Periodic cell problems and correctors

For each coordinate direction eⱼ, seek a periodic H¹ function χⱼ whose cell mean is zero and whose corrected field eⱼ+∇χⱼ has divergence-free A-weighted flux. The weak equation tests against all periodic H¹ functions, or equivalently their mean-zero representatives. A Poincaré inequality on the torus and uniform ellipticity give existence and uniqueness in the mean-zero space by Lax–Milgram. Constants would otherwise be invisible because only the gradient enters the cell equation.

The corrected field has average eⱼ since the integral of a periodic gradient is zero. Its local magnitude can vary strongly between phases even when the macroscopic gradient is constant. The effective tensor's jth column is the average of A(eⱼ+∇χⱼ). This is an average of a correlated coefficient and field, rather than the average coefficient acting on eⱼ. Changing χⱼ by a constant leaves this tensor unchanged, but it changes the displacement approximation and its boundary mismatch, so a fixed normalization remains useful.

When A is symmetric, the corrected field minimizes the cell energy among fields p+∇w with periodic mean-zero w. The minimum is p·A hom p. Taking w=0 gives an arithmetic upper bound, while coercivity and the mean of the gradient give a lower ellipticity bound. Polarization and the cell equations imply that A hom is symmetric. These energy statements should not be used to reconstruct the full nonsymmetric tensor from a quadratic form, which cannot see an antisymmetric part.

The minimizer for a general imposed vector p is the linear combination of coordinate correctors with coefficients pⱼ. This follows from linearity of the cell equation, not from assuming that the microscopic gradient is constant. The effective tensor therefore comes from d cell solves, while its energy can be checked on arbitrary test directions. For anisotropic or coupled coefficients, scalar harmonic formulas no longer replace those cell problems. They survive only in special geometries whose conservation equations can be solved explicitly.

The first-order expansion is u₀(x)+εΣχⱼ(x/ε)∂ⱼu₀(x). Its amplitude is small when correctors are bounded, but differentiation produces an order-one oscillatory term because the fast variable contributes 1/ε. Thus strong convergence of potentials can coexist with gradients that do not converge strongly to ∇u₀ alone. To justify an expansion quantitatively, one must check regularity of u₀ and χ and address the boundary trace. The formal expression does not automatically satisfy the original Dirichlet condition.

For the scalar real elliptic equation considered here, even measurable bounded uniformly elliptic coefficients give correctors with bounded Hölder representatives. Indeed, χⱼ+yⱼ is locally A-harmonic, so scalar De Giorgi–Nash regularity applies; periodicity then bounds χⱼ on a cell. Smooth derivatives still require additional coefficient regularity. For elliptic systems, the scalar boundedness conclusion is not automatic. A quantitative reconstruction must therefore specify its equation class and the regularity needed for each derivative or product, rather than treating every H¹ corrector as smooth.