Duality and compactness results in high-contrast homogenization of incompressible two-dimensional elasticity problems

We study incompressible two-dimensional elasticity problems with high-contrast coefficients. The Keller–Dykhne duality relations are extended to the case of Hooke's laws which are equicoercive and uniformly bounded in L 1 but not in L ∞. A compactness result is obtained for Hooke's laws which are uniformly bounded from above and such that their inverses are bounded in L 1 but not in L ∞, with a refinement in the periodic case. Moreover, we establish a compactness result in $L^2_\mathrm{loc}$ for a sequence of two-dimensional vector-valued functions in $H^1_0$ which are only bounded in L 2 .