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.].

DOI: 10.1051/cocv/2015007

Keywords: Calculus of variations, rank one convexity, quasiconvexity, polyconvexity, exterior convexity, exterior form, differential form

^{1}; Sil, Swarnendu

^{2}

@article{COCV_2016__22_2_338_0, author = {Bandyopadhyay, Saugata and Sil, Swarnendu}, title = {Exterior convexity and classical calculus of variations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {338--354}, publisher = {EDP-Sciences}, volume = {22}, number = {2}, year = {2016}, doi = {10.1051/cocv/2015007}, mrnumber = {3491773}, zbl = {1343.49022}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2015007/} }

TY - JOUR AU - Bandyopadhyay, Saugata AU - Sil, Swarnendu TI - Exterior convexity and classical calculus of variations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 338 EP - 354 VL - 22 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2015007/ DO - 10.1051/cocv/2015007 LA - en ID - COCV_2016__22_2_338_0 ER -

%0 Journal Article %A Bandyopadhyay, Saugata %A Sil, Swarnendu %T Exterior convexity and classical calculus of variations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 338-354 %V 22 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2015007/ %R 10.1051/cocv/2015007 %G en %F COCV_2016__22_2_338_0

Bandyopadhyay, Saugata; Sil, Swarnendu. Exterior convexity and classical calculus of variations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 2, pp. 338-354. doi : 10.1051/cocv/2015007. http://archive.numdam.org/articles/10.1051/cocv/2015007/

Calculus of variations with differential forms, J. Eur. Math. Soc. 17 (2015) 1009–1039. | DOI | MR | Zbl

, and ,S. Bandyopadhyay and S. Sil, Characterization of functions affine in the direction of one-divisible forms. In preparation.

B. Dacorogna, Direct methods in the calculus of variations. In vol. 78 of Appl. Math. Sci. 2nd edition. Springer, New York (2008). | MR | Zbl

S. Sil, Ph.D. thesis.

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