G-convergence and the weak operator topology

We show that a bounded sequence (a_n)_n of symmetric d × d-matrix valued functions is G-convergent if and only if ((ι^∗︁a_nι)⁻¹ )_n converges in the weak operator topology. Here ι: R(grad₀) ↪ L²(Ω)^d denotes the (canonical) embedding from the range of the weak gradient grad₀ defined on H¹₀(Ω) into L²(Ω)_d, where Ω ⊆ ℝ^d is open and bounded. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)