We investigate the homogenization through $ \Gamma $-convergence for the $ L^2({\Omega}) $-weak topology of the conductivity functional with a zero-order term where the matrix-valued conductivity is assumed to be non strongly elliptic. Under proper assumptions, we show that the homogenized matrix $ A^\ast $ is provided by the classical homogenization formula. We also give algebraic conditions for two and three dimensional $ 1 $-periodic rank-one laminates such that the homogenization result holds. For this class of laminates, an explicit expression of $ A^\ast $ is provided which is a generalization of the classical laminate formula. We construct a two-dimensional counter-example which shows an anomalous asymptotic behaviour of the conductivity functional.
Citation: Lorenza D'Elia. $ \Gamma $-convergence of quadratic functionals with non uniformly elliptic conductivity matrices[J]. Networks and Heterogeneous Media, 2022, 17(1): 15-45. doi: 10.3934/nhm.2021022
We investigate the homogenization through $ \Gamma $-convergence for the $ L^2({\Omega}) $-weak topology of the conductivity functional with a zero-order term where the matrix-valued conductivity is assumed to be non strongly elliptic. Under proper assumptions, we show that the homogenized matrix $ A^\ast $ is provided by the classical homogenization formula. We also give algebraic conditions for two and three dimensional $ 1 $-periodic rank-one laminates such that the homogenization result holds. For this class of laminates, an explicit expression of $ A^\ast $ is provided which is a generalization of the classical laminate formula. We construct a two-dimensional counter-example which shows an anomalous asymptotic behaviour of the conductivity functional.
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Lorenza D'Elia. $ \Gamma $-convergence of quadratic functionals with non uniformly elliptic conductivity matrices[J]. Networks and Heterogeneous Media, 2022, 17(1): 15-45. doi: 10.3934/nhm.2021022
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