Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T11:51:50.246Z Has data issue: false hasContentIssue false
  • English
  • Français

New convexity conditions in the calculus of variationsand compensated compactness theory

Published online by Cambridge University Press:  15 December 2005

Krzysztof Chełmiński  and
Agnieszka Kałamajska
Krzysztof Chełmiński
Affiliation:
Cardinal Stefan Wyszyński University, ul. Dewajtis 5, 01-815 Warszawa, Poland; chelminski@uksw.edu.pl University of Constance, Universitätsstr. 10, 78464 Konstanz, Germany
Agnieszka Kałamajska
Affiliation:
Institute of Mathematics, Warsaw University, ul. Banacha 2, 02–097 Warszawa, Poland; kalamajs@mimuw.edu.pl
Get access

Abstract

We consider the lower semicontinuous functional of the form $I_f(u)=\int_\Omega f(u){\rm d}x$ where u satisfies a given conservation law defined by differential operator of degree one with constant coefficients. We show that under certain constraints the well known Murat and Tartar's Λ-convexity condition for the integrand f extends to the new geometric conditions satisfied on four dimensional symplexes. Similar conditions on three dimensional symplexes were recently obtained by the second author. New conditions apply to quasiconvex functions.

Keywords

Quasiconvexityrank-one convexitysemicontinuity.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 1 , January 2006 , pp. 64 - 92
Copyright
© EDP Sciences, SMAI, 2006

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

Alibert, J.J. and Dacorogna, B., An example of a quasiconvex function that is not polyconvex in two dimensions two. Arch. Ration. Mech. Anal. 117 (1992) 155–166. CrossRef
Agmon, S., Maximum theorems for solutions of higher order elliptic equations. Bull. Am. Math. Soc. 66 (1960) 77–80. CrossRef
Agmon, S., Nirenberg, L. and Protter, M.H., A maximum principle for a class of hyperbolic equations and applications to equations of mixed elliptic–hyperbolic type. Commun. Pure Appl. Math. 6 (1953) 455-470. CrossRef
K. Astala, Analytic aspects of quasiconformality. Doc. Math. J. DMV, Extra Volume ICM, Vol. II (1998) 617–626.
Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1977) 337–403. CrossRef
J.M. Ball, Constitutive inequalities and existence theorems in nonlinear elastostatics. Nonlin. Anal. Mech., Heriot–Watt Symp. Vol. I, R. Knops Ed. Pitman, London (1977) 187–241.
Ball, J.M. and James, R.D., Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100 (1987) 13–52. CrossRef
J.M. Ball and R.D. James, Proposed experimental tests of a theory of fine microstructure and the two–well problem. Philos. Trans. R. Soc. Lond. 338(A) (1992) 389–450.
Ball, J.M., Kirchheim, B. and Kristensen, J., Regularity of quasiconvex envelopes. Calc. Var. Partial Differ. Equ. 11 (2000) 333–359. CrossRef
Ball, J.M. and Murat, F., Remarks on rank-one convexity and quasiconvexity. Ordinary and Partial Differential Equations, B.D. Sleeman and R.J. Jarvis Eds. Vol. III, Longman, New York. Pitman Res. Notes Math. Ser. 254 (1991) 25–37.
Ball, J.M., Currie, J.C. and Olver, P.J., Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal. 41 (1981) 135–174. CrossRef
Bitsadze, A.V., A system of nonlinear partial differential equations. Differ. Uravn. 15 (1979) 1267–1270 (in Russian).
Canfora, A., Teorema del massimo modulo e teorema di esistenza per il problema di Dirichlet relativo ai sistemi fortemente ellittici. Ric. Mat. 15 (1966) 249–294.
Casadio-Tarabusi, E., An algebraic characterization of quasiconvex functions. Ric. Mat. 42 (1993) 11–24.
Chipot, M. and Kinderlehrer, D., Equilibrium configurations of crystals. Arch. Ration. Mech. Anal. 103 (1988) 237–277. CrossRef
B. Dacorogna, Weak continuity and weak lower semicontinuity for nonlinear functionals. Berlin-Heidelberg-New York, Springer. Lect. Notes Math. 922 (1982).
B. Dacorogna, Direct methods in the calculus of variations. Springer, Berlin (1989).
Dacorogna, B., Douchet, J., Gangbo, W. and Rappaz, J., Some examples of rank–one convex functions in dimension two. Proc. R. Soc. Edinb. 114 (1990) 135–150. CrossRef
Dacorogna, B. and Haeberly, J.-P., Some numerical methods for the study of the convexity notions arising in the calculus of variations. M2AN 32 (1998) 153–175. CrossRef
Dolzmann, G., Numerical computation of rank–one convex envelopes. SIAM J. Numer. Anal. 36 (1999) 1621–1635. CrossRef
G. Dolzmann, Variational methods for crystalline microstructure–analysis and computation. Springer-Verlag, Berlin. Lect. Notes Math. 1803 (2003).
Dolzmann, G., Kirchheim, B. and Kristensen, J., Conditions for equality of hulls in the calculus of variations. Arch. Ration. Mech. Anal. 154 (2000) 93–100. CrossRef
Edelen, D.G.B., The null set of the Euler–Lagrange operator. Arch. Ration. Mech. Anal 11 (1962) 117–121. CrossRef
H. Federer, Geometric measure theory. Springer-Verlag, New York, Heldelberg (1969).
Fonseca, I. and Müller, S., A–quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal. 30 (1999) 1355–1390. CrossRef
L.E. Fraenkel, An introduction to maximum principles and symmetry in elliptic problems. Cambridge University Press, Cambridge (2000).
M. Giaquinta and E. Giusti, Quasi–minima, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1 (1984) 79–107.
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin–Heidelberg–New York (1977).
Iwaniec, T., Nonlinear Cauchy–Riemann operators in $\xR^{n}$ . Trans. Am. Math. Soc. 354 (2002) 1961–1995. CrossRef
T. Iwaniec, Integrability theory of the Jacobians. Lipshitz Lectures, preprint Univ. Bonn Sonderforschungsbereich 256 (1995).
T. Iwaniec, Nonlinear differential forms. Series in Lectures at the International School in Jyväskylä, published by Math. Inst. Univ. Jyväskylä (1998) 1–207.
Iwaniec, T. and Lutoborski, A., Integral estimates for null–lagrangians. Arch. Ration. Mech. Anal. 125 (1993) 25–79. CrossRef
Kałamajska, A., On $\Lambda $ –convexity conditions in the theory of lower semicontinuous functionals. J. Convex Anal. 10 (2003) 419–436.
Kałamajska, A., On new geometric conditions for some weakly lower semicontinuous functionals with applications to the rank-one conjecture of Morrey. Proc. R. Soc. Edinb. A 133 (2003) 1361–1377. CrossRef
B. Kirchheim, S. Müller and V. Šverák, Studing nonlinear pde by geometry in matrix space, in Geometric Analysis and Nonlinear Differential Equations, H. Karcher and S. Hildebrandt Eds. Springer (2003) 347–395.
Kohn, V. and Strang, G., Optimal design and relaxation of variational problems I. Commun. Pure Appl. Math. 39 (1986) 113–137. CrossRef
Kohn, V. and Strang, G., Optimal design and relaxation of variational problems II. Commun. Pure Appl. Math. 39 (1986) 139–182. CrossRef
J. Kolář, Non–compact lamination convex hulls. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20 (2003) 391–403.
J. Kristensen, On the non–locality of quasiconvexity. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16 (1999) 1–13.
Kružík, M., On the composition of quasiconvex functions and the transposition. J. Convex Anal. 6 (1999) 207–213.
Kružík, M., Bauer's maximum principle and hulls of sets. Calc. Var. Partial Differ. Equ. 11 (2000) 321–332.
S. Lang, Algebra. Addison–Wesley Publishing Company, New York (1965).
Le Dret, H. and Raoult, A., The quasiconvex envelope of the Saint–Venant–Kirchhoff stored energy function. Proc. R. Soc. Edinb. 125 (1995) 1179–1192. CrossRef
Leonetti, F., Maximum principle for vector–valued minimizers of some integral functionals. Boll. Unione Mat. Ital. 7 (1991) 51–56.
Lions, P.L., Jacobians and Hardy spaces. Ric. Mat. Suppl. 40 (1991) 255–260.
Luskin, M., On the computation of crystalline microstructure. Acta Numerica 5 (1996) 191–257. CrossRef
Manfredi, J.J., Weakly monotone functions. J. Geom. Anal. 4 (1994) 393–402. CrossRef
Marcellini, P., Quasiconvex quadratic forms in two dimensions. Appl. Math. Optimization 11 (1984) 183–189. CrossRef
M. Miranda, Maximum principles and minimal surfaces. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) XXV (1997) 667–681.
Morrey, C.B., Quasi–convexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2 (1952) 25–53. CrossRef
C.B. Morrey, Multiple integrals in the calculus of variations. Springer-Verlag, Berlin–Heidelberg–New York (1966).
F. Murat, Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978) 489–507.
F. Murat, A survey on compensated compactness. Contributions to modern calculus of variations, L. Cesari Ed. Longman, Harlow, Pitman Res. Notes Math. Ser. 148 (1987) 145–183.
F. Murat, Compacité par compensation; condition nécessaire et suffisante de continuité faible sous une hypothése de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981) 69–102.
Müller, S., A surprising higher integrability property of mappings with positive determinant. Bull. Am. Math. Soc. 21 (1989) 245–248. CrossRef
Müller, S., Variational models for microstructure and phase transitions, Collection: Calculus of variations and geometric evolution problems (Cetraro 1996), Springer, Berlin. Lect. Notes Math. 1713 (1999) 85–210. CrossRef
Müller, S., Rank–one convexity implies quasiconvexity on diagonal matrices. Int. Math. Res. Not. 20 (1999) 1087–1095.
S. Müller, Quasiconvexity is not invariant under transposition, in Proc. R. Soc. Edinb. 130 (2000) 389–395.
S. Müller and V. Šverák, Attainment results for the two–well problem by convex integration. Geom. Anal. and the Calc. Variations, J. Jost Ed. International Press (1996) 239–251.
S. Müller and V. Šverák, Unexpected solutions of first and second order partial differential equations. Doc. Math. J. DMV, Special volume Proc. ICM, Vol. II (1998) 691–702.
S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. of Math. (2) 157 (2003) 715–742.
S. Müller and V. Šverák, Convex integration with constrains and applications to phase transitions and partial differential equations. J. Eur. Math. Soc. (JEMS) 1/4 (1999) 393–422.
S. Müller and M.O. Rieger, V. Šverák, Parabolic systems with nowhere smooth solutions, preprint, http://www.math.cmu.edu/~nwOz/publications/02-CNA-014/014abs/
Parry, G.P., On the planar rank–one convexity condition. Proc. R. Soc. Edinb. A 125 (1995) 247–264. CrossRef
P. Pedregal, Parametrized measures and variational principles. Birkhäuser (1997).
Pedregal, P., Weak continuity and weak lower semicontinuity for some compensation operators. Proc. R. Soc. Edinb. A 113 (1989) 267–279. CrossRef
Pedregal, P., Laminates and microstructure. Eur. J. Appl. Math. 4 (1993) 121–149. CrossRef
Pedregal, P., Some remarks on quasiconvexity and rank–one convexity. Proc. R. Soc. Edinb. A 126 (1996) 1055–1065. CrossRef
Pedregal, P. and Šverák, V., A note on quasiconvexity and rank–one convexity for $2\times 2$ Matrices. J. Convex Anal. 5 (1998) 107–117.
Pipkin, A.C., Elastic materials with two preferred states. Q. J. Mech. Appl. Math. 44 (1991) 1–15. CrossRef
Robbin, J., Rogers, R.C. and Temple, B., On weak continuity and Hodge decomposition. Trans. Am. Math. Soc. 303 (1987) 609–618. CrossRef
T. Roubíuek, Relaxation in optimization theory and variational calculus. Berlin, W. de Gruyter (1997).
J. Sivaloganathan, Implications of rank one convexity. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5, 2 (1988) 99–118.
R. Stefaniuk, Numerical verification of certain property of quasiconvex function. MSC thesis, Warsaw University (2004).
Šverák, V., Examples of rank–one convex functions. Proc. R. Soc. Edinb. A 114 (1990) 237–242. CrossRef
V. Šverák, Quasiconvex functions with subquadratic growth. Proc. R. Soc. Lond. A 433 (1991), 723–725.
Šverák, V., Rank–one convexity does not imply quasiconvexity. Proc. R. Soc. Edinb. 120 (1992) 185–189. CrossRef
V. Šverák, Lower semicontinuity of variational integrals and compensated compactness, in Proc. of the Internaional Congress of Mathematicians, Zürich, Switzerland 1994, Birkhäuser Verlag, Basel, Switzerland (1995) 1153–1158.
V. Šverák, On the problem of two wells, in Microstructures and phase transitions, D. Kinderlehrer, R.D. James, M. Luskin and J. Ericksen Eds. Springer, IMA Vol. Appl. Math. 54 (1993) 183–189.
Tartar, L., Compensated compactness and applications to partial differential equations. Nonlinear Analysis and Mechanics: Heriot–Watt Symp., Vol. IV, R. Knops Ed. Pitman Res. Notes Math. 39 (1979) 136–212.
L. Tartar, The compensated compactness method applied to systems of conservation laws. Systems of Nonlinear Partial Differential Eq., J.M. Ball Ed. Reidel (1983) 263–285.
L. Tartar, Some remarks on separately convex functions, Microstructure and Phase Transitions, D. Kinderlehrer, R.D. James, M. Luskin and J.L. Ericksen Eds. Springer, IMA Vol. Math. Appl. 54 (1993) 191–204.
Yan, B., On rank–one convex and polyconvex conformal energy functions with slow growth. Proc. R. Soc. Edinb. 127 (1997) 651–663. CrossRef
K.W. Zhang, A construction of quasiconvex functions with linear growth at infinity. Ann. Sc. Norm. Super. Pisa, Cl. Sci. Ser. IV XIX (1992) 313–326.
K.W. Zhang, Biting theorems for Jacobians and their applications. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7 (1990) 345–365.
Zhang, K.W., On various semiconvex hulls in the calculus of variations. Calc. Var. Partial Differ. Equ. 6 (1998) 143–160. CrossRef