Correctors and field fluctuations for the $p_{\epsilon }\left(x\right)$-laplacian with rough exponents : The sublinear growth case

Jimenez, Silvia

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 2, pp. 349-375.

A corrector theory for the strong approximation of gradient fields inside periodic composites made from two materials with different power law behavior is provided. Each material component has a distinctly different exponent appearing in the constitutive law relating gradient to flux. The correctors are used to develop bounds on the local singularity strength for gradient fields inside micro-structured media. The bounds are multi-scale in nature and can be used to measure the amplification of applied macroscopic fields by the microstructure. The results in this paper are developed for materials having power law exponents strictly between  -1 and zero.

[1] B. Amaziane, S.N. Antontsev, L. Pankratov and A. Piatnitski, Γ-convergence and homogenization of functionals in Sobolev spaces with variable exponents. J. Math. Anal. Appl. 342 (2008) 1192-1202. | MR

[2] S.N. Antontsev and J.F. Rodrigues, On stationary thermo-rheological viscous flows. Ann. Univ. Ferrara Sez. VII Sci. Mat. 52 (2006) 19-36. | MR | Zbl

[3] C. Atkinson and C.R. Champion, Some boundary-value problems for the equation ∇·( | ∇ϕ | N∇ϕ) = 0. Quart. J. Mech. Appl. Math. 37 (1984) 401-419. | MR | Zbl

[4] L.C. Berselli, L. Diening and M. Ružička, Existence of strong solutions for incompressible fluids with shear dependent viscosities. J. Math. Fluid Mech. 12 (2010) 101-132. | MR | Zbl

[5] A. Braides, V. Chiadò Piat and A. Defranceschi, Homogenization of almost periodic monotone operators. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 9 (1992) 399-432. | Numdam | MR | Zbl

[6] J. Byström, Correctors for some nonlinear monotone operators. J. Nonlinear Math. Phys. 8 (2001) 8-30. | MR | Zbl

[7] J. Byström, Sharp constants for some inequalities connected to the p-Laplace operator. JIPAM. J. Inequal. Pure Appl. Math. 6 (2005). Article 56 (electronic) 8. | MR | Zbl

[8] B. Dacorogna, Direct methods in the calculus of variations, in Appl. Math. Sci. Springer-Verlag, Berlin 78 (1989). | MR | Zbl

[9] G. Dal Maso and A. Defranceschi, Correctors for the homogenization of monotone operators. Differ. Integral Equ. 3 (1990) 1151-1166. | MR | Zbl

[10] N. Fusco and G. Moscariello, On the homogenization of quasilinear divergence structure operators. Ann. Mat. Pura Appl. 146 (1987) 1-13. | MR | Zbl

[11] A. Garroni and R.V. Kohn, Some three-dimensional problems related to dielectric breakdown and polycrystal plasticity. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 459 (2003) 2613-2625. | MR | Zbl

[12] A. Garroni, V. Nesi and M. Ponsiglione, Dielectric breakdown : optimal bounds. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 457 (2001) 2317-2335. | MR | Zbl

[13] R. Glowinski and J. Rappaz, Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology. ESAIM : M2AN 37 (2003) 175-186. | Numdam | MR | Zbl

[14] M. Idiart, The macroscopic behavior of power-law and ideally plastic materials with elliptical distribution of porosity. Mech. Res. Commun. 35 (2008) 583-588. | Zbl

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

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

[17] S. Levine, J. Stanich and Y. Chen, Image restoration via nonstandard diffusion. Technical report (2004).

[18] O. Levy and R.V. Kohn, Duality relations for non-Ohmic composites, with applications to behavior near percolation. J. Stat. Phys. 90 (1998) 159-189. | MR | Zbl

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

[20] F. Murat and L. Tartar, H-convergence. In Topics in the mathematical modelling of composite materials, Progr. Nonlinear Diff. Equ. Appl. Birkhäuser Boston, Boston, MA 31 (1997) 21-43. | MR | Zbl

[21] P. Pedregal, Parametrized measures and variational principles, in Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Verlag, Basel (1997). | MR | Zbl

[22] P. Pedregal and H. Serrano, Homogenization of periodic composite power-law materials through young measures, in Multi scale problems and asymptotic analysis. GAKUTO Int. Ser. Math. Sci. Appl. Gakkōtosho, Tokyo 24 (2006) 305-310. | MR

[23] P. Ponte Castañeda and P. Suquet, Nonlinear composties. Adv. Appl. Mech. 34 (1997) 171-302. | Zbl

[24] P. Ponte Castañeda and J.R. Willis, Variational second-order estimates for nonlinear composites. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 455 (1999) 1799-1811. | MR | Zbl

[25] M. Ružička, Electrorheological fluids : modeling and mathematical theory. Lect. Notes Math. 1748 (2000). | MR | Zbl

[26] P. Suquet, Overall potentials and extremal surfaces of power law or ideally plastic composites. J. Mech. Phys. Solids 41 (1993) 981-1002. | MR | Zbl

[27] D.R.S. Talbot and J.R. Willis, Upper and lower bounds for the overall properties of a nonlinear elastic composite dielectric I. random microgeometry. Proc. R. Soc. Lond. A 447 (1994) 365-384. | MR | Zbl

[28] D.R.S. Talbot and J.R. Willis, Upper and lower bounds for the overall properties of a nonlinear elastic composite dielectric II. periodic microgeometry. Proc. R. Soc. Lond. A 447 (1994) 385-396. | MR | Zbl

[29] A.C. Zaanen, An introduction to the theory of integration. Publishing Company, North-Holland, Amsterdam (1958). | MR | Zbl

[30] V.V. Zhikov, S.M. Kozlov and O.A. Oleinik, Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin (1994). Translated from the Russian by G.A. Yosifian. | MR | Zbl