American Institute of Mathematical Sciences

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April  2011, 4(2): 467-482. doi: 10.3934/dcdss.2011.4.467

On certain convex compactifications for relaxation in evolution problems

Tomáš Roubíček 1,

1. 

Mathematical Institute, Charles University, Sokolovská 83, CZ-186 75 Praha 8

Received  March 2009 Revised  July 2009 Published  November 2010

A general-topological construction of limits of inverse systems is applied to convex compactifications and furthermore to special convex compactifications of Lebesgue-space-valued functions parameterized by time. Application to relaxation of quasistatic evolution in phase-change-type problems is outlined.
Keywords: generalized Young measures, Limits of inverse systems, Helly principle..
Mathematics Subject Classification: Primary: 54D35; Secondary: 26A45, 49J4.
Citation: Tomáš Roubíček. On certain convex compactifications for relaxation in evolution problems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 467-482. doi: 10.3934/dcdss.2011.4.467
References:
[1]

P. Alexandroff, Untersuchungen über gestalt und lage abgeschlossener mengen beliebiger dimension,, Math. Anal., 30 (1929), 101.   Google Scholar

[2]

S. Aubri, M. Fago and M. Ortiz, A constrained sequential-lamination algorithm for the simulation of sub-grid microstructure in martensitic materials,, Comp. Meth. in Appl. Mech. Engr., 192 (2003), 2823.   Google Scholar

[3]

V. Barbu and T. Precupanu, "Convexity and Optimization in Banach Spaces,", D. Reidel Publ., (1986).   Google Scholar

[4]

S. Bartels, C. Carstensen, K. Hackl and U. Hoppe, Effective relaxation for microstructure simulations: Algorithms and applications,, Comput. Methods Appl. Mech. Engrg., 193 (2004), 5143.  doi: 10.1016/j.cma.2003.12.065.  Google Scholar

[5]

S. A. Belov and V. V. Chistyakov, A selection principle or mappings of bounded variation,, J. Math. Anal. Appl., 249 (2000), 351.  doi: 10.1006/jmaa.2000.6844.  Google Scholar

[6]

F. Cagnetti and R. Toader, Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: A Young measure approach,, SISSA, ().   Google Scholar

[7]

C. Carstensen, K. Hackl and A. Mielke, Non-convex potentials and microstructures in finite-strain plasticity,, Proc. Royal Soc. London, 458 (2002), 299.   Google Scholar

[8]

V. V. Chistyakov, Mappings of bounded variations,, J. Dyn. Cont. Syst., 3 (1997), 261.  doi: 10.1007/BF02465896.  Google Scholar

[9]

V. V. Chistyakov and O. E. Galkin, Mappings of bounded $\Phi$-variation with arbitrary function $\Phi$,, J. Dyn. Cont. Syst., 4 (1998), 217.  doi: 10.1023/A:1022889902536.  Google Scholar

[10]

G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, Time-dependent systems of generalized Young measures,, Netw. Heterog. Media, 2 (2007), 1.   Google Scholar

[11]

G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening,, Arch. Rational Mech. Anal., 189 (2007), 469.  doi: 10.1007/s00205-008-0117-5.  Google Scholar

[12]

G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, Globally stable quasistatic evolution in plasticity with softening,, Netw. Heterog. Media, 3 (2008), 567.   Google Scholar

[13]

R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations,, Comm. Math. Phys., 108 (1987), 667.  doi: 10.1007/BF01214424.  Google Scholar

[14]

S. Eilenberg and N. Steenrod, "Foundation of Algebraic Topology,", Princeton, (1952).   Google Scholar

[15]

R. Engelking, "General Topology,", PWN, (1977).   Google Scholar

[16]

A. Fiaschi, A Young measure approach to a quasistatic evolution for a class of material models with nonconvex elastic energies,, ESAIM: Control, 15 (2009), 245.  doi: 10.1051/cocv:2008030.  Google Scholar

[17]

A. Fiaschi, A vanishing viscosity approach to a quasistatic evolution problem with nonconvex energy,, Ann. Inst. H. Poincaré, 26 (2009), 1055.   Google Scholar

[18]

A. Fiaschi, Rate-independent phase transitions in elastic materials: A Young-measure approach,, (preprint SISSA, ().   Google Scholar

[19]

S. Govindjee, A. Mielke and G. J. Hall, Free-energy of mixing for $n$-variant martensitic phase transformations using quasi-convex analysis,, J. Mech. Physics Solids, 50 (2002), 1897.  doi: 10.1016/S0022-5096(02)00009-1.  Google Scholar

[20]

K. Hackl and D. M. Kochmann, Relaxed potentials and evolution equations for inelastic microstructures,, in, (2008), 27.  doi: 10.1007/978-1-4020-9090-5_3.  Google Scholar

[21]

B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés., J. Mécanique, 14 (1975), 39.   Google Scholar

[22]

E. Helly, Über lineare funktionaloperationen,, Sitzungsberichte der Math.-Natur. Klasse der Kaiserlichen Akademie der Wissenschaften, 121 (1912), 265.   Google Scholar

[23]

M. Kružík, A. Mielke and T. Roubíček, Modelling of microstructure and its evolution in shape-memory-alloy single-crystals, in particular in CuAlNi,, Meccanica, 40 (2005), 389.  doi: 10.1007/s11012-005-2106-1.  Google Scholar

[24]

M. Kružík and A. Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism,, SIAM Rev., 48 (2006), 439.  doi: 10.1137/S0036144504446187.  Google Scholar

[25]

M. Kružík and J. Zimmer, Evolutionary problems in non-reflexive spaces,, (preprint no. 5/07, (2007).  doi: 10.1051/cocv:2008060.  Google Scholar

[26]

M. Kružík and J. Zimmer, A model of shape-memory alloys accounting for plasiticity., (preprint no. 20/08, (2008).   Google Scholar

[27]

M. Kružík and J. Zimmer, A note on time-dependent DiPerna-Majda measures,, (preprint no. 19/08, (2008).   Google Scholar

[28]

S. Lefschetz, On compact spaces,, Math. Anal., 32 (1931), 521.  doi: 10.2307/1968249.  Google Scholar

[29]

A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems,, Calc. Var. PDEs, 22 (2005), 73.   Google Scholar

[30]

A. Mielke, Deriving new evolution equations for microstructures via relaxation of variational incremental problems,, Comput. Methods Appl. Mech. Engrg., 193 (2004), 5095.  doi: 10.1016/j.cma.2004.07.003.  Google Scholar

[31]

A. Mielke, Evolution of rate-independent systems,, in, (2005), 461.   Google Scholar

[32]

A. Mielke, A mathematical framework for generalized standard materials in the rate-independent case,, in, 28 (2006), 351.   Google Scholar

[33]

A. Mielke and T. Roubíček, A rate-independent model for inelastic behavior of shape-memory alloys,, Multiscale Model. Simul., 1 (2003), 571.  doi: 10.1137/S1540345903422860.  Google Scholar

[34]

A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonlin. Diff. Eq. Appl., 11 (2004), 151.   Google Scholar

[35]

A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle,, Archive Rat. Mech. Anal., 162 (2002), 137.  doi: 10.1007/s002050200194.  Google Scholar

[36]

T. Roubíček, Convex compactifications and special extensions of optimization problems,, Nonlinear Anal., 16 (1991), 1117.  doi: 10.1016/0362-546X(91)90199-B.  Google Scholar

[37]

T. Roubíček, "Relaxation in Optimization Theory and Variational Calculus,", W. de Gruyter, (1997).   Google Scholar

[38]

T. Roubíček, Convex locally compact extensions of Lebesgue spaces and their applications,, in:, (1999), 237.   Google Scholar

[39]

T. Roubíček and K.-H. Hoffmann, About the concept of measure-valued solutions to distributed parameter systems,, Math. Methods Appl. Sci., 18 (1995), 671.  doi: 10.1002/mma.1670180902.  Google Scholar

[40]

T. Roubíček and M. Kružík, Microstructure evolution model in micromagnetics,, Z. angew. Math. Physik, 55 (2004), 159.   Google Scholar

[41]

A. Tychonoff, Über die topologische Erweiterung von Räumen,, Math. Annalen, 102 (1930), 544.  doi: 10.1007/BF01782364.  Google Scholar

show all references

References:
[1]

P. Alexandroff, Untersuchungen über gestalt und lage abgeschlossener mengen beliebiger dimension,, Math. Anal., 30 (1929), 101.   Google Scholar

[2]

S. Aubri, M. Fago and M. Ortiz, A constrained sequential-lamination algorithm for the simulation of sub-grid microstructure in martensitic materials,, Comp. Meth. in Appl. Mech. Engr., 192 (2003), 2823.   Google Scholar

[3]

V. Barbu and T. Precupanu, "Convexity and Optimization in Banach Spaces,", D. Reidel Publ., (1986).   Google Scholar

[4]

S. Bartels, C. Carstensen, K. Hackl and U. Hoppe, Effective relaxation for microstructure simulations: Algorithms and applications,, Comput. Methods Appl. Mech. Engrg., 193 (2004), 5143.  doi: 10.1016/j.cma.2003.12.065.  Google Scholar

[5]

S. A. Belov and V. V. Chistyakov, A selection principle or mappings of bounded variation,, J. Math. Anal. Appl., 249 (2000), 351.  doi: 10.1006/jmaa.2000.6844.  Google Scholar

[6]

F. Cagnetti and R. Toader, Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: A Young measure approach,, SISSA, ().   Google Scholar

[7]

C. Carstensen, K. Hackl and A. Mielke, Non-convex potentials and microstructures in finite-strain plasticity,, Proc. Royal Soc. London, 458 (2002), 299.   Google Scholar

[8]

V. V. Chistyakov, Mappings of bounded variations,, J. Dyn. Cont. Syst., 3 (1997), 261.  doi: 10.1007/BF02465896.  Google Scholar

[9]

V. V. Chistyakov and O. E. Galkin, Mappings of bounded $\Phi$-variation with arbitrary function $\Phi$,, J. Dyn. Cont. Syst., 4 (1998), 217.  doi: 10.1023/A:1022889902536.  Google Scholar

[10]

G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, Time-dependent systems of generalized Young measures,, Netw. Heterog. Media, 2 (2007), 1.   Google Scholar

[11]

G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening,, Arch. Rational Mech. Anal., 189 (2007), 469.  doi: 10.1007/s00205-008-0117-5.  Google Scholar

[12]

G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, Globally stable quasistatic evolution in plasticity with softening,, Netw. Heterog. Media, 3 (2008), 567.   Google Scholar

[13]

R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations,, Comm. Math. Phys., 108 (1987), 667.  doi: 10.1007/BF01214424.  Google Scholar

[14]

S. Eilenberg and N. Steenrod, "Foundation of Algebraic Topology,", Princeton, (1952).   Google Scholar

[15]

R. Engelking, "General Topology,", PWN, (1977).   Google Scholar

[16]

A. Fiaschi, A Young measure approach to a quasistatic evolution for a class of material models with nonconvex elastic energies,, ESAIM: Control, 15 (2009), 245.  doi: 10.1051/cocv:2008030.  Google Scholar

[17]

A. Fiaschi, A vanishing viscosity approach to a quasistatic evolution problem with nonconvex energy,, Ann. Inst. H. Poincaré, 26 (2009), 1055.   Google Scholar

[18]

A. Fiaschi, Rate-independent phase transitions in elastic materials: A Young-measure approach,, (preprint SISSA, ().   Google Scholar

[19]

S. Govindjee, A. Mielke and G. J. Hall, Free-energy of mixing for $n$-variant martensitic phase transformations using quasi-convex analysis,, J. Mech. Physics Solids, 50 (2002), 1897.  doi: 10.1016/S0022-5096(02)00009-1.  Google Scholar

[20]

K. Hackl and D. M. Kochmann, Relaxed potentials and evolution equations for inelastic microstructures,, in, (2008), 27.  doi: 10.1007/978-1-4020-9090-5_3.  Google Scholar

[21]

B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés., J. Mécanique, 14 (1975), 39.   Google Scholar

[22]

E. Helly, Über lineare funktionaloperationen,, Sitzungsberichte der Math.-Natur. Klasse der Kaiserlichen Akademie der Wissenschaften, 121 (1912), 265.   Google Scholar

[23]

M. Kružík, A. Mielke and T. Roubíček, Modelling of microstructure and its evolution in shape-memory-alloy single-crystals, in particular in CuAlNi,, Meccanica, 40 (2005), 389.  doi: 10.1007/s11012-005-2106-1.  Google Scholar

[24]

M. Kružík and A. Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism,, SIAM Rev., 48 (2006), 439.  doi: 10.1137/S0036144504446187.  Google Scholar

[25]

M. Kružík and J. Zimmer, Evolutionary problems in non-reflexive spaces,, (preprint no. 5/07, (2007).  doi: 10.1051/cocv:2008060.  Google Scholar

[26]

M. Kružík and J. Zimmer, A model of shape-memory alloys accounting for plasiticity., (preprint no. 20/08, (2008).   Google Scholar

[27]

M. Kružík and J. Zimmer, A note on time-dependent DiPerna-Majda measures,, (preprint no. 19/08, (2008).   Google Scholar

[28]

S. Lefschetz, On compact spaces,, Math. Anal., 32 (1931), 521.  doi: 10.2307/1968249.  Google Scholar

[29]

A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems,, Calc. Var. PDEs, 22 (2005), 73.   Google Scholar

[30]

A. Mielke, Deriving new evolution equations for microstructures via relaxation of variational incremental problems,, Comput. Methods Appl. Mech. Engrg., 193 (2004), 5095.  doi: 10.1016/j.cma.2004.07.003.  Google Scholar

[31]

A. Mielke, Evolution of rate-independent systems,, in, (2005), 461.   Google Scholar

[32]

A. Mielke, A mathematical framework for generalized standard materials in the rate-independent case,, in, 28 (2006), 351.   Google Scholar

[33]

A. Mielke and T. Roubíček, A rate-independent model for inelastic behavior of shape-memory alloys,, Multiscale Model. Simul., 1 (2003), 571.  doi: 10.1137/S1540345903422860.  Google Scholar

[34]

A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonlin. Diff. Eq. Appl., 11 (2004), 151.   Google Scholar

[35]

A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle,, Archive Rat. Mech. Anal., 162 (2002), 137.  doi: 10.1007/s002050200194.  Google Scholar

[36]

T. Roubíček, Convex compactifications and special extensions of optimization problems,, Nonlinear Anal., 16 (1991), 1117.  doi: 10.1016/0362-546X(91)90199-B.  Google Scholar

[37]

T. Roubíček, "Relaxation in Optimization Theory and Variational Calculus,", W. de Gruyter, (1997).   Google Scholar

[38]

T. Roubíček, Convex locally compact extensions of Lebesgue spaces and their applications,, in:, (1999), 237.   Google Scholar

[39]

T. Roubíček and K.-H. Hoffmann, About the concept of measure-valued solutions to distributed parameter systems,, Math. Methods Appl. Sci., 18 (1995), 671.  doi: 10.1002/mma.1670180902.  Google Scholar

[40]

T. Roubíček and M. Kružík, Microstructure evolution model in micromagnetics,, Z. angew. Math. Physik, 55 (2004), 159.   Google Scholar

[41]

A. Tychonoff, Über die topologische Erweiterung von Räumen,, Math. Annalen, 102 (1930), 544.  doi: 10.1007/BF01782364.  Google Scholar

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