Evolutionary problems in non-reflexive spaces

Martin Kružík; Johannes Zimmer

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 16, Issue: 1, page 1-22
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Kružík, Martin, and Zimmer, Johannes. "Evolutionary problems in non-reflexive spaces." ESAIM: Control, Optimisation and Calculus of Variations 16.1 (2010): 1-22. <http://eudml.org/doc/250727>.

@article{Kružík2010,
abstract = { 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. },
author = {Kružík, Martin, Zimmer, Johannes},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Concentrations; energetic solution; energies with linear growth; oscillations; relaxation; concentrations; energies with linear growth},
language = {eng},
month = {1},
number = {1},
pages = {1-22},
publisher = {EDP Sciences},
title = {Evolutionary problems in non-reflexive spaces},
url = {http://eudml.org/doc/250727},
volume = {16},
year = {2010},
}

TY - JOUR
AU - Kružík, Martin
AU - Zimmer, Johannes
TI - Evolutionary problems in non-reflexive spaces
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/1//
PB - EDP Sciences
VL - 16
IS - 1
SP - 1
EP - 22
AB - 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.
LA - eng
KW - Concentrations; energetic solution; energies with linear growth; oscillations; relaxation; concentrations; energies with linear growth
UR - http://eudml.org/doc/250727
ER -

References

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  1. L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000).  
  2. J.M. Ball, 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.  
  3. J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal.100 (1987) 13–52.  
  4. S. Conti and M. Ortiz, Dislocation microstructures and the effective behavior of single crystals. Arch. Ration. Mech. Anal.176 (2005) 103–147.  
  5. G. Dal Maso, A. DeSimone, M.G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening. Technical report, Scuola Normale Superiore, Pisa (2006).  
  6. G. Dal Maso, A. DeSimone, M.G. Mora and M. Morini, Time-dependent systems of generalized Young measures. Netw. Heterog. Media2 (2007) 1–36 (electronic).  
  7. R.J. DiPerna and A.J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Phys.108 (1987) 667–689.  
  8. R. Engelking, General topology. Translated from the Polish by the author, Monografie Matematyczne60 [Mathematical Monographs]. PWN – Polish Scientific Publishers, Warsaw (1977).  
  9. L.C. Evans, Partial differential equations, Graduate Studies in Mathematics19. American Mathematical Society, Providence, USA (1998).  
  10. L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, USA (1992).  
  11. G.B. Folland, Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics. John Wiley & Sons Inc., New York, first edition (1999); Wiley-Interscience, second edition.  
  12. G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math.595 (2006) 55–91.  
  13. A. Kałamajska and M. Kružík, Oscillations and concentrations in sequences of gradients. ESAIM: COCV14 (2008) 71–104.  
  14. M. Kružík and T. Roubíček, On the measures of DiPerna and Majda. Math. Bohem.122 (1997) 383–399.  
  15. A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems. Calc. Var. Partial Differential Equations22 (2005) 73–99.  
  16. A. Mielke, Evolution of rate-independent systems, in Evolutionary equationsII, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam (2005) 461–559.  
  17. A. Mielke and T. Roubíček, A rate-independent model for inelastic behavior of shape-memory alloys. Multiscale Model. Simul.1 (2003) 571–597 (electronic).  
  18. A. Mielke, F. Theil and V.I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Ration. Mech. Anal.162 (2002) 137–177.  
  19. M. Ortiz and E.A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids47 (1999) 397–462.  
  20. T. Roubíček, Relaxation in optimization theory and variational calculus, de Gruyter Series in Nonlinear Analysis and Applications4. Walter de Gruyter & Co., Berlin (1997).  
  21. M.E. Schonbek, Convergence of solutions to nonlinear dispersive equations. Comm. Partial Differential Equations7 (1982) 959–1000.  
  22. J. Souček, Spaces of functions on domain Ω , whose k -th derivatives are measures defined on Ω ¯ . Časopis Pěst. Mat.97 (1972) 10–46.  
  23. L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics: Heriot-Watt SymposiumIV, Pitman, Boston, USA (1979) 136–212.  

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