IN CALCULUS OF VARIATIONS
Anno di pubblicazione: 2020
Digital Object Identifier (DOI): http://dx.doi.org/10.1515/acv-2019-0094
Handle: http://hdl.handle.net/11572/279461
Citazione: Rigidity and trace properties of divergence-measure vector fields / Leonardi, Gian Paolo; Saracco, Giorgio. - In: ADVANCES IN CALCULUS OF VARIATIONS. - ISSN 1864-8266. - STAMPA. - 2020(2020).
Appare nelle tipologie:03.1 Articolo su rivista (Journal article)

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