Exterior convexity and classical calculus of variations

• Published in: ESAIM. Control, optimisation and calculus of variations Vol. 22; no. 2; pp. 338 - 354
• Main Authors: Bandyopadhyay, Saugata, Sil, Swarnendu
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 01.04.2016
• Subjects: 49-XX; Calculus of variations; Convexity; differential form; exterior convexity; exterior form; Functional Analysis; Mathematical analysis; Mathematics; polyconvexity; quasiconvexity; rank one convexity; Tensors; differential form; exterior form; calculus of variations; vexity; polycon; quasiconvexity; exterior convexity; 2010 Mathematics Subject Classification: 49-XX; rank one convexity
• ISSN: 1292-8119; 1262-3377
• DOI: 10.1051/cocv/2015007

We study the relation between various notions of exterior convexity introduced in [S. Bandyopadhyay, B. Dacorogna and S. Sil, J. Eur. Math. Soc. 17 (2015) 1009–1039.] with the classical notions of rank one convexity, quasiconvexity and polyconvexity. To this end, we introduce a projection map, which generalizes the alternating projection for two-tensors in a new way and study the algebraic properties of this map. We conclude with a few simple consequences of this relation which yields new proofs for some of the results discussed in [S. Bandyopadhyay, B. Dacorogna and S. Sil, J. Eur. Math. Soc. 17 (2015) 1009–1039.].