Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices

Carl-Friedrich Kreiner; Johannes Zimmer

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 2, page 253-270
  • ISSN: 1292-8119

Abstract

top
Continuing earlier work by Székelyhidi, we describe the topological and geometric structure of so-called T4-configurations which are the most prominent examples of nontrivial rank-one convex hulls. It turns out that the structure of T4-configurations in 2 × 2 is very rich; in particular, their collection is open as a subset of ( 2 × 2 ) 4 . Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm to detect T4-configurations.

How to cite

top

Kreiner, Carl-Friedrich, and Zimmer, Johannes. "Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices." ESAIM: Control, Optimisation and Calculus of Variations 12.2 (2006): 253-270. <http://eudml.org/doc/249663>.

@article{Kreiner2006,
abstract = { Continuing earlier work by Székelyhidi, we describe the topological and geometric structure of so-called T4-configurations which are the most prominent examples of nontrivial rank-one convex hulls. It turns out that the structure of T4-configurations in $\mathbb\{R\}^\{2\times 2\}$ is very rich; in particular, their collection is open as a subset of $(\mathbb\{R\}^\{2\times 2\})^\{4\}$. Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm to detect T4-configurations. },
author = {Kreiner, Carl-Friedrich, Zimmer, Johannes},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Rank-one convexity; T4-configurations.; -configurations},
language = {eng},
month = {3},
number = {2},
pages = {253-270},
publisher = {EDP Sciences},
title = {Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices},
url = {http://eudml.org/doc/249663},
volume = {12},
year = {2006},
}

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
DA - 2006/3//
PB - EDP Sciences
VL - 12
IS - 2
SP - 253
EP - 270
AB - Continuing earlier work by Székelyhidi, we describe the topological and geometric structure of so-called T4-configurations which are the most prominent examples of nontrivial rank-one convex hulls. It turns out that the structure of T4-configurations in $\mathbb{R}^{2\times 2}$ is very rich; in particular, their collection is open as a subset of $(\mathbb{R}^{2\times 2})^{4}$. Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm to detect T4-configurations.
LA - eng
KW - Rank-one convexity; T4-configurations.; -configurations
UR - http://eudml.org/doc/249663
ER -

References

top
  1. E. Aranda and P. Pedregal, On the computation of the rank-one convex hull of a function. SIAM J. Sci. Comput.22 (2000) 1772–1790 (electronic).  Zbl0989.49015
  2. S. Aubry, M. Fago and M. Ortiz, A constrained sequential-lamination algorithm for the simulation of sub-grid microstructure in martensitic materials. Comput. Methods Appl. Mech. Engrg.192 (2003) 2823–2843.  Zbl1054.74697
  3. M. Chlebík and B. Kirchheim, Rigidity for the four gradient problem. J. Reine Angew. Math.551 (2002) 1–9.  Zbl1019.49022
  4. B. Dacorogna, Direct methods in the calculus of variations. Applied Mathematical Sciences, Springer-Verlag, Berlin 78 (1989).  Zbl0703.49001
  5. G. Dolzmann, Numerical computation of rank-one convex envelopes. SIAM J. Numer. Anal.36 (1999) 1621–1635 (electronic).  Zbl0941.65062
  6. Da.R. Grayson and M.E. Stillman, Macaulay 2, a software system for research in algebraic geometry. Available at  URIhttp://www.math.uiuc.edu/Macaulay2/
  7. J. Harris, Algebraic geometry. Springer-Verlag, New York (1995). A first course, Corrected reprint of the 1992 original.  Zbl0779.14001
  8. B. Kirchheim, Rigidity and geometry of microstructures. Lecture notes 16/2003, Max Planck Institute for Mathematics in the Sciences, Leipzig (2003).  
  9. B. Kirchheim, S. Müller and V. Šverák, Studying nonlinear pde by geometry in matrix space, in Geometric analysis and nonlinear partial differential equations. Springer, Berlin (2003) 347–395.  Zbl1290.35097
  10. C.-F. Kreiner, Algebraic methods for convexity notions in the calculus of variations. Master's thesis, Technische Universität München, Zentrum Mathematik (2003).  
  11. C.-F. Kreiner, J. Zimmer and I. Chenchiah, Towards the efficient computation of effective properties of microstructured materials. Comptes Rendus Mecanique332 (2004) 169–174.  
  12. J. Matoušek and P. Plecháč, On functional separately convex hulls. Discrete Comput. Geom.19 (1998) 105–130.  Zbl0892.68102
  13. S. Müller, 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.  
  14. S. Müller and V. Šverák, 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.  Zbl0896.35029
  15. L. Råde and B. Westergren, Mathematics handbook for science and engineering. Springer-Verlag, Berlin, fourth edition (1999).  Zbl0915.00005
  16. V. Scheffer, Regularity and irregularity of solutions to nonlinear second order elliptic systems of partial differential equations and inequalities. Ph.D. thesis, Princeton University (1974).  
  17. V. Šverák, Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh Sect. A120 (1992) 185–189.  Zbl0777.49015
  18. L. Székelyhidi Jr, Rank-one convex hulls in 2 × 2 . Calc. Var. Partial Differ. Equ.22 (2005) 253–281.  
  19. L. Tartar, Some remarks on separately convex functions, in Microstructure and phase transition. Springer, New York, IMA Vol. Math. Appl.54 (1993) 191–204.  Zbl0823.26008

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.