Abstract
In [8 Chapter 4.3] Kirchheim and Preiss gave an example of a set K consisting of five 2 × 2 symmetric matrices without rank-one connections, for which there exists a Lipschitz mapping u satisfying \({Du \in K}\) . In the present paper we construct the rank-one convex hull of K. As a corollary we obtain that for each \({F \in {\rm int}\,K^{rc}}\) there exists a Lipschitz mapping u satisfying
Moreover, we show that the rank-one convex hull of K and the quasiconvex hull of K are equal.
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Astala K., Faraco D., Székelyhidi L. Jr. : Convex integration and the L p theory of elliptic equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 7(1), – (2008)
Ball J.M.: Sets of gradients with no rank-one connections. J. Math. Pures et Appliquees 69, 241–259 (1990)
Ball J.M., James R.D. : Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100, 13–52 (1987)
Chlebik M., Kirchheim B.: Rigidity for the four gradient problem. J. Reine Angew. Math. 551, 1–9 (2002)
Conti S., Theil F. : Single-slip elastoplastic microstructures. Arch. Rat. Mech. Anal. 178, 125–148 (2005)
Kinderlehrer D., Pedregal P. : Characterization of Young measures generated by gradients. Arch. Ration. Mech. Anal. 115, 329–365 (1991)
Kirchheim B.: Deformations with finitely many gradients and stability of quasiconvex hulls. C.R. Acad. Sci. Paris Sr. I Math. 332, 289–294 (2001)
Kirchheim, B.: Rigidity and geometry of microstructures. MPI-MIS Lecture notes 16, (2003). Available at http://www.mis.mpg.de/publications/other-series/ln/lecturenote-1603.html
Kirchheim, B., Müller, S., Šverák, V.: In: Hildebrandt, S., Karcher, H. (eds.)Studying Nonlinear Pde by Geometry in Matrix Space, Geometric Analysis and Nonlinear Partial Differential Equations, pp. 347–395. Springer-Verlag (2003)
de Lellis, C., Székelyhidi, L.: The Euler equations as a differential inclusion. Preprint (2007)
Meyer, P.-A.: Probability and Potentials. Blaisdell (1966)
Morey C.B. : Quasi-convexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2, 25–53 (1952)
Müller, S.: Variational models for microstructure and phase transitions. In: Bethuel, F., Huisken, G., Müller, S., Steffen, K., Hildebrandt, S., Struwe,M. (eds.) Calculus of variations and geometric evolution problems, Proc. C.I.M.E. summer school, Cetraro, 1996. Springer LNM 1713, (1999)
Müller S., Šverák V.: Convex integration with constraints and applications to phase transitions and partial differential equations. J. Eur. Math. Soc. (JEMS) 1, 393–442 (1999)
Pompe, W.: Explicit construction of piecewise affine mappings with constraints. Preprint (2008)
Šverák V. : New examples of quasiconvex functions. Arch. Ration. Mech. Anal. 119, 293–300 (1992)
Šverák V.: Rank-one convexity does not imply quasiconvexity. Proc. R. Soc. Edinb. 120, 185–189 (1992)
Sychev M.A.: A few remarks on differential inclusions. Proc. R. Soc. Edinb., Sect. A, Math. 136(3), 649–668 (2006)
Zhang K. : Multiwell problems and restrictions on microstructure. Proc. CMA, ANU 33, 259–271 (1994)
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This work is supported by the KBN grant nr 1 PO3 A 008 29.