On the Extreme Points of the Unit Ball in the Space of Solenoidal Vector Measures on the Plane
DOI:
https://doi.org/10.5890/DNC.2020.12.005Abstract
We consider the space of finite divergence-free Borel vector measures on~$\mathbb R^d$, endowed with the total variation norm. For $d=2$ we present a characterization of the extreme points of the unit ball in this space. This allows one to decompose (for $d=2$) any finite divergence-free vector measure into measures induced by closed Lipschitz curves. The results are based on a joint work with P. Bonicatto.References
[1] Stepanov, E. and Trevisan, D. (2017), Three superposition principles: Currents, continuity equations and curves of measures, { Journal of Functional Analysis}, 272(3), 1044-1103.
[2] Ambrosio, L. and Crippa, G. (2008), { Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields}, pages 3-57, Springer Berlin Heidelberg, Berlin, Heidelberg.
[3] Bianchini, S., Bonicatto, P., and Gusev, N.A. (2016), Renormalization for autonomous nearly incompressible {BV} vector fields in two dimensions, { {SIAM} Journal on Mathematical Analysis}, 48(1), 1-33.
[4] Smirnov, S.K. (1993), Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows, { Algebra i Analiz}, 5, 206-238.
[5] Paolini, E. and Stepanov, E. (2012), Decomposition of acyclic normal currents in a metric space, { Journal of Functional Analysis}, 263(11), 3358-3390.
[6] Paolini, and Stepanov, E. (2013), Structure of metric cycles and normal one-dimensional currents, { Journal of Functional Analysis}, 264(6), 1269-1295.
[7] Phelps, R.R. (2001), { Lectures on Choquet's Theorem}, Lecture Notes in Mathematics, Springer Berlin Heidelberg.
[8] Bonicatto, P. and Gusev, N.A. (2019), On the structure of divergence-free measures on the plane, { In preparation}.
[9] Ambrosio, L., Caselles, V., Masnou, S., and Morel, J.M. (2001), Connected components of sets of finite perimeter and applications to image processing, { Journal of the European Mathematical Society}, 3(1), 39-92.
[10] Bianchini, S. and Gusev, N.A. (2016), Steady nearly incompressible vector fields in two-dimension: Chain rule and renormalization, { Archive for Rational Mechanics and Analysis}, 222(2), 451-505.
Article Metrics
Usage tracking begins September 1, 2026.