High-contrast homogenization: Setup

Imported from the Apache-2.0 HighContrastHomogenization development at commit 7a13dbcd8d6609264a713373f5c69ceeac870472.

Setup for the high-contrast polynomial-scale theorems

This is the object layer of the setup in s.introduction: the coefficient space and its local sigma-fields, the integer translation action, the coarse block response and its Schur data, the reference and annealed contrasts, and the triadic and adapted geometry. The two structural assumptions on the law and the coarse ellipticity assumption are frozen declarations and live under HCPoly.Frozen.

Scope of the coefficient class. The coefficient space used here is the set of measurable fields that are locally uniformly elliptic: on every bounded subset of ℝ^d a pair of ellipticity constants serves almost every value of the field there, the pair being attached to the field and to the set rather than fixed in advance. No constant appearing in any statement below depends on such a pair. The paper's standing condition e.qualitative.ellipticity requires local essential bounds on s, s⁻¹ and kᵗ s⁻¹ k; this is its formulation of local uniform ellipticity. The class used here also implies the weaker local integrability conditions needed by the analytic library. All quantitative content — the reference aspect ratio Π, the concentration gauge and its growth witness, the annealed contrast Θ_m, and every dimensional constant — is unchanged.

Module layout

• HCPoly.Setup.CoefficientSpace — the class AEUniformlyEllipticField, the carrier CoeffSpace, the translation action, and pointwise representatives.
• HCPoly.Setup.LocalSigmaFields — the generators of F(U), the canonical measurable structure, unit separation, and measurability of the translation.
• HCPoly.Setup.BlockAlgebra — Schur data, the scalar Loewner bound, the intrinsic contrast, the reference aspect ratio, the matrix square root.
• HCPoly.Setup.SpectralBound — specBound is the least scalar Loewner bound; matSqrt is the unique positive semidefinite square root.
• HCPoly.Setup.Contrast — the conjugation identity and the printed form of the intrinsic contrast and of Λ_0.
• HCPoly.Setup.Geometry — triadic cubes, the rounding convention, adapted cells, ellipsoids.
• HCPoly.Setup.Response — the coarse response, the annealed block, Θ_m, and the identities that pin the annealed block to an expectation.
• HCPoly.Setup.ResponsePositivity — positive definiteness of the coarse response on the coefficient space.