Rate-independent problems are considered, where the stored energy density is a function of the gradient. The stored energy density may not be quasiconvex and is assumed to grow linearly. Moreover, arbitrary behaviour at infinity is allowed. In particular, the stored energy density is not required to coincide at infinity with a positively 1-homogeneous function. The existence of a rate-independent process is shown in the so-called energetic formulation.
Keywords: concentrations, energetic solution, energies with linear growth, oscillations, relaxation
@article{COCV_2010__16_1_1_0,
author = {Kru\v{z}{\'\i}k, Martin and Zimmer, Johannes},
title = {Evolutionary problems in non-reflexive spaces},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1--22},
year = {2010},
publisher = {EDP Sciences},
volume = {16},
number = {1},
doi = {10.1051/cocv:2008060},
mrnumber = {2598085},
zbl = {1183.49015},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv:2008060/}
}
TY - JOUR AU - Kružík, Martin AU - Zimmer, Johannes TI - Evolutionary problems in non-reflexive spaces JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 1 EP - 22 VL - 16 IS - 1 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv:2008060/ DO - 10.1051/cocv:2008060 LA - en ID - COCV_2010__16_1_1_0 ER -
%0 Journal Article %A Kružík, Martin %A Zimmer, Johannes %T Evolutionary problems in non-reflexive spaces %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 1-22 %V 16 %N 1 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv:2008060/ %R 10.1051/cocv:2008060 %G en %F COCV_2010__16_1_1_0
Kružík, Martin; Zimmer, Johannes. Evolutionary problems in non-reflexive spaces. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 1, pp. 1-22. doi: 10.1051/cocv:2008060
[1] , and , Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000). | Zbl
[2] , A version of the fundamental theorem for Young measures, in PDEs and continuum models of phase transitions (Nice, 1988), M. Rascle, D. Serre and M. Slemrod Eds., Springer, Berlin (1989) 207-215. | Zbl
[3] and , Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100 (1987) 13-52. | Zbl
[4] and , Dislocation microstructures and the effective behavior of single crystals. Arch. Ration. Mech. Anal. 176 (2005) 103-147. | Zbl | MR
[5] , , and , A vanishing viscosity approach to quasistatic evolution in plasticity with softening. Technical report, Scuola Normale Superiore, Pisa (2006). | Zbl
[6] , , and , Time-dependent systems of generalized Young measures. Netw. Heterog. Media 2 (2007) 1-36 (electronic). | Zbl | MR
[7] and , Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Phys. 108 (1987) 667-689. | Zbl | MR
[8] , General topology. Translated from the Polish by the author, Monografie Matematyczne 60 [Mathematical Monographs]. PWN - Polish Scientific Publishers, Warsaw (1977). | Zbl | MR
[9] , Partial differential equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence, USA (1998). | Zbl | MR
[10] and , Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, USA (1992). | Zbl
[11] , Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics. John Wiley & Sons Inc., New York, first edition (1999); Wiley-Interscience, second edition. | Zbl
[12] and , Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math. 595 (2006) 55-91. | Zbl
[13] and , Oscillations and concentrations in sequences of gradients. ESAIM: COCV 14 (2008) 71-104. | Zbl | Numdam
[14] and , On the measures of DiPerna and Majda. Math. Bohem. 122 (1997) 383-399. | Zbl
[15] and , Existence results for energetic models for rate-independent systems. Calc. Var. Partial Differential Equations 22 (2005) 73-99. | Zbl
[16] , Evolution of rate-independent systems, in Evolutionary equations II, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam (2005) 461-559. | Zbl
[17] and , A rate-independent model for inelastic behavior of shape-memory alloys. Multiscale Model. Simul. 1 (2003) 571-597 (electronic). | Zbl
[18] , and , A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Ration. Mech. Anal. 162 (2002) 137-177. | Zbl
[19] and , Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids 47 (1999) 397-462. | Zbl
[20] , Relaxation in optimization theory and variational calculus, de Gruyter Series in Nonlinear Analysis and Applications 4. Walter de Gruyter & Co., Berlin (1997). | Zbl
[21] , Convergence of solutions to nonlinear dispersive equations. Comm. Partial Differential Equations 7 (1982) 959-1000. | Zbl
[22] , Spaces of functions on domain , whose -th derivatives are measures defined on . Časopis Pěst. Mat. 97 (1972) 10-46. | Zbl
[23] , Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics: Heriot-Watt Symposium IV, Pitman, Boston, USA (1979) 136-212. | Zbl
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