# Representation formulas for L∞ norms of weakly convergent sequences of gradient fields in homogenization

• Volume: 46, Issue: 5, page 1121-1146
• ISSN: 0764-583X

top

## Abstract

top
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.

## How to cite

top

Lipton, Robert, and Mengesha, Tadele. "Representation formulas for L∞ norms of weakly convergent sequences of gradient fields in homogenization." ESAIM: Mathematical Modelling and Numerical Analysis 46.5 (2012): 1121-1146. <http://eudml.org/doc/222152>.

@article{Lipton2012,
abstract = {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.},
author = {Lipton, Robert, Mengesha, Tadele},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {L∞norms; nonlinear composition; weak limits; material design; homogenization; -convergence; piecewise constant coefficient matrix; local corrector; periodic homogenization; laminate composite; graded microstructure; optimal design},
language = {eng},
month = {2},
number = {5},
pages = {1121-1146},
publisher = {EDP Sciences},
title = {Representation formulas for L∞ norms of weakly convergent sequences of gradient fields in homogenization},
url = {http://eudml.org/doc/222152},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Lipton, Robert
TI - Representation formulas for L∞ norms of weakly convergent sequences of gradient fields in homogenization
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/2//
PB - EDP Sciences
VL - 46
IS - 5
SP - 1121
EP - 1146
AB - 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.
LA - eng
KW - L∞norms; nonlinear composition; weak limits; material design; homogenization; -convergence; piecewise constant coefficient matrix; local corrector; periodic homogenization; laminate composite; graded microstructure; optimal design
UR - http://eudml.org/doc/222152
ER -

## References

top
1. G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal.23, (1992) 1482–1518.  Zbl0770.35005
2. M. Avellaneda and F.-H. Lin, Compactness methods in the theory of homogenization. Comm. Pure Appl. Math.40 (1987) 803–847.  Zbl0632.35018
3. A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications5. North-Holland, Amsterdam (1978)  Zbl0404.35001
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.  Zbl0947.35044
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.
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.
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.  Zbl1245.74066
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.  Zbl1161.49018
9. M. Chipot, D. Kinderlehrer and L. Vergara-Caffarelli, Smoothness of linear laminates. Arch. Rational Mech. Anal.96 (1985) 81–96.  Zbl0617.73062
10. E. De Giorgi and S. Spagnolo, Sulla convergenza degli integrali dell’ energia peroperatori ellittici del secondo ordine. Boll. UMI8 (1973) 391–411.  Zbl0274.35002
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.  Zbl0924.73158
12. D. Faraco, Milton’s conjecture on the regularity of solutions to isotropic equations. Ann. Inst. Henri Poincare, Nonlinear Analysis20 (2003) 889–909.  Zbl1029.30012
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.  Zbl0994.74055
14. D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin, New York (2001).  Zbl1042.35002
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).
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.  Zbl1198.35268
18. A. Kelly and N.H. Macmillan, Strong Solids. Monographs on the Physics and Chemistry of Materials. Clarendon Press, Oxford, (1986).
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.  Zbl0869.35019
20. Y.Y. Li and L. Nirenberg, Estimates for elliptic systems from composite material. Comm. Pure Appl. Math.LVI (2003) 892–925.  Zbl1125.35339
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.  Zbl0958.35060
22. R. Lipton, Assessment of the local stress state through macroscopic variables. Phil. Trans. R. Soc. Lond. Ser. A361 (2003) 921–946.  Zbl1079.74054
23. R. Lipton, Bounds on the distribution of extreme values for the stress in composite materials. J. Mech. Phys. Solids52 (2004) 1053–1069.  Zbl1070.74041
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.  Zbl1320.74091
25. R. Lipton, Homogenization and field concentrations in heterogeneous media. SIAM J. Math. Anal.38 (2006) 1048–1059.  Zbl1151.35007
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.  Zbl1087.74046
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.  Zbl1245.74007
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. Pisa17 (1963) 189–206.  Zbl0127.31904
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 Applications1 (1986) 150–174.
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 France57 (1985) 319–369.
33. G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal.20 (1989) 608–623.  Zbl0688.35007
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 Materials593 (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 Bolyai27 (1981) 835–873.
37. E. Sanchez-Palencia, Non Homogeneous Media and Vibration Theory. Springer, Heidelberg (1980).  Zbl0432.70002
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.

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.