Geodesic
Rank 1 convex hulls of isotropic functions in dimension 2 by 2
Mathematica Bohemica, Tome 126 (2001) no. 2, pp. 521-529

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Let $f$ be a rotationally invariant (with respect to the proper orthogonal group) function defined on the set $\text{M}^{2\times 2}$ of all $2$ by $2$ matrices. Based on conditions for the rank 1 convexity of $f$ in terms of signed invariants of $\mathbb{A}$ (to be defined below), an iterative procedure is given for calculating the rank 1 convex hull of a rotationally invariant function. A special case in which the procedure terminates after the second step is determined and examples of the actual calculations are given.
Let $f$ be a rotationally invariant (with respect to the proper orthogonal group) function defined on the set $\text{M}^{2\times 2}$ of all $2$ by $2$ matrices. Based on conditions for the rank 1 convexity of $f$ in terms of signed invariants of $\mathbb{A}$ (to be defined below), an iterative procedure is given for calculating the rank 1 convex hull of a rotationally invariant function. A special case in which the procedure terminates after the second step is determined and examples of the actual calculations are given.
DOI : 10.21136/MB.2001.134029
Classification : 49J45, 74G65, 74N99
Keywords: rank 1 convexity; relaxation; stored energies 
Šilhavý, M. Rank 1 convex hulls of isotropic functions in dimension 2 by 2. Mathematica Bohemica, Tome 126 (2001) no. 2, pp. 521-529. doi: 10.21136/MB.2001.134029
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[1] Ball, J. M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977), 337–403. | DOI | MR | Zbl

[2] Buttazzo, G., Dacorogna, B., Gangbo, W.: On the envelopes of functions depending on singular values of matrices. Bolletino U. M. I. 7 (1994), 17–35. | MR

[3] Dacorogna, B.: Direct Methods in the Calculus of Variations. Springer, Berlin, 1990. | MR

[4] Kohn, R. V.: The relaxation of a double-well energy. Continuum Mech. Thermodyn. 3 (1991), 3–236. | DOI | MR | Zbl

[5] Kohn, R. V., Strang, G.: Optimal design and relaxation of variational problems, I, II, III. Comm. Pure Appl. Math. 39 (1986), 113–137, 139–182, 353–377. | DOI | MR

[6] Morrey, Jr, C. B.: Multiple Integrals in the Calculus of Variations. Springer, New York, 1966. | MR

[7] Rosakis, P.: Characterizat on of convex isotropic functions. J. Elasticity 49 (1997), 257–267. | DOI | MR

[8] Roubíček, T.: Relaxation in optimization theory and variational calculus. W. de Gruyter, Berlin, 1997. | MR

[9] Šilhavý, M.: The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin, 1997. | MR

[10] Šilhavý, M.: Convexity conditions for rotationally invariant functions in two dimensions. Applied Nonlinear Analysis, A. Sequeira et al. (eds.), Kluwer Academic, New York, 1999, pp. 513–530. | MR

[11] Šilhavý, M.: Rank 1 convex hulls of rotationally invariant functions. In preparation.

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