Continuing earlier work by Székelyhidi, we describe the topological and geometric structure of so-called -configurations which are the most prominent examples of nontrivial rank-one convex hulls. It turns out that the structure of -configurations in is very rich; in particular, their collection is open as a subset of . Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm to detect -configurations.
Keywords: rank-one convexity, $T_4$-configurations
@article{COCV_2006__12_2_253_0, author = {Kreiner, Carl-Friedrich and Zimmer, Johannes}, title = {Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {253--270}, publisher = {EDP-Sciences}, volume = {12}, number = {2}, year = {2006}, doi = {10.1051/cocv:2005036}, mrnumber = {2209353}, zbl = {1108.49010}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2005036/} }
TY - JOUR AU - Kreiner, Carl-Friedrich AU - Zimmer, Johannes TI - Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2006 SP - 253 EP - 270 VL - 12 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2005036/ DO - 10.1051/cocv:2005036 LA - en ID - COCV_2006__12_2_253_0 ER -
%0 Journal Article %A Kreiner, Carl-Friedrich %A Zimmer, Johannes %T Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices %J ESAIM: Control, Optimisation and Calculus of Variations %D 2006 %P 253-270 %V 12 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2005036/ %R 10.1051/cocv:2005036 %G en %F COCV_2006__12_2_253_0
Kreiner, Carl-Friedrich; Zimmer, Johannes. Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices. ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 2, pp. 253-270. doi : 10.1051/cocv:2005036. http://archive.numdam.org/articles/10.1051/cocv:2005036/
[1] On the computation of the rank-one convex hull of a function. SIAM J. Sci. Comput. 22 (2000) 1772-1790 (electronic). | Zbl
and ,[2] A constrained sequential-lamination algorithm for the simulation of sub-grid microstructure in martensitic materials. Comput. Methods Appl. Mech. Engrg. 192 (2003) 2823-2843. | Zbl
, and ,[3] Rigidity for the four gradient problem. J. Reine Angew. Math. 551 (2002) 1-9. | Zbl
and ,[4] Direct methods in the calculus of variations. Applied Mathematical Sciences, Springer-Verlag, Berlin 78 (1989). | MR | Zbl
,[5] Numerical computation of rank-one convex envelopes. SIAM J. Numer. Anal. 36 (1999) 1621-1635 (electronic). | Zbl
,[6] Da.R. Grayson and M.E. Stillman, Macaulay 2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/
[7] Algebraic geometry. Springer-Verlag, New York (1995). A first course, Corrected reprint of the 1992 original. | MR | Zbl
,[8] Rigidity and geometry of microstructures. Lecture notes 16/2003, Max Planck Institute for Mathematics in the Sciences, Leipzig (2003).
,[9] Studying nonlinear pde by geometry in matrix space, in Geometric analysis and nonlinear partial differential equations. Springer, Berlin (2003) 347-395.
, and ,[10] Algebraic methods for convexity notions in the calculus of variations. Master's thesis, Technische Universität München, Zentrum Mathematik (2003).
,[11] Towards the efficient computation of effective properties of microstructured materials. Comptes Rendus Mecanique 332 (2004) 169-174.
, and ,[12] On functional separately convex hulls. Discrete Comput. Geom. 19 (1998) 105-130. | Zbl
and ,[13] Variational models for microstructure and phase transitions, in Calculus of variations and geometric evolution problems (Cetraro, 1996). Springer, Berlin, Lect. Notes Math. 1713 (1999) 85-210. | Zbl
,[14] Unexpected solutions of first and second order partial differential equations, in Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Extra Vol. II (1998) 691-702. | Zbl
and ,[15] Mathematics handbook for science and engineering. Springer-Verlag, Berlin, fourth edition (1999). | MR | Zbl
and ,[16] Regularity and irregularity of solutions to nonlinear second order elliptic systems of partial differential equations and inequalities. Ph.D. thesis, Princeton University (1974).
,[17] Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh Sect. A 120 (1992) 185-189. | Zbl
,[18] Rank-one convex hulls in . Calc. Var. Partial Differ. Equ. 22 (2005) 253-281. | Zbl
,[19] Some remarks on separately convex functions, in Microstructure and phase transition. Springer, New York, IMA Vol. Math. Appl. 54 (1993) 191-204. | Zbl
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