Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-23T03:55:24.674Z Has data issue: false hasContentIssue false
  • English
  • Français

A-Quasiconvexity: Relaxation and Homogenization

Published online by Cambridge University Press:  15 August 2002

Andrea Braides ,
Irene Fonseca  and
Giovanni Leoni
Andrea Braides
Affiliation:
SISSA, Trieste, Italy; [email protected].
Irene Fonseca
Affiliation:
Department of MathematicalSciences, Carnegie-Mellon University, Pittsburgh, PA, U.S.A.; [email protected].
Giovanni Leoni
Affiliation:
Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale, Alessandria, Italy; [email protected].
Get access

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.

Keywords

${\cal A}$-quasiconvexityequi-integrabilityYoung measurerelaxationΓ-convergencehomogenization.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 5 , 2000 , pp. 539 - 577
Copyright
© EDP Sciences, SMAI, 2000

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

Acerbi, E. and Fusco, N., Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984) 125 -145. CrossRef
Amar, M. and De Cicco, V., Relaxation of quasi-convex integrals of arbitrary order. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 927-946. CrossRef
Ambrosio, L., Mortola, S. and Tortorelli, V.M., Functionals with linear growth defined on vector valued BV functions. J. Math. Pures Appl. 70 (1991) 269-323.
Balder, E.J., A general approach to lower semicontinuity and lower closure in optimal control theory. SIAM J. Control Optim. 22 (1984) 570-598. CrossRef
Ball, J.M., 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. CrossRef
Ball, J.M. and Murat, F., Remarks on Chacon's biting lemma. Proc. Amer. Math. Soc. 107 (1989) 655-663.
Berliocchi, H. and Lasry, J.M., Intégrands normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France 101 (1973) 129-184. CrossRef
Braides, A., A homogenization theorem for weakly almost periodic functionals, Rend. Accad. Naz. Sci. XL Mem. Sci. Fis. Natur. (5) 104 (1986) 261-281.
Braides, A., Relaxation of functionals with constraints on the divergence. Ann. Univ. Ferrara Ser. VII (N.S.) 33 (1987) 157-177.
A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Clarendon Press, Oxford (1998).
Braides, A., Defranceschi, A. and Vitali, E., Homogenization of free-discontinuity problems. Arch. Rational Mech. Anal. 135 (1996) 297-356. CrossRef
G. Buttazzo, Semicontinuity, relaxation and integral representation problems in the Calculus of Variations. Longman, Harlow, Pitman Res. Notes Math. Ser. 207 (1989).
B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin (1989).
B. Dacorogna, Weak Continuity and Weak Lower Semicontinuity for Nonlinear Functionals. Springer-Verlag, Berlin, Lecture Notes in Math. 922 (1982).
G. Dal Maso, An Introduction to Γ-Convergence. Birkhäuser, Boston (1993).
G. Dal Maso, A. Defranceschi and E. Vitali (private communication).
De Simone, A., Energy minimizers for large ferromagnetic bodies. Arch. Rational Mech. Anal. 125 (1993) 99-143. CrossRef
Fonseca, I., The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures Appl. 67 (1988) 175-195.
I. Fonseca, G. Leoni, J. Malý and R. Paroni (in preparation).
Fonseca, I. and Müller, S., Quasiconvex integrands and lower semicontinuity in L 1. SIAM J. Math. Anal. 23 (1992) 1081-1098. CrossRef
Fonseca, I. and Müller, S., Relaxation of quasiconvex functionals in $BV(\Omega,\mathbb R ^p)$ for integrands $f(x,u,\nabla u)$ . Arch. Rational Mech. Anal. 123 (1993) 1-49. CrossRef
Fonseca, I. and Müller, S., A-quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal. 30 (1999) 1355-1390. CrossRef
Fusco, N., Quasi-convessitá e semicontinuitá per integrali di ordine superiore. Ricerche Mat. 29 (1980) 307-323.
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.
M. Guidorzi and L. Poggiolini, Lower semicontinuity for quasiconvex integrals of higher order. NoDEA Nonlinear Differential Equations Appl. 6 (1999) 227-246.
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).
Marcellini, P., Approximation of quasiconvex functions and semicontinuity of multiple integrals. Manuscripta Math. 51 (1985) 1-28. CrossRef
Marcellini, P. and Sbordone, C., Semicontinuity problems in the Calculus of Variations. Nonlinear Anal. 4 (1980) 241-257. CrossRef
Meyers, N.G., Quasi-convexity and lower semi-continuity of multiple variational integrals of any order. Trans. Amer. Math. Soc. 119 (1965) 125-149. CrossRef
C.B. Morrey, Multiple Integrals in the calculus of Variations. Springer-Verlag, Berlin (1966).
Müller, S., 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. CrossRef
Murat, F., 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.
P. Pedregal, Parametrized Measures and Variational Principles. Birkhäuser, Boston (1997).
Tartar, L., 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.
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).
Tartar, L., Étude des oscillations dans les équations aux dérivées partielles nonlinéaires. Springer-Verlag, Berlin, Lectures Notes in Phys. 195 (1984) 384-412. CrossRef
Tartar, L., H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 115 (1990) 193-230. CrossRef
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).
Tartar, L., 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.
L.C. Young, Lectures on Calculus of Variations and Optimal Control Theory. W.B. Saunders (1969).