New convexity conditions in the calculus of variations and compensated compactness theory

Chełmiński, Krzysztof; Kałamajska, Agnieszka

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 1, pp. 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.

Classification : 49J10, 49J45
Mots clés : quasiconvexity, rank-one convexity, semicontinuity

[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

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

[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

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

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

[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

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

[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

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

[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

[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

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

[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

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

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

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

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

[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

[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

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

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

[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

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

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

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

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

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

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

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

[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

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

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

[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

[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.

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

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

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

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

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

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

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

[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

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

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

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

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

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

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

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

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

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

[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

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

[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

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

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

[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

[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

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

[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

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

[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

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

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

[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

[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

[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

[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

[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

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

[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

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

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