Interior Regularity Estimates in High Conductivity Homogenization and Application

In this paper, uniform pointwise regularity estimates for the solutions of conductivity equations are obtained in a unit conductivity medium reinforced by an ε-periodic lattice of highly conducting thin rods. The estimates are derived only at a distance ε1+τ (for some τ > 0) away from the fibres. This distance constraint is rather sharp since the gradients of the solutions are shown to be unbounded locally in Lp as soon as p > 2. One key ingredient is the derivation in dimension two of regularity estimates to the solutions of the equations deduced from a Fourier series expansion with respect to the fibres’ direction, and weighted by the high-contrast conductivity. The dependence on powers of ε of these two-dimensional estimates is shown to be sharp. The initial motivation for this work comes from imaging, and enhanced resolution phenomena observed experimentally in the presence of micro-structures (Lerosey et al., Science 315:1120–1124, 2007). We use these regularity estimates to characterize the signature of low volume fraction heterogeneities in the fibred reinforced medium, assuming that the heterogeneities stay at a distance ε1+τ away from the fibres.

[1]  V. Marchenko,et al.  Homogenization of Partial Differential Equations , 2005 .

[2]  E. Ya. Khruslov,et al.  Homogenized Models of Composite Media , 1991 .

[3]  Nicoletta Tchou,et al.  Fibered microstructures for some nonlocal Dirichlet forms , 2001 .

[4]  M. Vogelius,et al.  Gradient Estimates for Solutions to Divergence Form Elliptic Equations with Discontinuous Coefficients , 2000 .

[5]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[6]  H. Ammari,et al.  Reconstruction of Small Inhomogeneities from Boundary Measurements , 2005 .

[7]  Yves Capdeboscq,et al.  A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction , 2003 .

[8]  Marc Briane.  Homogenization of High-Conductivity Periodic Problems: Application to a General Distribution of One-Directional Fibers , 2003, SIAM J. Math. Anal..

[9]  Biao Yin,et al.  Gradient Estimates for the Perfect Conductivity Problem , 2006 .

[10]  R. Wheeden,et al.  Weighted Inequalities for Fractional Integrals on Euclidean and Homogeneous Spaces , 1992 .

[11]  Yves Capdeboscq,et al.  A review of some recent work on impedance imaging for inhomogeneities of low volume fraction , 2004 .

[12]  Louis Nirenberg,et al.  Estimates for elliptic systems from composite material , 2003 .

[13]  V. Nesi,et al.  Quasiconformal solutions to certain first order systems and the proof of a conjecture of G. W. Milton , 1997 .

[14]  Ronald F. Boisvert,et al.  NIST Handbook of Mathematical Functions , 2010 .

[15]  G. Lerosey,et al.  Focusing Beyond the Diffraction Limit with Far-Field Time Reversal , 2007, Science.

[16]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[17]  Gerald Kaiser,et al.  Homogenization of Partial Differential Equations , 2005 .

[18]  Eric Bonnetier,et al.  Enhanced Resolution in Structured Media , 2009, SIAM J. Appl. Math..

[19]  Eric Bonnetier,et al.  An asymptotic formula for the voltage potential in a perturbed ε-periodic composite medium containing misplaced inclusions of size ε , 2006, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[20]  D. Cioranescu,et al.  Homogenization in open sets with holes , 1979 .

[21]  Graeme W. Milton,et al.  Theory of Composites. Cambridge Monographs on Applied and Computational Mathematics , 2003 .

[22]  Habib Ammari,et al.  Gradient estimates for solutions to the conductivity problem , 2005 .

[23]  N. Meyers.  An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations , 1963 .

[24]  Robert V. Kohn,et al.  Topics in the Mathematical Modelling of Composite Materials , 1997 .

[25]  Semi-strong convergence of sequences satisfying a variational inequality , 2008 .

[26]  Guy Bouchitté,et al.  Homogenization of the 3D Maxwell system near resonances and artificial magnetism , 2009 .

[27]  G. Minty.  On the extension of Lipschitz, Lipschitz-Hölder continuous, and monotone functions , 1970 .

[28]  Jacques-Louis Lions,et al.  Nonlinear partial differential equations and their applications , 1998 .

[29]  Robert Lipton,et al.  Inequalities for electric and elastic polarization tensors with applications to random composites , 1993 .

[30]  Yves Capdeboscq,et al.  Pointwise polarization tensor bounds, and applications to voltage perturbations caused by thin inhomogeneities , 2006, Asymptot. Anal..

[31]  Guy Bouchitté,et al.  HOMOGENIZATION OF ELLIPTIC PROBLEMS IN A FIBER REINFORCED STRUCTURE. NON LOCAL EFFECTS , 1998 .

[32]  G. Milton.  The Theory of Composites , 2002 .

[33]  Ali Sili.  A diffusion equation through a highly heterogeneous medium , 2010 .

[34]  Compactness of sequences of two-dimensional energies with a zero-order term. Application to three-dimensional nonlocal effects , 2008 .