New convexity conditions in the calculus of variations and compensated compactness theory
ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 1, p. 64-92
We consider the lower semicontinuous functional of the form ${I}_{f}\left(u\right)={\int }_{\Omega }f\left(u\right)\mathrm{d}x$ 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 $\Lambda$-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.
DOI : https://doi.org/10.1051/cocv:2005034
Classification:  49J10,  49J45
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},
publisher = {EDP-Sciences},
volume = {12},
number = {1},
year = {2006},
pages = {64-92},
doi = {10.1051/cocv:2005034},
zbl = {1114.49019},
language = {en},
url = {http://www.numdam.org/item/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://www.numdam.org/item/COCV_2006__12_1_64_0/

[1] J.J. Alibert and B. Dacorogna, An example of a quasiconvex function that is not polyconvex in two dimensions two. Arch. Ration. Mech. Anal. 117 (1992) 155-166. | Zbl 0761.26009

[2] S. Agmon, Maximum theorems for solutions of higher order elliptic equations. Bull. Am. Math. Soc. 66 (1960) 77-80. | Zbl 0091.27301

[3] S. Agmon, L. Nirenberg and M.H. Protter, 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 0090.07401

[4] K. Astala, Analytic aspects of quasiconformality. Doc. Math. J. DMV, Extra Volume ICM, Vol. II (1998) 617-626. | Zbl 0906.30019

[5] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1977) 337-403. | Zbl 0368.73040

[6] 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. | Zbl 0377.73043

[7] J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100 (1987) 13-52. | Zbl 0629.49020

[8] 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. | Zbl 0758.73009

[9] J.M. Ball, B. Kirchheim and J. Kristensen, Regularity of quasiconvex envelopes. Calc. Var. Partial Differ. Equ. 11 (2000) 333-359. | Zbl 0972.49024

[10] J.M. Ball and F. Murat, 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 0751.49005

[11] J.M. Ball, J.C. Currie and P.J. Olver, Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal. 41 (1981) 135-174. | Zbl 0459.35020

[12] A.V. Bitsadze, A system of nonlinear partial differential equations. Differ. Uravn. 15 (1979) 1267-1270 (in Russian). | Zbl 0418.35029

[13] A. Canfora, 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 0152.11003

[14] E. Casadio-Tarabusi, An algebraic characterization of quasiconvex functions. Ric. Mat. 42 (1993) 11-24. | Zbl 0883.26011

[15] M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Ration. Mech. Anal. 103 (1988) 237-277. | Zbl 0673.73012

[16] B. Dacorogna, Weak continuity and weak lower semicontinuity for nonlinear functionals. Berlin-Heidelberg-New York, Springer. Lect. Notes Math. 922 (1982). | MR 658130 | Zbl 0484.46041

[17] B. Dacorogna, Direct methods in the calculus of variations. Springer, Berlin (1989). | MR 990890 | Zbl 0703.49001

[18] B. Dacorogna, J. Douchet, W. Gangbo and J. Rappaz, Some examples of rank-one convex functions in dimension two. Proc. R. Soc. Edinb. 114 (1990) 135-150. | Zbl 0722.49018

[19] B. Dacorogna and J.-P. Haeberly, Some numerical methods for the study of the convexity notions arising in the calculus of variations. M2AN 32 (1998) 153-175. | Numdam | Zbl 0905.65075

[20] G. Dolzmann, Numerical computation of rank-one convex envelopes. SIAM J. Numer. Anal. 36 (1999) 1621-1635. | Zbl 0941.65062

[21] G. Dolzmann, Variational methods for crystalline microstructure-analysis and computation. Springer-Verlag, Berlin. Lect. Notes Math. 1803 (2003). | Zbl 1016.74002

[22] G. Dolzmann, B. Kirchheim and J. Kristensen, Conditions for equality of hulls in the calculus of variations. Arch. Ration. Mech. Anal. 154 (2000) 93-100. | Zbl 0970.49014

[23] D.G.B. Edelen, The null set of the Euler-Lagrange operator. Arch. Ration. Mech. Anal 11 (1962) 117-121. | Zbl 0125.33002

[24] H. Federer, Geometric measure theory. Springer-Verlag, New York, Heldelberg (1969). | MR 257325 | Zbl 0176.00801

[25] I. Fonseca and S. Müller, $A$-quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal. 30 (1999) 1355-1390. | Zbl 0940.49014

[26] L.E. Fraenkel, An introduction to maximum principles and symmetry in elliptic problems. Cambridge University Press, Cambridge (2000). | MR 1751289 | Zbl 0947.35002

[27] M. Giaquinta and E. Giusti, Quasi-minima, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1 (1984) 79-107. | Numdam | Zbl 0541.49008

[28] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin-Heidelberg-New York (1977). | Zbl 0361.35003

[29] T. Iwaniec, Nonlinear Cauchy-Riemann operators in ${ℝ}^{n}$. Trans. Am. Math. Soc. 354 (2002) 1961-1995. | Zbl 1113.35068

[30] T. Iwaniec, Integrability theory of the Jacobians. Lipshitz Lectures, preprint Univ. Bonn Sonderforschungsbereich 256 (1995).

[31] T. Iwaniec, Nonlinear differential forms. Series in Lectures at the International School in Jyväskylä, published by Math. Inst. Univ. Jyväskylä (1998) 1-207. | Zbl 0919.30001

[32] T. Iwaniec and A. Lutoborski, Integral estimates for null-lagrangians. Arch. Ration. Mech. Anal. 125 (1993) 25-79. | Zbl 0793.58002

[33] A. Kałamajska, On $\Lambda$-convexity conditions in the theory of lower semicontinuous functionals. J. Convex Anal. 10 (2003) 419-436. | Zbl 1040.49019

[34] A. Kałamajska, 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 1054.49019

[35] 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. | Zbl pre01944370

[36] V. Kohn and G. Strang, Optimal design and relaxation of variational problems I. Commun. Pure Appl. Math. 39 (1986) 113-137. | Zbl 0609.49008

[37] V. Kohn and G. Strang, Optimal design and relaxation of variational problems II. Commun. Pure Appl. Math. 39 (1986) 139-182. | Zbl 0621.49008

[38] J. Kolář, Non-compact lamination convex hulls. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20 (2003) 391-403. | Numdam | Zbl 1038.26008

[39] J. Kristensen, On the non-locality of quasiconvexity. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16 (1999) 1-13. | Numdam | Zbl 0932.49015

[40] M. Kružík, On the composition of quasiconvex functions and the transposition. J. Convex Anal. 6 (1999) 207-213. | Zbl 0944.49011

[41] M. Kružík, Bauer's maximum principle and hulls of sets. Calc. Var. Partial Differ. Equ. 11 (2000) 321-332. | Zbl 0981.49010

[42] S. Lang, Algebra. Addison-Wesley Publishing Company, New York (1965). | Zbl 0193.34701

[43] H. Le Dret and A. Raoult, The quasiconvex envelope of the Saint-Venant-Kirchhoff stored energy function. Proc. R. Soc. Edinb. 125 (1995) 1179-1192. | Zbl 0843.73016

[44] F. Leonetti, Maximum principle for vector-valued minimizers of some integral functionals. Boll. Unione Mat. Ital. 7 (1991) 51-56. | Zbl 0729.49015

[45] P.L. Lions, Jacobians and Hardy spaces. Ric. Mat. Suppl. 40 (1991) 255-260. | Zbl 0846.46033

[46] M. Luskin, On the computation of crystalline microstructure. Acta Numerica 5 (1996) 191-257. | Zbl 0867.65033

[47] J.J. Manfredi, Weakly monotone functions. J. Geom. Anal. 4 (1994) 393-402. | Zbl 0805.35013

[48] P. Marcellini, Quasiconvex quadratic forms in two dimensions. Appl. Math. Optimization 11 (1984) 183-189. | Zbl 0567.49007

[49] M. Miranda, Maximum principles and minimal surfaces. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) XXV (1997) 667-681. | Numdam | Zbl 1015.49028

[50] C.B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2 (1952) 25-53. | Zbl 0046.10803

[51] C.B. Morrey, Multiple integrals in the calculus of variations. Springer-Verlag, Berlin-Heidelberg-New York (1966). | Zbl 0142.38701

[52] F. Murat, Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978) 489-507. | Numdam | Zbl 0399.46022

[53] 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.

[54] 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. | Numdam | Zbl 0464.46034

[55] S. Müller, A surprising higher integrability property of mappings with positive determinant. Bull. Am. Math. Soc. 21 (1989) 245-248. | Zbl 0689.49006

[56] S. Müller, 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 0968.74050

[57] S. Müller, Rank-one convexity implies quasiconvexity on diagonal matrices. Int. Math. Res. Not. 20 (1999) 1087-1095. | Zbl 1055.49506

[58] S. Müller, Quasiconvexity is not invariant under transposition, in Proc. R. Soc. Edinb. 130 (2000) 389-395. | Zbl 0980.49017

[59] 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. | Zbl 0930.35038

[60] 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. | Zbl 0896.35029

[61] S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. of Math. (2) 157 (2003) 715-742. | Zbl 1083.35032

[62] 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. | Zbl 0953.35042

[63] 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/ | MR 2187312 | Zbl 1116.35059

[64] G.P. Parry, On the planar rank-one convexity condition. Proc. R. Soc. Edinb. A 125 (1995) 247-264. | Zbl 0841.49008

[65] P. Pedregal, Parametrized measures and variational principles. Birkhäuser (1997). | MR 1452107 | Zbl 0879.49017

[66] P. Pedregal, Weak continuity and weak lower semicontinuity for some compensation operators. Proc. R. Soc. Edinb. A 113 (1989) 267-279. | Zbl 0702.35054

[67] P. Pedregal, Laminates and microstructure. Eur. J. Appl. Math. 4 (1993) 121-149. | Zbl 0779.73050

[68] P. Pedregal, Some remarks on quasiconvexity and rank-one convexity. Proc. R. Soc. Edinb. A 126 (1996) 1055-1065. | Zbl 0867.49012

[69] P. Pedregal and V. Šverák, A note on quasiconvexity and rank-one convexity for $2×2$ Matrices. J. Convex Anal. 5 (1998) 107-117. | Zbl 0918.49012

[70] A.C. Pipkin, Elastic materials with two preferred states. Q. J. Mech. Appl. Math. 44 (1991) 1-15. | Zbl 0735.73032

[71] J. Robbin, R.C. Rogers and B. Temple, On weak continuity and Hodge decomposition. Trans. Am. Math. Soc. 303 (1987) 609-618. | Zbl 0634.35005

[72] T. Roubíuek, Relaxation in optimization theory and variational calculus. Berlin, W. de Gruyter (1997). | MR 1458067 | Zbl 0880.49002

[73] J. Sivaloganathan, Implications of rank one convexity. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5, 2 (1988) 99-118. | Numdam | Zbl 0664.73006

[74] R. Stefaniuk, Numerical verification of certain property of quasiconvex function. MSC thesis, Warsaw University (2004).

[75] V. Šverák, Examples of rank-one convex functions. Proc. R. Soc. Edinb. A 114 (1990) 237-242. | Zbl 0714.49024

[76] V. Šverák, Quasiconvex functions with subquadratic growth. Proc. R. Soc. Lond. A 433 (1991), 723-725. | Zbl 0741.49016

[77] V. Šverák, Rank-one convexity does not imply quasiconvexity. Proc. R. Soc. Edinb. 120 (1992) 185-189. | Zbl 0777.49015

[78] 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. | Zbl 0852.49010

[79] 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. | Zbl 0797.73079

[80] L. Tartar, 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 0437.35004

[81] 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. | Zbl 0536.35003

[82] 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. | Zbl 0823.26008

[83] B. Yan, On rank-one convex and polyconvex conformal energy functions with slow growth. Proc. R. Soc. Edinb. 127 (1997) 651-663. | Zbl 0896.49004

[84] 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. | Numdam | Zbl 0778.49015

[85] K.W. Zhang, Biting theorems for Jacobians and their applications. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7 (1990) 345-365. | Numdam | Zbl 0717.49012

[86] K.W. Zhang, On various semiconvex hulls in the calculus of variations. Calc. Var. Partial Differ. Equ. 6 (1998) 143-160. | Zbl 0896.49005