Representation formulas for ${L}^{\infty }$ norms of weakly convergent sequences of gradient fields in homogenization
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 5, p. 1121-1146

We examine the composition of the L∞ norm with weakly convergent sequences of gradient fields associated with the homogenization of second order divergence form partial differential equations with measurable coefficients. Here the sequences of coefficients are chosen to model heterogeneous media and are piecewise constant and highly oscillatory. We identify local representation formulas that in the fine phase limit provide upper bounds on the limit superior of the L∞ norms of gradient fields. The local representation formulas are expressed in terms of the weak limit of the gradient fields and local corrector problems. The upper bounds may diverge according to the presence of rough interfaces. We also consider the fine phase limits for layered microstructures and for sufficiently smooth periodic microstructures. For these cases we are able to provide explicit local formulas for the limit of the L∞ norms of the associated sequence of gradient fields. Local representation formulas for lower bounds are obtained for fields corresponding to continuously graded periodic microstructures as well as for general sequences of oscillatory coefficients. The representation formulas are applied to problems of optimal material design.

DOI : https://doi.org/10.1051/m2an/2011049
Classification:  35J15,  49N60
Keywords: L∞norms, nonlinear composition, weak limits, material design, homogenization
@article{M2AN_2012__46_5_1121_0,
author = {Lipton, Robert and Mengesha, Tadele},
title = {Representation formulas for $L^\infty$ norms of weakly convergent sequences of gradient fields in homogenization},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {46},
number = {5},
year = {2012},
pages = {1121-1146},
doi = {10.1051/m2an/2011049},
zbl = {1273.35038},
language = {en},
url = {http://www.numdam.org/item/M2AN_2012__46_5_1121_0}
}

Lipton, Robert; Mengesha, Tadele. Representation formulas for $L^\infty$ norms of weakly convergent sequences of gradient fields in homogenization. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 5, pp. 1121-1146. doi : 10.1051/m2an/2011049. http://www.numdam.org/item/M2AN_2012__46_5_1121_0/

[1] G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal. 23, (1992) 1482-1518. | MR 1185639 | Zbl 0770.35005

[2] M. Avellaneda and F.-H. Lin, Compactness methods in the theory of homogenization. Comm. Pure Appl. Math. 40 (1987) 803-847. | MR 910954 | Zbl 0632.35018

[3] A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications 5. North-Holland, Amsterdam (1978) | MR 503330 | Zbl 0404.35001

[4] E. Bonnetier and M. Vogelius, An elliptic regularity result for a composite medium with “touching” fibers of circular cross-section. SIAM J. Math. Anal. 31 (2000) 651-677. | MR 1745481 | Zbl 0947.35044

[5] B.V. Boyarsky, Generalized solutions of a system of differential equations of the first order of elliptic type with discontinuous coefficients. Mat. Sb. N. S. 43 (1957) 451-503. | MR 106324 | Zbl 1173.35403

[6] L.A. Caffarelli and I. Peral, On W1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998) 1-21. | MR 1486629 | Zbl 0906.35030

[7] J. Carlos-Bellido, A. Donoso and P. Pedregal, Optimal design in conductivity under locally constrained heat flux. Arch. Rational Mech. Anal. 195 (2010) 333-351. | MR 2564477 | Zbl 1245.74066

[8] J. Casado-Diaz, J. Couce-Calvo and J.D. Martin-Gomez, Relaxation of a control problem in the coefficients with a functional of quadratic growth in the gradient. SIAM J. Control Optim. 47 (2008) 1428-1459. | MR 2407023 | Zbl 1161.49018

[9] M. Chipot, D. Kinderlehrer and L. Vergara-Caffarelli, Smoothness of linear laminates. Arch. Rational Mech. Anal. 96 (1985) 81-96. | MR 853976 | Zbl 0617.73062

[10] E. De Giorgi and S. Spagnolo, Sulla convergenza degli integrali dell' energia peroperatori ellittici del secondo ordine. Boll. UMI 8 (1973) 391-411. | MR 348255 | Zbl 0274.35002

[11] P. Duysinx and M.P. Bendsoe, Topology optimization of continuum structures with local stress constraints. Internat. J. Numer. Math. Engrg. 43 (1998) 1453-1478. | MR 1658541 | Zbl 0924.73158

[12] D. Faraco, Milton's conjecture on the regularity of solutions to isotropic equations. Ann. Inst. Henri Poincare, Nonlinear Analysis 20 (2003) 889-909. | Numdam | MR 1995506 | Zbl 1029.30012

[13] D. Fujii, B.C. Chen and N. Kikuchi, Composite material design of two-dimensional structures using the homogenization design method. Internat. J. Numer. Methods Engrg. 50 (2001) 2031-2051. | MR 1818050 | Zbl 0994.74055

[14] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin, New York (2001). | MR 1814364 | Zbl 0361.35003

[15] J.H. Gosse and S. Christensen, Strain invariant failure criteria for polymers in composite materials. AIAA (2001) 1184.

[16] V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin, New York (1994). | MR 1329546 | Zbl 0801.35001

[17] S. Jimenez and R. Lipton, Correctors and field fluctuations for the pϵ(x)-Laplacian with rough exponents. J. Math. Anal. Appl. 372 (2010) 448-469. | MR 2678875 | Zbl 1198.35268

[18] A. Kelly and N.H. Macmillan, Strong Solids. Monographs on the Physics and Chemistry of Materials. Clarendon Press, Oxford, (1986). | Zbl 0052.42502

[19] F. Leonetti and V. Nesi, Quasiconformal solutions to certain first order systems and the proof of a conjecture of G.W. Milton. J. Math. Pures. Appl. 76 (1997) 109-124. | MR 1432370 | Zbl 0869.35019

[20] Y.Y. Li and L. Nirenberg, Estimates for elliptic systems from composite material. Comm. Pure Appl. Math. LVI (2003) 892-925. | MR 1990481 | Zbl 1125.35339

[21] Y.Y. Li and M. Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients. Arch. Rational Mech. Anal. 153 (2000) 91-151. | MR 1770682 | Zbl 0958.35060

[22] R. Lipton, Assessment of the local stress state through macroscopic variables. Phil. Trans. R. Soc. Lond. Ser. A 361 (2003) 921-946. | MR 1995443 | Zbl 1079.74054

[23] R. Lipton, Bounds on the distribution of extreme values for the stress in composite materials. J. Mech. Phys. Solids 52 (2004) 1053-1069. | MR 2050209 | Zbl 1070.74041

[24] R. Lipton, Homogenization and design of functionally graded composites for stiffness and strength, in Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, edited by P.P. Castaneda et al., Kluwer Academic Publishers, Netherlands (2004) 169-192. | MR 2268904

[25] R. Lipton, Homogenization and field concentrations in heterogeneous media. SIAM J. Math. Anal. 38 (2006) 1048-1059. | MR 2274473 | Zbl 1151.35007

[26] R. Lipton and M. Stuebner, Inverse homogenization and design of microstructure for point wise stress control. Quart. J. Mech. Appl. Math. 59 (2006) 139-161. | MR 2204835 | Zbl 1087.74046

[27] R. Lipton and M. Stuebner, Optimal design of composite structures for strength and stiffness : an inverse homogenization approach. Struct. Multidisc. Optim. 33 (2007) 351-362. | MR 2310589 | Zbl 1245.74007

[28] R. Lipton and M. Stuebner, A new method for design of composite structures for strength and stiffness, 12th AIAA/ISSMO Multidisciplinary Analysis & Optimization Conference. American Institute of Aeronautics and Astronautics Paper AIAA, Victoria British Columbia, Canada (2008) 5986.

[29] A.J. Markworth, K.S. Ramesh and W.P. Parks, Modelling studies applied to functionally graded materials. J. Mater. Sci. 30 (1995) 2183-2193.

[30] N. Meyers, An Lp-Estimate for the gradient of solutions of second order elliptic divergence equations. Annali della Scuola Norm. Sup. Pisa 17 (1963) 189-206. | Numdam | MR 159110 | Zbl 0127.31904

[31] G.W. Milton, Modeling the properties of composites by laminates, edited by J. Erickson, D. Kinderleher, R.V. Kohn and J.L. Lions. Homogenization and Effective Moduli of Materials and Media, IMA Volumes in Mathematics and Its Applications 1 (1986) 150-174. | MR 859415 | Zbl 0631.73011

[32] F. Murat and L. Tartar, Calcul des Variations et Homogénéisation, Les Méthodes de l'Homogénéisation : Théorie et Applications en Physique, edited by D. Bergman et al. Collection de la Direction des Études et Recherches d'Electricité de France 57 (1985) 319-369. | MR 844873

[33] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608-623. | MR 990867 | Zbl 0688.35007

[34] R.J. Nuismer and J.M. Whitney, Uniaxial failure of composite laminates containing stress concentrations, in Fracture Mechanics of Composites, ASTM Special Technical Publication, American Society for Testing and Materials 593 (1975) 117-142.

[35] Y. Ootao, Y. Tanigawa and O. Ishimaru, Optimization of material composition of functionally graded plate for thermal stress relaxation using a genetic algorithim. J. Therm. Stress. 23 (2000) 257-271.

[36] G. Papanicolaou and S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random fields, Rigorous results in statistical mechanics and quantum field theory, Esztergom 1979. Colloq. Math. Soc. Janos Bolyai 27 (1981) 835-873. | MR 712714 | Zbl 0499.60059

[37] E. Sanchez-Palencia, Non Homogeneous Media and Vibration Theory. Springer, Heidelberg (1980). | Zbl 0432.70002

[38] S. Spagnolo, Convergence in Energy for Elliptic Operators, Proceedings of the Third Symposium on Numerical Solutions of Partial Differential Equations, edited by B. Hubbard. College Park (1975); Academic Press, New York (1976) 469-498. | MR 477444 | Zbl 0347.65034