The theory of compensated compactness of Murat and Tartar links the algebraic condition of rank-$r$ convexity with the analytic condition of weak lower semicontinuity. The former is an algebraic condition and therefore it is, in principle, very easy to use. However, in applications of this theory, the need for an efficient classification of rank-$r$ convex forms arises. In the present paper, we define the concept of extremal $2$-forms and characterize them in the rotationally invariant jointly rank-$r$ convex case.

Keywords: compensated compactness, rank-$r$ convexity, effective conductivity, quadratic forms

@article{COCV_2007__13_1_1_0, author = {Nesi, Vincenzo and Rogora, Enrico}, title = {A complete characterization of invariant jointly rank-r convex quadratic forms and applications to composite materials}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1--34}, publisher = {EDP-Sciences}, volume = {13}, number = {1}, year = {2007}, doi = {10.1051/cocv:2007002}, mrnumber = {2282100}, zbl = {1107.74039}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2007002/} }

TY - JOUR AU - Nesi, Vincenzo AU - Rogora, Enrico TI - A complete characterization of invariant jointly rank-r convex quadratic forms and applications to composite materials JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2007 SP - 1 EP - 34 VL - 13 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007002/ DO - 10.1051/cocv:2007002 LA - en ID - COCV_2007__13_1_1_0 ER -

%0 Journal Article %A Nesi, Vincenzo %A Rogora, Enrico %T A complete characterization of invariant jointly rank-r convex quadratic forms and applications to composite materials %J ESAIM: Control, Optimisation and Calculus of Variations %D 2007 %P 1-34 %V 13 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2007002/ %R 10.1051/cocv:2007002 %G en %F COCV_2007__13_1_1_0

Nesi, Vincenzo; Rogora, Enrico. A complete characterization of invariant jointly rank-r convex quadratic forms and applications to composite materials. ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 1, pp. 1-34. doi : 10.1051/cocv:2007002. http://www.numdam.org/articles/10.1051/cocv:2007002/

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