2017/08/31 by Gian Paolo Leonardi, Giorgio Saracco · 19 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Composite material #Computer science #Data mining #Geometry #Materials science #Mathematical analysis #Mathematics #Measure (data warehouse) #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Pure mathematics #Rigidity (electromagnetism) #TRACE (psycholinguistics) #Vector field #math.AP #msc:26B20 #msc:28A75 #msc:35L65
paper · pdf · open access · doi:10.1515/acv-2019-0094
published in Advances in Calculus of Variations 15(1), 133-149 (De Gruyter) · 19 pages, 3 figures
arxiv created 2020/02/11 · openalex publication_date 2020/02/11 · arxiv updated 2022/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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) = ct2) 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.