High-contrast homogenization: Annealed.Integrability #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Integrability of the coarse response over the law #
Under e.coarse.ellipticity every entry of the coarse response on a centered
cube is integrable over the law, at every scale. The assumption run at the
sample's own burn scale bounds the response by an affine function of the source,
and the gauge tail makes the source integrable; the measurability of the coarse
observable for the local sigma-fields holds on the coefficient space itself.
theorem
HCPolySupport.HighContrast.hasIntegrableCoarseBlock_of_coarseEllipticityDagger
{d : ℕ}
{P : MeasureTheory.Measure (CoeffSpace d)}
[MeasureTheory.IsProbabilityMeasure P]
{g : ℝ}
{E : BlockMat d}
{Ψ : ℝ → ℝ}
{K : ℝ}
{S : CoeffSpace d → ℝ}
(hdag : CoarseEllipticityDagger P g E Ψ K S)
(m : ℤ)
:
HasIntegrableCoarseBlock P (centeredCube d m)
The integrability of the annealed block. Under
e.coarse.ellipticity every entry of the coarse response on a centered cube
is integrable over the law, at every scale.