Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T02:06:19.239Z Has data issue: false hasContentIssue false
  • English
  • Français

Evolutionary problems in non-reflexive spaces

Published online by Cambridge University Press:  21 October 2008

Martin Kružík  and
Johannes Zimmer
Martin Kružík
Affiliation:
Corresponding address: Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 4, 182 08 Praha 8, Czech Republic. Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 166  29 Praha 6, Czech Republic. [email protected]
Johannes Zimmer
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. [email protected]
Get access

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.

Keywords

Concentrationsenergetic solutionenergies with linear growthoscillationsrelaxation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 1 , January 2010 , pp. 1 - 22
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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).
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.
Ball, J.M. and James, R.D., Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100 (1987) 13–52. CrossRef
Conti, S. and Ortiz, M., Dislocation microstructures and the effective behavior of single crystals. Arch. Ration. Mech. Anal. 176 (2005) 103–147. CrossRef
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).
Dal Maso, G., DeSimone, A., Mora, M.G. and Morini, M., Time-dependent systems of generalized Young measures. Netw. Heterog. Media 2 (2007) 1–36 (electronic).
DiPerna, R.J. and Majda, A.J., Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Phys. 108 (1987) 667–689. CrossRef
R. Engelking, General topology. Translated from the Polish by the author, Monografie Matematyczne 60 [Mathematical Monographs]. PWN – Polish Scientific Publishers, Warsaw (1977).
L.C. Evans, Partial differential equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence, USA (1998).
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, USA (1992).
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.
Francfort, G. and Mielke, A., Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math. 595 (2006) 55–91.
Kałamajska, A. and Kružík, M., Oscillations and concentrations in sequences of gradients. ESAIM: COCV 14 (2008) 71–104. CrossRef
Kružík, M. and Roubíček, T., On the measures of DiPerna and Majda. Math. Bohem. 122 (1997) 383–399.
Mainik, A. and Mielke, A., Existence results for energetic models for rate-independent systems. Calc. Var. Partial Differential Equations 22 (2005) 73–99. CrossRef
A. Mielke, Evolution of rate-independent systems, in Evolutionary equations II, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam (2005) 461–559.
Mielke, A. and Roubíček, T., A rate-independent model for inelastic behavior of shape-memory alloys. Multiscale Model. Simul. 1 (2003) 571–597 (electronic). CrossRef
Mielke, A., Theil, F. and Levitas, V.I., A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Ration. Mech. Anal. 162 (2002) 137–177. CrossRef
Ortiz, M. and Repetto, E.A., Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids 47 (1999) 397–462. CrossRef
T. Roubíček, Relaxation in optimization theory and variational calculus, de Gruyter Series in Nonlinear Analysis and Applications 4. Walter de Gruyter & Co., Berlin (1997).
Schonbek, M.E., Convergence of solutions to nonlinear dispersive equations. Comm. Partial Differential Equations 7 (1982) 959–1000.
Souček, J., Spaces of functions on domain $\Omega $ , whose $k$ -th derivatives are measures defined on $\bar \Omega $ . Časopis Pěst. Mat. 97 (1972) 10–46.
L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics: Heriot-Watt Symposium IV, Pitman, Boston, USA (1979) 136–212.