On the Composition of Quasiconvex Functions and the Transposition

Martin Kružík

doi:10.68381/jca06013

Abstract

The paper shows that the composition of a quasiconvex function (taking the value +∞) with the transposition is not necessarily a quasiconvex function.

Author information

Contact details are reproduced from the original publication and may be historical.

Martin Kružík

Inst. of Information Theory and Automation, Czech Academy of Sciences, Pod vodárenskou věží 4, 18208 Praha 8, Czech Republic

Keywords

Polyconvexity
quasiconvexity
rank-one convexity

Mathematics Subject Classification

26B25; 49R99

References

E. Acerbi, N. Fusco: Semicontinuity problems in the calculus of variations, Arch. Rat. Mech. Anal. 86 (1986) 125-145.https://doi.org/10.1007/BF00275731 Google Scholar
J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 63 (1977) 337-403.https://doi.org/10.1007/BF00279992 Google Scholar
J. M. Ball: Sets of gradients with no rank-one connections, J. Math. Pures et Appl. 69 (1990) 241-259.Google Scholar
B. Dacorogna: Direct Methods in the Calculus of Variations, Springer, Berlin, 1989.https://doi.org/10.1007/978-3-642-51440-1 Google Scholar
B. Dacorogna, J. Douchet, W. Gangbo, J. Rappaz: Some examples of rank-one convex functions in dimension two, Proc. Roy. Soc. Edinburgh 114 (1990) 135-150.https://doi.org/10.1017/S0308210500024318 Google Scholar
C. B. Morrey, Jr.: Multiple Integrals in the Calculus of Variations, Springer, Berlin, 1966.https://doi.org/10.1007/978-3-540-69952-1 Google Scholar
S. Müller: A sharp version of Zhang's theorem on truncating sequences of gradients, Max Planck Institute preprint No. 18, Leipzig, 1997.Google Scholar
S. Müller: Quasiconvexity is not invariant under transposition, MPI preprint 19, 1998.Google Scholar
G. P. Parry: On the planar rank-one convexity condition, Proc. Roy. Soc. Edinburgh 125 (1995) 247-264.https://doi.org/10.1017/S030821050002802X Google Scholar
P. Pedregal: Some remarks on quasiconvexity and rank-one convexity, Proc. Roy. Soc. Edinburgh 126 (1996) 1055-1066.https://doi.org/10.1017/S0308210500023258 Google Scholar
P. Pedregal: Parametrized Measures and Variational Principles, Birkhäuser, Basel, 1997.https://doi.org/10.1007/978-3-0348-8886-8 Google Scholar
T. Roubíček: Relaxation in Optimization Theory and Variational Calculus, W. de Gruyter, Berlin, 1997.https://doi.org/10.1515/9783110811919 Google Scholar
V. Šverák: Quasiconvex functions with subquadratic growth, Proc. Roy. Soc. London 433 (1991) 723-725.https://doi.org/10.1098/rspa.1991.0073 Google Scholar
L. Tartar: Compensated compactness and applications to partial differential equations, In: Nonlinear analysis and mechanics, Herriot-Watt Symposium IV, Pitman Research Notes in Math. 39, 136-212, Pitman, London, 1979.Google Scholar
F. J. Terpstra: Die Darstellung der biquadratische Formen als Summen von Quadraten mit Anwendung auf die Variationsrechnung, Math. Ann. 116 (1938) 166-180.https://doi.org/10.1007/BF01597353 Google Scholar

Suggested citation

Published by Heldermann Verlag, 1999. Rights now held by Banach Press LLC.

[1] E. Acerbi, N. Fusco: Semicontinuity problems in the calculus of variations, Arch. Rat. Mech. Anal. 86 (1986) 125-145.https://doi.org/10.1007/BF00275731 Google Scholar

[2] J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 63 (1977) 337-403.https://doi.org/10.1007/BF00279992 Google Scholar

[3] J. M. Ball: Sets of gradients with no rank-one connections, J. Math. Pures et Appl. 69 (1990) 241-259.Google Scholar

[4] B. Dacorogna: Direct Methods in the Calculus of Variations, Springer, Berlin, 1989.https://doi.org/10.1007/978-3-642-51440-1 Google Scholar

[5] B. Dacorogna, J. Douchet, W. Gangbo, J. Rappaz: Some examples of rank-one convex functions in dimension two, Proc. Roy. Soc. Edinburgh 114 (1990) 135-150.https://doi.org/10.1017/S0308210500024318 Google Scholar

[6] C. B. Morrey, Jr.: Multiple Integrals in the Calculus of Variations, Springer, Berlin, 1966.https://doi.org/10.1007/978-3-540-69952-1 Google Scholar

[7] S. Müller: A sharp version of Zhang's theorem on truncating sequences of gradients, Max Planck Institute preprint No. 18, Leipzig, 1997.Google Scholar

[8] S. Müller: Quasiconvexity is not invariant under transposition, MPI preprint 19, 1998.Google Scholar

[9] G. P. Parry: On the planar rank-one convexity condition, Proc. Roy. Soc. Edinburgh 125 (1995) 247-264.https://doi.org/10.1017/S030821050002802X Google Scholar

[10] P. Pedregal: Some remarks on quasiconvexity and rank-one convexity, Proc. Roy. Soc. Edinburgh 126 (1996) 1055-1066.https://doi.org/10.1017/S0308210500023258 Google Scholar

[11] P. Pedregal: Parametrized Measures and Variational Principles, Birkhäuser, Basel, 1997.https://doi.org/10.1007/978-3-0348-8886-8 Google Scholar

[12] T. Roubíček: Relaxation in Optimization Theory and Variational Calculus, W. de Gruyter, Berlin, 1997.https://doi.org/10.1515/9783110811919 Google Scholar

[13] V. Šverák: Quasiconvex functions with subquadratic growth, Proc. Roy. Soc. London 433 (1991) 723-725.https://doi.org/10.1098/rspa.1991.0073 Google Scholar

[14] L. Tartar: Compensated compactness and applications to partial differential equations, In: Nonlinear analysis and mechanics, Herriot-Watt Symposium IV, Pitman Research Notes in Math. 39, 136-212, Pitman, London, 1979.Google Scholar

[15] F. J. Terpstra: Die Darstellung der biquadratische Formen als Summen von Quadraten mit Anwendung auf die Variationsrechnung, Math. Ann. 116 (1938) 166-180.https://doi.org/10.1007/BF01597353 Google Scholar