A-Quasiconvexity: Relaxation and Homogenization

Andrea Braides; Irene Fonseca; Giovanni Leoni

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

  • Volume: 5, page 539-577
  • ISSN: 1292-8119

Abstract

top
Integral representation of relaxed energies and of Γ-limits of functionals ( u , v ) ↦ ∫ Ω f ( x , u ( x ) , v ( x ) ) d x are obtained when sequences of fields v may develop oscillations and are constrained to satisfy a system of first order linear partial differential equations. This framework includes the treatement of divergence-free fields, Maxwell's equations in micromagnetics, and curl-free fields. In the latter case classical relaxation theorems in W1,p, are recovered.

How to cite

top

Braides, Andrea, Fonseca, Irene, and Leoni, Giovanni. "A-Quasiconvexity: Relaxation and Homogenization." ESAIM: Control, Optimisation and Calculus of Variations 5 (2010): 539-577. <http://eudml.org/doc/197360>.

@article{Braides2010,
abstract = { Integral representation of relaxed energies and of Γ-limits of functionals $$ (u,v)\mapsto \int\_\Omega f( x,u(x),v(x))\,dx $$ are obtained when sequences of fields v may develop oscillations and are constrained to satisfy a system of first order linear partial differential equations. This framework includes the treatement of divergence-free fields, Maxwell's equations in micromagnetics, and curl-free fields. In the latter case classical relaxation theorems in W1,p, are recovered. },
author = {Braides, Andrea, Fonseca, Irene, Leoni, Giovanni},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {$\{\cal A\}$-quasiconvexity; equi-integrability; Young measure; relaxation; Γ-convergence; homogenization.; Young measures; Gamma-convergence; differential constraint; integral representation of the relaxed energy},
language = {eng},
month = {3},
pages = {539-577},
publisher = {EDP Sciences},
title = {A-Quasiconvexity: Relaxation and Homogenization},
url = {http://eudml.org/doc/197360},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Braides, Andrea
AU - Fonseca, Irene
AU - Leoni, Giovanni
TI - A-Quasiconvexity: Relaxation and Homogenization
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 5
SP - 539
EP - 577
AB - Integral representation of relaxed energies and of Γ-limits of functionals $$ (u,v)\mapsto \int_\Omega f( x,u(x),v(x))\,dx $$ are obtained when sequences of fields v may develop oscillations and are constrained to satisfy a system of first order linear partial differential equations. This framework includes the treatement of divergence-free fields, Maxwell's equations in micromagnetics, and curl-free fields. In the latter case classical relaxation theorems in W1,p, are recovered.
LA - eng
KW - ${\cal A}$-quasiconvexity; equi-integrability; Young measure; relaxation; Γ-convergence; homogenization.; Young measures; Gamma-convergence; differential constraint; integral representation of the relaxed energy
UR - http://eudml.org/doc/197360
ER -

References

top
  1. E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal.86 (1984) 125 -145.  
  2. M. Amar and V. De Cicco, Relaxation of quasi-convex integrals of arbitrary order. Proc. Roy. Soc. Edinburgh Sect. A124 (1994) 927-946.  
  3. L. Ambrosio, S. Mortola and V.M. Tortorelli, Functionals with linear growth defined on vector valued BV functions. J. Math. Pures Appl.70 (1991) 269-323.  
  4. E.J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory. SIAM J. Control Optim.22 (1984) 570-598.  
  5. J.M. Ball, A version of the fundamental theorem for Young measures, in PDE's and Continuum Models of Phase Transitions, edited by M. Rascle, D. Serre and M. Slemrod. Springer-Verlag, Berlin, Lecture Notes in Phys.344 (1989) 207-215.  
  6. J.M. Ball and F. Murat, Remarks on Chacon's biting lemma. Proc. Amer. Math. Soc.107 (1989) 655-663.  
  7. H. Berliocchi and J.M. Lasry, Intégrands normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France101 (1973) 129-184.  
  8. A. Braides, A homogenization theorem for weakly almost periodic functionals, Rend. Accad. Naz. Sci. XL Mem. Sci. Fis. Natur. (5)104 (1986) 261-281.  
  9. A. Braides, Relaxation of functionals with constraints on the divergence. Ann. Univ. Ferrara Ser. VII (N.S.)33 (1987) 157-177.  
  10. A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Clarendon Press, Oxford (1998).  
  11. A. Braides, A. Defranceschi and E. Vitali, Homogenization of free-discontinuity problems. Arch. Rational Mech. Anal.135 (1996) 297-356.  
  12. G. Buttazzo, Semicontinuity, relaxation and integral representation problems in the Calculus of Variations. Longman, Harlow, Pitman Res. Notes Math. Ser.207 (1989).  
  13. B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin (1989).  
  14. B. Dacorogna, Weak Continuity and Weak Lower Semicontinuity for Nonlinear Functionals. Springer-Verlag, Berlin, Lecture Notes in Math.922 (1982).  
  15. G. Dal Maso, An Introduction to Γ-Convergence. Birkhäuser, Boston (1993).  
  16. G. Dal Maso, A. Defranceschi and E. Vitali (private communication).  
  17. A. De Simone, Energy minimizers for large ferromagnetic bodies. Arch. Rational Mech. Anal.125 (1993) 99-143.  
  18. I. Fonseca, The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures Appl.67 (1988) 175-195.  
  19. I. Fonseca, G. Leoni, J. Malý and R. Paroni (in preparation).  
  20. I. Fonseca and S. Müller, Quasiconvex integrands and lower semicontinuity in L1. SIAM J. Math. Anal.23 (1992) 1081-1098.  
  21. I. Fonseca and S. Müller, Relaxation of quasiconvex functionals in B V ( Ω , ℝ p ) for integrands f ( x , u , ∇ u ) . Arch. Rational Mech. Anal.123 (1993) 1-49.  
  22. I. Fonseca and S. Müller, A-quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal.30 (1999) 1355-1390.  
  23. N. Fusco, Quasi-convessitá e semicontinuitá per integrali di ordine superiore. Ricerche Mat.29 (1980) 307-323.  
  24. M. Giaquinta and G. Modica, Regularity results for some classes of higher order non linear elliptic systems. J. reine angew. Math.311/312 (1979) 145-169.  
  25. M. Guidorzi and L. Poggiolini, Lower semicontinuity for quasiconvex integrals of higher order. NoDEA Nonlinear Differential Equations Appl.6 (1999) 227-246.  
  26. J. Kristensen, Finite functionals and Young measures generated by gradients of Sobolev functions. Mathematical Institute, Technical University of Denmark, Mat-Report No. 1994-34 (1994).  
  27. P. Marcellini, Approximation of quasiconvex functions and semicontinuity of multiple integrals. Manuscripta Math.51 (1985) 1-28.  
  28. P. Marcellini and C. Sbordone, Semicontinuity problems in the Calculus of Variations. Nonlinear Anal.4 (1980) 241-257.  
  29. N.G. Meyers, Quasi-convexity and lower semi-continuity of multiple variational integrals of any order. Trans. Amer. Math. Soc.119 (1965) 125-149.  
  30. C.B. Morrey, Multiple Integrals in the calculus of Variations. Springer-Verlag, Berlin (1966).  
  31. S. Müller, Variational models for microstructures and phase transitions, in Calculus of Variations and Geometric Evolution Problems, edited by S. Hildebrant et al. Springer-Verlag, Berlin, Lecture Notes in Math.1713 (1999) 85-210.  
  32. F. Murat, Compacité par compensation : condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sw. (4)8 (1981) 68-102.  
  33. P. Pedregal, Parametrized Measures and Variational Principles. Birkhäuser, Boston (1997).  
  34. L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, edited by R. Knops. Longman, Harlow, Pitman Res. Notes Math. Ser.39 (1979) 136-212.  
  35. L. Tartar, The compensated compactness method applied to systems of conservation laws, in Systems of Nonlinear Partial Differential Eq., edited by J.M. Ball. Riedel (1983).  
  36. L. Tartar, Étude des oscillations dans les équations aux dérivées partielles nonlinéaires. Springer-Verlag, Berlin, Lectures Notes in Phys.195 (1984) 384-412.  
  37. L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A115 (1990) 193-230.  
  38. L. Tartar, On mathematical tools for studying partial differential equations of continuum physics: H-measures and Young measures, in Developments in Partial Differential Equations and Applications to Mathematical Physics, edited by Buttazzo, Galdi and Zanghirati. Plenum, New York (1991).  
  39. L. Tartar, Some remarks on separately convex functions, in Microstructure and Phase Transitions, edited by D. Kinderlehrer, R.D. James, M. Luskin and J.L. Ericksen. Springer-Verlag, IMA J. Math. Appl.54 (1993) 191-204.  
  40. L.C. Young, Lectures on Calculus of Variations and Optimal Control Theory. W.B. Saunders (1969).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.