Assuming H-convergence of the diffusion tensors, for small homogenization parameters there exists exactly one weak solution close to a given non-degenerate weak solution of the homogenized semilinear elliptic system.
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Meyers, A
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An H-convergence-based implicit function theorem for homogenization of nonlinear non-smooth elliptic systems
Assuming H-convergence of the diffusion tensors, for small homogenization parameters there exists exactly one weak solution close to a given non-degenerate weak solution of the homogenized semilinear elliptic system.