We consider the lower semicontinuous functional of the form where 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 -convexity condition for the integrand 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: quasiconvexity, rank-one convexity, semicontinuity
@article{COCV_2006__12_1_64_0, author = {Che{\l}mi\'nski, Krzysztof and Ka{\l}amajska, Agnieszka}, title = {New convexity conditions in the calculus of variations and compensated compactness theory}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {64--92}, publisher = {EDP-Sciences}, volume = {12}, number = {1}, year = {2006}, doi = {10.1051/cocv:2005034}, zbl = {1114.49019}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2005034/} }
TY - JOUR AU - Chełmiński, Krzysztof AU - Kałamajska, Agnieszka TI - New convexity conditions in the calculus of variations and compensated compactness theory JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2006 SP - 64 EP - 92 VL - 12 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2005034/ DO - 10.1051/cocv:2005034 LA - en ID - COCV_2006__12_1_64_0 ER -
%0 Journal Article %A Chełmiński, Krzysztof %A Kałamajska, Agnieszka %T New convexity conditions in the calculus of variations and compensated compactness theory %J ESAIM: Control, Optimisation and Calculus of Variations %D 2006 %P 64-92 %V 12 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2005034/ %R 10.1051/cocv:2005034 %G en %F COCV_2006__12_1_64_0
Chełmiński, Krzysztof; Kałamajska, Agnieszka. New convexity conditions in the calculus of variations and compensated compactness theory. ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 1, pp. 64-92. doi : 10.1051/cocv:2005034. http://archive.numdam.org/articles/10.1051/cocv:2005034/
[1] An example of a quasiconvex function that is not polyconvex in two dimensions two. Arch. Ration. Mech. Anal. 117 (1992) 155-166. | Zbl
and ,[2] Maximum theorems for solutions of higher order elliptic equations. Bull. Am. Math. Soc. 66 (1960) 77-80. | Zbl
,[3] 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. | Zbl
, and ,[4] Analytic aspects of quasiconformality. Doc. Math. J. DMV, Extra Volume ICM, Vol. II (1998) 617-626. | Zbl
,[5] Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1977) 337-403. | Zbl
,[6] Constitutive inequalities and existence theorems in nonlinear elastostatics. Nonlin. Anal. Mech., Heriot-Watt Symp. Vol. I, R. Knops Ed. Pitman, London (1977) 187-241. | Zbl
,[7] Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100 (1987) 13-52. | Zbl
and ,[8] Proposed experimental tests of a theory of fine microstructure and the two-well problem. Philos. Trans. R. Soc. Lond. 338(A) (1992) 389-450. | Zbl
and ,[9] Regularity of quasiconvex envelopes. Calc. Var. Partial Differ. Equ. 11 (2000) 333-359. | Zbl
, and ,[10] 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. | Zbl
and ,[11] Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal. 41 (1981) 135-174. | Zbl
, and ,[12] A system of nonlinear partial differential equations. Differ. Uravn. 15 (1979) 1267-1270 (in Russian). | Zbl
,[13] Teorema del massimo modulo e teorema di esistenza per il problema di Dirichlet relativo ai sistemi fortemente ellittici. Ric. Mat. 15 (1966) 249-294. | Zbl
,[14] An algebraic characterization of quasiconvex functions. Ric. Mat. 42 (1993) 11-24. | Zbl
,[15] Equilibrium configurations of crystals. Arch. Ration. Mech. Anal. 103 (1988) 237-277. | Zbl
and ,[16] Weak continuity and weak lower semicontinuity for nonlinear functionals. Berlin-Heidelberg-New York, Springer. Lect. Notes Math. 922 (1982). | MR | Zbl
,[17] Direct methods in the calculus of variations. Springer, Berlin (1989). | MR | Zbl
,[18] Some examples of rank-one convex functions in dimension two. Proc. R. Soc. Edinb. 114 (1990) 135-150. | Zbl
, , and ,[19] Some numerical methods for the study of the convexity notions arising in the calculus of variations. M2AN 32 (1998) 153-175. | Numdam | Zbl
and ,[20] Numerical computation of rank-one convex envelopes. SIAM J. Numer. Anal. 36 (1999) 1621-1635. | Zbl
,[21] Variational methods for crystalline microstructure-analysis and computation. Springer-Verlag, Berlin. Lect. Notes Math. 1803 (2003). | Zbl
,[22] Conditions for equality of hulls in the calculus of variations. Arch. Ration. Mech. Anal. 154 (2000) 93-100. | Zbl
, and ,[23] The null set of the Euler-Lagrange operator. Arch. Ration. Mech. Anal 11 (1962) 117-121. | Zbl
,[24] Geometric measure theory. Springer-Verlag, New York, Heldelberg (1969). | MR | Zbl
,[25] -quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal. 30 (1999) 1355-1390. | Zbl
and ,[26] An introduction to maximum principles and symmetry in elliptic problems. Cambridge University Press, Cambridge (2000). | MR | Zbl
,[27] Quasi-minima, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1 (1984) 79-107. | Numdam | Zbl
and ,[28] Elliptic partial differential equations of second order. Springer-Verlag, Berlin-Heidelberg-New York (1977). | Zbl
and ,[29] Nonlinear Cauchy-Riemann operators in . Trans. Am. Math. Soc. 354 (2002) 1961-1995. | Zbl
,[30] Integrability theory of the Jacobians. Lipshitz Lectures, preprint Univ. Bonn Sonderforschungsbereich 256 (1995).
,[31] Nonlinear differential forms. Series in Lectures at the International School in Jyväskylä, published by Math. Inst. Univ. Jyväskylä (1998) 1-207. | Zbl
,[32] Integral estimates for null-lagrangians. Arch. Ration. Mech. Anal. 125 (1993) 25-79. | Zbl
and ,[33] On -convexity conditions in the theory of lower semicontinuous functionals. J. Convex Anal. 10 (2003) 419-436. | Zbl
,[34] 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. | Zbl
,[35] 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.
, and ,[36] Optimal design and relaxation of variational problems I. Commun. Pure Appl. Math. 39 (1986) 113-137. | Zbl
and ,[37] Optimal design and relaxation of variational problems II. Commun. Pure Appl. Math. 39 (1986) 139-182. | Zbl
and ,[38] Non-compact lamination convex hulls. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20 (2003) 391-403. | Numdam | Zbl
,[39] On the non-locality of quasiconvexity. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16 (1999) 1-13. | Numdam | Zbl
,[40] On the composition of quasiconvex functions and the transposition. J. Convex Anal. 6 (1999) 207-213. | Zbl
,[41] Bauer's maximum principle and hulls of sets. Calc. Var. Partial Differ. Equ. 11 (2000) 321-332. | Zbl
,[42] Algebra. Addison-Wesley Publishing Company, New York (1965). | Zbl
,[43] The quasiconvex envelope of the Saint-Venant-Kirchhoff stored energy function. Proc. R. Soc. Edinb. 125 (1995) 1179-1192. | Zbl
and ,[44] Maximum principle for vector-valued minimizers of some integral functionals. Boll. Unione Mat. Ital. 7 (1991) 51-56. | Zbl
,[45] Jacobians and Hardy spaces. Ric. Mat. Suppl. 40 (1991) 255-260. | Zbl
,[46] On the computation of crystalline microstructure. Acta Numerica 5 (1996) 191-257. | Zbl
,[47] Weakly monotone functions. J. Geom. Anal. 4 (1994) 393-402. | Zbl
,[48] Quasiconvex quadratic forms in two dimensions. Appl. Math. Optimization 11 (1984) 183-189. | Zbl
,[49] Maximum principles and minimal surfaces. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) XXV (1997) 667-681. | Numdam | Zbl
,[50] Quasi-convexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2 (1952) 25-53. | Zbl
,[51] Multiple integrals in the calculus of variations. Springer-Verlag, Berlin-Heidelberg-New York (1966). | Zbl
,[52] Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978) 489-507. | Numdam | Zbl
,[53] 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.
,[54] 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. | Numdam | Zbl
,[55] A surprising higher integrability property of mappings with positive determinant. Bull. Am. Math. Soc. 21 (1989) 245-248. | Zbl
,[56] 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. | Zbl
,[57] Rank-one convexity implies quasiconvexity on diagonal matrices. Int. Math. Res. Not. 20 (1999) 1087-1095. | Zbl
,[58] Quasiconvexity is not invariant under transposition, in Proc. R. Soc. Edinb. 130 (2000) 389-395. | Zbl
,[59] Attainment results for the two-well problem by convex integration. Geom. Anal. and the Calc. Variations, J. Jost Ed. International Press (1996) 239-251. | Zbl
and ,[60] Unexpected solutions of first and second order partial differential equations. Doc. Math. J. DMV, Special volume Proc. ICM, Vol. II (1998) 691-702. | Zbl
and ,[61] Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. of Math. (2) 157 (2003) 715-742. | Zbl
and ,[62] Convex integration with constrains and applications to phase transitions and partial differential equations. J. Eur. Math. Soc. (JEMS) 1/4 (1999) 393-422. | Zbl
and ,[63] V. Šverák, Parabolic systems with nowhere smooth solutions, preprint, http://www.math.cmu.edu/~nwOz/publications/02-CNA-014/014abs/ | MR | Zbl
and ,[64] On the planar rank-one convexity condition. Proc. R. Soc. Edinb. A 125 (1995) 247-264. | Zbl
,[65] Parametrized measures and variational principles. Birkhäuser (1997). | MR | Zbl
,[66] Weak continuity and weak lower semicontinuity for some compensation operators. Proc. R. Soc. Edinb. A 113 (1989) 267-279. | Zbl
,[67] Laminates and microstructure. Eur. J. Appl. Math. 4 (1993) 121-149. | Zbl
,[68] Some remarks on quasiconvexity and rank-one convexity. Proc. R. Soc. Edinb. A 126 (1996) 1055-1065. | Zbl
,[69] A note on quasiconvexity and rank-one convexity for Matrices. J. Convex Anal. 5 (1998) 107-117. | Zbl
and ,[70] Elastic materials with two preferred states. Q. J. Mech. Appl. Math. 44 (1991) 1-15. | Zbl
,[71] On weak continuity and Hodge decomposition. Trans. Am. Math. Soc. 303 (1987) 609-618. | Zbl
, and ,[72] Relaxation in optimization theory and variational calculus. Berlin, W. de Gruyter (1997). | MR | Zbl
,[73] Implications of rank one convexity. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5, 2 (1988) 99-118. | Numdam | Zbl
,[74] Numerical verification of certain property of quasiconvex function. MSC thesis, Warsaw University (2004).
,[75] Examples of rank-one convex functions. Proc. R. Soc. Edinb. A 114 (1990) 237-242. | Zbl
,[76] Quasiconvex functions with subquadratic growth. Proc. R. Soc. Lond. A 433 (1991), 723-725. | Zbl
,[77] Rank-one convexity does not imply quasiconvexity. Proc. R. Soc. Edinb. 120 (1992) 185-189. | Zbl
,[78] 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. | Zbl
,[79] 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. | Zbl
,[80] 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. | Zbl
,[81] The compensated compactness method applied to systems of conservation laws. Systems of Nonlinear Partial Differential Eq., J.M. Ball Ed. Reidel (1983) 263-285. | Zbl
,[82] 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. | Zbl
,[83] On rank-one convex and polyconvex conformal energy functions with slow growth. Proc. R. Soc. Edinb. 127 (1997) 651-663. | Zbl
,[84] A construction of quasiconvex functions with linear growth at infinity. Ann. Sc. Norm. Super. Pisa, Cl. Sci. Ser. IV XIX (1992) 313-326. | Numdam | Zbl
,[85] Biting theorems for Jacobians and their applications. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7 (1990) 345-365. | Numdam | Zbl
,[86] On various semiconvex hulls in the calculus of variations. Calc. Var. Partial Differ. Equ. 6 (1998) 143-160. | Zbl
,Cited by Sources: