| Title:
|
On semiconvexity properties of rotationally invariant functions in two dimensions (English) |
| Author:
|
Šilhavý, M. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
54 |
| Issue:
|
3 |
| Year:
|
2004 |
| Pages:
|
559-571 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $f$ be a function defined on the set ${\mathbf M}^{2\times 2}$ of all $2$ by $2$ matrices that is invariant with respect to left and right multiplications of its argument by proper orthogonal matrices. The function $f$ can be represented as a function $\tilde{f}$ of the signed singular values of its matrix argument. The paper expresses the ordinary convexity, polyconvexity, and rank 1 convexity of $f$ in terms of its representation $\tilde{f}.$ (English) |
| Keyword:
|
semiconvexity |
| Keyword:
|
rank 1 convexity |
| Keyword:
|
polyconvexity |
| Keyword:
|
convexity |
| Keyword:
|
rotational invariance |
| MSC:
|
26B25 |
| MSC:
|
49J10 |
| MSC:
|
49J45 |
| MSC:
|
74B20 |
| MSC:
|
74G65 |
| idZBL:
|
Zbl 1080.49013 |
| idMR:
|
MR2086716 |
| . |
| Date available:
|
2009-09-24T11:15:25Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127911 |
| . |
| Reference:
|
[1] J. J. Alibert and B. Dacorogna: An example of a quasiconvex function that is not polyconvex in two dimensions.Arch. Rational Mech. Anal. 117 (1992), 155–166. MR 1145109, 10.1007/BF00387763 |
| Reference:
|
[2] G. Aubert: On a counterexample of a rank 1 convex function which is not polyconvex in the case $N=2$.Proc. Roy. Soc. Edinburgh 106A (1987), 237–240. MR 0906209 |
| Reference:
|
[3] G. Aubert: Necessary and sufficient conditions for isotropic rank-one convex functions in dimension $2$.J. Elasticity 39 (1995), 31–46. Zbl 0828.73015, MR 1343149, 10.1007/BF00042440 |
| Reference:
|
[4] G. Aubert and R. Tahraoui: Sur la faible fermeture de certains ensembles de contrainte en élasticité nonlinéaire plane.C. R. Acad. Sci. Paris 290 (1980), 537–540. MR 0573804 |
| Reference:
|
[5] G. Aubert and R. Tahraoui: Sur la faible fermeture de certains ensembles de contrainte en élasticite nonlinéaire plane.Arch. Rational Mech. Anal. 97 (1987), 33–58. MR 0856308, 10.1007/BF00279845 |
| Reference:
|
[6] J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity.Arch. Rational Mech. Anal. 63 (1977), 337–403. Zbl 0368.73040, MR 0475169 |
| Reference:
|
[7] B. Dacorogna: Direct Methods in the Calculus of Variations.Springer, Berlin, 1989. Zbl 0703.49001, MR 0990890 |
| Reference:
|
[8] B. Dacorogna and H. Koshigoe: On the different notions of convexity for rotationally invariant functions.Ann. Fac. Sci. Toulouse II (1993), 163–184. MR 1253387 |
| Reference:
|
[9] B. Dacorogna and P. Marcellini: A counterexample in the vectorial calculus of variations.In: Material Instabilities in Continuum Mechanics, J. M. Ball (ed.), Clarendon Press, Oxford, 1985/1986, pp. 77–83. MR 0970519 |
| Reference:
|
[10] B. Dacorogna and P. Marcellini: Implicit Partial Differential Equations.Birkhäuser, Basel, 1999. MR 1702252 |
| Reference:
|
[11] C. B. Morrey, Jr.: Multiple Integrals in the Calculus of Variations.Springer, New York, 1966. Zbl 0142.38701, MR 0202511 |
| Reference:
|
[12] P. Rosakis: Characterization of convex isotropic functions.J. Elasticity 49 (1998), 257–267. Zbl 0906.73018, MR 1633494 |
| Reference:
|
[13] M. Šilhavý: The Mechanics and Thermodynamics of Continuous Media.Springer, Berlin, 1997. MR 1423807 |
| Reference:
|
[14] M. Šilhavý: On isotropic rank 1 convex functions.Proc. Roy. Soc. Edinburgh 129A (1999), 1081–1105. MR 1719253 |
| Reference:
|
[15] M. Šilhavý: Convexity conditions for rotationally invariant functions in two dimensions.In: Applied Nonlinear Analysis, A. Sequeira et al. (ed.), Kluwer Academic, New York, 1999, pp. 513–530. MR 1727470 |
| Reference:
|
[16] M. Šilhavý: Rotationally invariant rank 1 convex functions.Appl. Math. Optim. 44 (2001), 1–15. MR 1833365, 10.1007/s00245-001-0012-z |
| Reference:
|
[17] M. Šilhavý: Monotonicity of rotationally invariant convex and rank 1 convex functions.Proc. Royal Soc. Edinburgh 132A (2002), 419–435. MR 1899830 |
| Reference:
|
[18] M. Šilhavý: Rank 1 Convex hulls of isotropic functions in dimension 2 by 2.Math. Bohem. 126 (2001), 521–529. MR 1844288 |
| Reference:
|
[19] M. Šilhavý: An $O(n)$ invariant rank 1 convex function that is not polyconvex.Theor. Appl. Mech. 28–29 (2002), 325–336. Zbl 1055.26006, MR 2025155 |
| . |
