Gradient Young measures generated by sequences in Sobolev spaces. (English) Zbl 0808.46046

Kinderlehrer, David; Pedregal, Pablo

J. Geom. Anal. 4, No. 1, 59-90 (1994).

Summary: Oscillatory properties of a weak convergent sequence of functions bounded in \(L^ p\), \(1\leq p\leq \infty\), may be summarized by the parametrized measure it generates. When such a measure is generated by the gradients of a sequence of functions bounded in \(H^{1,p}\), it must have special properties. The purpose of this paper is to characterize such parametrized measures as the ones that obey Jensen’s inequality for all quasiconvex functions with the appropriate growth at infinity. We have found subtle differences between the cases \(p< \infty\) and \(p= \infty\). A consequence is that any measure determined by biting convergence is in fact generated by a sequence convergent in a stronger sense. We also give a few applications.

MSC:

46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E27	Spaces of measures
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
26B25	Convexity of real functions of several variables, generalizations
35J20	Variational methods for second-order elliptic equations
74B20	Nonlinear elasticity

Keywords:

weak convergence; biting convergence; lower semicontinuity; Young measure; calculus of variations; oscillatory properties; Jensen’s inequality; quasiconvex functions

References:

[1]	Acerbi, E.; Fusco, N., Semicontinuity problems in the calculus of variations, Arch. Rat. Mech. Anal., 86, 125-145 (1984) · Zbl 0565.49010 · doi:10.1007/BF00275731
[2]	Acerbi, E.; Fusco, N.; Ball, J. M., An approximation lemma forW^1,p functions, Proc. Symp. Material Instabilities in Continuum Mechanics, 1-5 (1988), Oxford: Heriot-Watt, Oxford · Zbl 0644.46026
[3]	Ball, J. M. A version of the fundamental theorem for Young measures. InPDE’s and Continuum Models of Phase Transitions, Lecture Notes in Physics, vol. 344, edited by M. Rascle, D. Serre, and M. Slemrod, pp. 207-215. Springer. · Zbl 0991.49500
[4]	Ball, J. M., Sets of gradients with no rank-one connections, J. Math Pures et Appl., 69, 241-259 (1990) · Zbl 0644.49011
[5]	Ball, J. M.; James, R., Fine phase mixtures as minimizers of energy, Arch. Rat. Mech. Anal., 100, 15-52 (1987) · Zbl 0629.49020 · doi:10.1007/BF00281246
[6]	Ball, J. M.; James, R., Proposed experimental tests of a theory of fine microstructure and the two well problem, Phil. Trans. R. Soc. London A, 338, 389-450 (1992) · Zbl 0758.73009 · doi:10.1098/rsta.1992.0013
[7]	Ball, J. M.; Murat, F., W^1,p-quasiconvexity and variational problems for multiple integrals, J. Fin. Anal., 58, 225-253 (1984) · Zbl 0549.46019 · doi:10.1016/0022-1236(84)90041-7
[8]	Ball, J. M.; Murat, F., Remarks on Chacon’s biting lemma, Proc. Am. Math. Soc., 107, 655-663 (1989) · Zbl 0678.46023 · doi:10.2307/2048162
[9]	Ball, J. M., and Murat, F. Remarks on rank-one convexity and quasiconvexity, to appear. · Zbl 0751.49005
[10]	Ball, J. M.; Zhang, K., Lower semicontinuity of multiple integrals and the biting lemma, Proc. R. Soc. Edinburgh, 114A, 367-379 (1990) · Zbl 0716.49011
[11]	Battacharya, K., Wedge-like microstructure in martensite, Acta Metal., 39, 2431-2444 (1991) · doi:10.1016/0956-7151(91)90023-T
[12]	Battacharya, K., Self accomodation in martensite, Arch. Rat. Mech. Anal., 120, 201-244 (1992) · Zbl 0771.73007 · doi:10.1007/BF00375026
[13]	Billingsley, P. 1986Probability and Measure. Wiley.
[14]	Brandon, D., and Rogers, R. Nonlocal regularization of L. C. Young’s tacking problem (to appear). · Zbl 0769.49016
[15]	Brooks, J. K.; Chacon, R. V., Continuity and compactness of measures, Adv. in Math., 37, 16-26 (1980) · Zbl 0463.28003 · doi:10.1016/0001-8708(80)90023-7
[16]	Chipot, M., Numerical analysis of oscillations in nonconvex problems, Num. Math., 59, 747-767 (1991) · Zbl 0712.65063 · doi:10.1007/BF01385808
[17]	Chipot, M.; Collins, C., Numerical approximation in variational problems with potential wells, SIAM J. Num. Anal., 29, 1002-1014 (1992) · Zbl 0763.65049 · doi:10.1137/0729061
[18]	Chipot, M.; Kinderlehrer, D., Equilibrium configurations of crystals, Arch. Rat. Mech. Anal., 103, 237-277 (1988) · Zbl 0673.73012 · doi:10.1007/BF00251759
[19]	Chipot, M.; Kinderlehrer, D.; Vergara-Caffarelli, G., Smoothness of linear laminates, Arch. Rat. Mech. and Anal., 96, 81-96 (1986) · Zbl 0617.73062
[20]	Coifman, P.-L.; Lions, Meyer; Semmes, S., Compacité par compensation et espaces de Hardy, CRAS Paris, 309, 945-949 (1989) · Zbl 0684.46044
[21]	Collins, C., and Luskin, M. The computation of the austenitic-martensitic phase transition. In:PDE’s and continuum models of phase transitions, Lecture Notes in Physics, vol. 344, edited by M. Rascle, D. Serre, and M. Slemrod, pp. 34-50. Springer 1989. · Zbl 0991.80502
[22]	Collins, C., and Luskin, M. Numerical modeling of the microstructure of crystals with symmetry-related variants. In:Proc. ARO US-Japan Workshop on Smart/Intelligent Materials and Systems. Technomic.
[23]	Collins, C.; Luskin, M., Optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem, Math. Comp., 57, 621-637 (1991) · Zbl 0735.65042 · doi:10.2307/2938708
[24]	Collins, C.; Kinderlehrer, D.; Luskin, M., Numerical approximation of the solution of a variational problem with a double well potential, SIAM J. Numer. Anal., 28, 321-333 (1991) · Zbl 0725.65067 · doi:10.1137/0728018
[25]	Dacorogna, B.Direct Methods in the Calculus of Variations. Springer 1989. · Zbl 0703.49001
[26]	Dal Maso, G., Integral representation onBV(Ω) of Γ-limits of variational integrals, Manuscripta Math., 30, 387-416 (1980) · Zbl 0435.49016 · doi:10.1007/BF01301259
[27]	Ericksen, J. L., On the symmetry of deformable crystals, Arch. Rat. Mech. Anal., 72, 1-13 (1979) · Zbl 0439.20031 · doi:10.1007/BF00250733
[28]	Ericksen, J. L., Some phase transitions in crystals, Arch. Rat. Mech. Anal., 73, 99-124 (1980) · Zbl 0429.73007 · doi:10.1007/BF00258233
[29]	Ericksen, J. L. Changes in symmetry in elastic crystals.IUTAM Symp. Finite Elasticity, edited by D. E. Carlson and R. T. Shield, pp. 167-177. M. Nijhoff 1981.
[30]	Ericksen, J. L., Some simpler cases of the Gibbs phenomenon for thermoelastic solids, J. of Thermal Stresses, 4, 13-30 (1981) · doi:10.1080/01495738108909949
[31]	Ericksen, J. L., Crystal lattices and sublattices, Rend. Sem. Mat. Padova, 68, 1-9 (1982) · Zbl 0523.73002
[32]	Ericksen, J. L. Ill posed problems in thermoelasticity theory. InSystems of Nonlinear Partial Differential Equations, edited by J. Ball, pp. 71-95. D. Reidel 1983.
[33]	Ericksen, J. L. The Cauchy and Born hypotheses for crystals. InPhase Transformations and Material Instabilities in Solids, edited by M. Gurtin, pp. 61-78. Academic Press 1984.
[34]	Ericksen, J. L., Constitutive theory for some constrained elastic crystals, Int. J. Solids Structures, 22, 951-964 (1986) · Zbl 0595.73001 · doi:10.1016/0020-7683(86)90030-2
[35]	Ericksen, J. L., Stable equilibrium configurations of elastic crystals, Arch. Rat. Mech. Anal., 94, 1-14 (1986) · Zbl 0597.73006 · doi:10.1007/BF00278240
[36]	Ericksen, J. L. Twinning of crystals I. InMetastability and Incompletely Posed Problems, IMA Vol. Math. Appl., vol. 3, edited by S. Antman, J. L. Ericksen, D. Kinderlehrer, and I. M ller, pp. 77-96. Springer 1987.
[37]	Ericksen, J. L. Some constrained elastic crystals. InMaterial Instabilities in Continuum Mechanics, edited by J. Ball, pp. 119-136. Oxford 1988.
[38]	Ericksen, J. L., Weak martensitic transformations in Bravais lattices, Arch. Rat. Mech. Anal., 107, 23-36 (1989) · Zbl 0697.73005 · doi:10.1007/BF00251425
[39]	Ericksen, J. L., James, R. D., Kinderlehrer, D., and Luskin, M. (eds.)Microstructure and Phase Transitions, IMA Volume 54. Springer 1993.
[40]	Evans, L. C.Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS 74. American Mathematical Society 1990.
[41]	Evans, L. C.; Gariepy, R. F., Blow-up, compactness, and partial regularity in the calculus of variations, Indiana Univ. Math. J., 36, 361-371 (1987) · Zbl 0626.49007 · doi:10.1512/iumj.1987.36.36021
[42]	Evans, L. C.; Gariepy, R. F., Some remarks on quasiconvexity and strong convergence, Proc. R. Soc. Edinburgh Sect. A, 106, 53-61 (1987) · Zbl 0628.49011
[43]	Firoozye, N., and Kohn, R. Geometric parameters and the relaxation of multiwell energies, IMA preprint Series 765 (1991). · Zbl 0797.73051
[44]	Fonseca, I., Variational methods for elastic crystals, Arch. Rat. Mech. Anal., 97, 189-220 (1985) · Zbl 0611.73023
[45]	Fonseca, I., The lower quasiconvex envelope of the stored energy function for an elastic crystal, J. Math. Pures et Appl., 67, 175-195 (1988) · Zbl 0718.73075
[46]	Fonseca, I., The Wulff Theorem revisited, Proc. R. Soc. Lond., 432, 125-145 (1991) · Zbl 0725.49017
[47]	Fonseca, I., Lower semicontinuity of surface energies, Proc. R. Acad. Edin. A, 120, 99-115 (1992) · Zbl 0757.49013
[48]	Fonseca, I.; Müller, S., Relaxation of Quasiconvex functionals inB V(ό; ℝ^p) for Integrandsf(x, y, ∇u), Arch. Rat. Mech. Anal., 123, 1-51 (1993) · Zbl 0788.49039 · doi:10.1007/BF00386367
[49]	Fonseca, I.; Müller, S., Quasiconvex integrands and lower semicontinuity inL^1, SIAM J. Math Anal., 23, 1081-1098 (1992) · Zbl 0764.49012 · doi:10.1137/0523060
[50]	Iwaniec, T.; Sbordone, C., On the integrability of the Jacobian under minimal hypothesis, Arch. Rat. Mech. Anal., 119, 129-144 (1992) · Zbl 0766.46016 · doi:10.1007/BF00375119
[51]	James, R. D. Microstructure and weak convergence. InProc. Symp. Material Instabilities in Continuum Mechanics,Heriot-Watt, edited by J. M. Ball, pp. 175-196. Oxford 1988. · Zbl 0645.73005
[52]	James, R. D., and Kinderlehrer, D. Theory of diffusionless phase transitions. InPDE’s and Continuum Models of Phase Transitions, Lecture Notes in Physics, vol. 344, edited by M. Rascle, D. Serre, and M. Slemrod, pp. 51-84. Springer 1989. · Zbl 0991.74504
[53]	James, R. D.; Kinderlehrer, D., Frustration in ferromagnetic materials, Cont. Mech. Therm., 2, 215-239 (1990) · doi:10.1007/BF01129598
[54]	James, R. D.; Kinderlehrer, D., A theory of magnetostriction with application to TbDyFe2, Phil. Mag. B, 68, 237-274 (1993)
[55]	James, R. D., and Kinderlehrer, D. Frustration and microstructure: An example in magnetostriction.Proc. First Europ. Conf. Elliptic Parab. Problems. Pont a Mousson 1990. · Zbl 0797.49037
[56]	Kinderlehrer, D. Remarks about the equilibrium configurations of crystals. InProc. Symp. Material Instabilities in Continuum Mechanics, Heriot-Watt, edited by J. M. Ball, pp. 217-242. Oxford 1988. · Zbl 0850.73037
[57]	Kinderlehrer, D.; Pedregal, P., Characterizations of Young measures generated by gradients, Arch. Rat. Mech. Anal., 115, 329-365 (1991) · Zbl 0754.49020 · doi:10.1007/BF00375279
[58]	Kinderlehrer, D.; Pedregal, P., Charactérisation des mesures de Young associées á un gradient, CRAS Paris, 313, 765-770 (1991) · Zbl 0759.49005
[59]	Kinderlehrer, D., and Pedregal, P. Gradient Young measures generated by sequences in Sobolev spaces. InNonlinear PDE and their Applications, Collége de France Seminar. Longman (to appear). · Zbl 0808.46046
[60]	Kinderlehrer, D., and Pedregal, P. Remarks about the analysis of gradient Young measures. InPDE and Related Subjects, edited by M. Miranda. Bitman/Longman Res. Notes269, pp. 125-150 (1992). · Zbl 0833.49012
[61]	Kinderlehrer, D.; Pedregal, P., Weak convergence of integrands and the Young measure representation, SIAM J. Math. Anal., 23, 1-19 (1992) · Zbl 0757.49014 · doi:10.1137/0523001
[62]	Kinderlehrer, D., and Pedregal, P. Remarks about Young measures supported on two wells (to appear). · Zbl 0833.49012
[63]	Kinderlehrer, D.; Vergara-Caffarelli, G., The relaxation of functionals with surface energies, Asymptotic Analysis, 2, 279-298 (1989) · Zbl 0696.49007
[64]	Kohn, R. V., The relaxation of a double-well energy, Cont. Mech. Therm., 3, 193-236 (1991) · Zbl 0825.73029 · doi:10.1007/BF01135336
[65]	Liu, F.-C., A Luzin type property of Sobolev functions, Ind. Math. J., 26, 645-651 (1977) · Zbl 0368.46036 · doi:10.1512/iumj.1977.26.26051
[66]	Luskin, M., and Ma, L. Analysis of the finite element approximation of microstructure in micromagnetics. UMSI report 90/164 (1990). · Zbl 0760.65113
[67]	Matos, J. Thesis, University of Minnesota (1990).
[68]	Matos, J., The absence of fine microstructure in α − β quartz, Eur. J. Appl. Math., 3, 31-54 (1992) · Zbl 0751.73003 · doi:10.1017/S095679250000067X
[69]	Morrey, C. B., Jr.Multiple Integrals in the Calculus of Variations. Springer 1966. · Zbl 0142.38701
[70]	Müller, S., Weak continuity of determinants an nonlinear elasticity, CRAS Paris, 307, 501-506 (1988) · Zbl 0679.34051
[71]	Müller, S., A surprising higher integrability property of mappings with positive determinant, Bull. AMS, 21, 245-248 (1989) · Zbl 0689.49006 · doi:10.1090/S0273-0979-1989-15818-7
[72]	Müller, S., Higher integrability of determinants and weak convergence inL^1, J. Reine Angew. Math., 412, 20-34 (1990) · Zbl 0713.49004
[73]	Nicolaides, R. A., and Walkington, N. Computation of microstructure utilizing Young measure representations.Proc. Recent Adv. Adaptive Sensory Materials Appl. (1990).
[74]	Pedregal, P., Weak continuity and weak lower semicontinuity for some compensation operators, Proc. R. Soc. Edin., 113, 267-279 (1989) · Zbl 0702.35054
[75]	Pedregal, P., Laminates and microstructure, Eur. J. Appl. Math., 4, 121-149 (1993) · Zbl 0779.73050
[76]	Slemrod, M., Dynamics of measure valued solutions to a backward-forward heat equation, J. Dyn. Diff. Eqns., 3, 1-28 (1991) · Zbl 0747.35013 · doi:10.1007/BF01049487
[77]	Sverak, V. On the regularity of the Monge-Ampére equation without convexity (to appear).
[78]	Sverak, V., Quasiconvex functions with subquadratic growth, Proc. R. Soc. London A, 433, 723-725 (1991) · Zbl 0741.49016
[79]	Sverak, V., Rank one convexity does not imply quasiconvexity, Proc. R. Soc. Edin. A, 120, 185-189 (1992) · Zbl 0777.49015
[80]	Tartar, L. Compensated compactness and applications to partial differential equations. InNonlinear analysis and mechanics: Heriot Watt Symposium, vol. IV,Pitman Research Notes in Math., vol. 39, edited by R. Knops, pp. 136-212. · Zbl 0437.35004
[81]	Tartar, L. The compensated compactness method applied to systems of conservation laws. InSystems of Nonlinear Partial Differential Equations, edited by J. M. Ball. Riedel 1983. · Zbl 0536.35003
[82]	Tartar, L. Étude des oscillations dans les équations aux dérivées partielles nonlinéaires.Springer Lect. Notes Physics, vol. 195, pp. 384-412. Springer 1984. · Zbl 0595.35012
[83]	Young, L. C.Lectures on Calculus of Variations and Optimal Control Theory. W. B. Saunders 1969. · Zbl 0177.37801
[84]	Zhang, K., Biting theorems for Jacobians and their applications, Anal. Nonlineare, 7, 345-366 (1990) · Zbl 0717.49012
[85]	Zhang, K., A construction of quasiconvex functions with linear growth at infinity, Ann. S.N.S. Pisa, 19, 313-326 (1992) · Zbl 0778.49015
[86]	Zhang, K. Rank-one connections and the three “well” problem.Trans. AMS (to appear).