High-contrast homogenization: Provider.Quenched.SmallContrastSymmetricSkew #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The symmetric Schur skew at small diagonal contrast #
The large-contrast version of the symmetric-skew estimate fails, while its
perturbative form is sufficient for the small-contrast centering estimate.
This file proves that form directly: if a
Schur block with a SYMMETRIC skew coordinate dominates its own sharp, then the
σ*-normalized skew is bounded by half of the normalized diagonal gap —
first order in θ − 1, with an absolute constant.
The proof tests the sharp ordering at the single doubled vector
(x, t·k x) with t = (θ+1)/(θ−1): at this shift the cross terms cancel and
the quadratic gain appears with the exact constant 4/(θ−1).
The symmetric Schur skew is first order in the diagonal gap. If the
Schur block of (S, S₊, k) with symmetric k dominates its sharp and
S ≤ θ·S₊, then the S₊⁻¹-quadratic of k is bounded by ((θ−1)/2)²
times the S₊-quadratic. This is the perturbative half of the printed
display HC (2.81), proved without the refuted large-contrast half.
Matrix-order form of the symmetric-skew bound.