High-contrast homogenization: Entry.Response.Cutoff.CanonicalCutoffPairingMeasurability #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Measurability of the canonical cutoff pairing #
The cutoff pairing of the response estimate e.response.cutoff.estimate is the volume average
of the cutoff-weighted Euclidean pairing of the potential and flux defects of the doubled
canonical optimizer state; its integrand expands into one genuinely quadratic term and three
linear terms. This file proves the almost-sure strong measurability of that pairing and of its
absolute value, for both the recentred and adjoint canonical coefficients, together with the
square-integrability of the flux slots the expansion needs. It also shows that a coefficient
field agreeing almost everywhere on a Chapter-2 domain with a field uniformly elliptic there has
the same subcell averages, for every response maximizer, as the Chapter-2 canonical maximizer for
that field.
Measurability of the canonical cutoff pairing #
The cutoff pairing of the response estimate e.response.cutoff.estimate is the volume average of
the cutoff-weighted Euclidean pairing of the potential defect and the flux defect of the doubled
canonical optimizer state. Its integrand expands into one genuinely quadratic term and three
linear terms,
φ · ⟨Z.1 - Y.1, Z.2 - Y.2⟩ = φ · ⟨Z.1, Z.2⟩ - φ · ⟨Z.1, Y.2⟩ - φ · ⟨Y.1, Z.2⟩ + φ · ⟨Y.1, Y.2⟩,
of which the last three are linear readouts of the canonical state against the fixed vector Y
and the constant ⟨Y.1, Y.2⟩.
This file records the algebraic expansion and the measurability of the three linear readouts, for
the two recentred response coefficient families of e.response.cutoff.estimate. The remaining
quadratic term is the volume average of the energy density of the canonical optimizer; it is not a
linear readout of the canonical optimizer state; it is carried below as an explicit hypothesis.
The algebraic expansion #
The Euclidean pairing of the potential defect A - B and the flux defect C - D expands into
one quadratic and three linear terms. This is the pointwise expansion of the integrand of the
cutoff pairing e.response.cutoff.estimate (AK.HC Lemma A.1, (A.4)).
Linear readouts of the canonical optimizer state #
The cutoff-weighted volume average of the potential component of the canonical optimizer state
paired against a fixed vector is measurable, on a family of Chapter-2 coefficient objects whose
weighted coordinate volume averages are measurable. This is the potential linear term in the
expansion of the cutoff pairing of e.response.cutoff.estimate (AK.HC Lemma A.1, (A.4)).
The cutoff-weighted volume average of the flux component of the canonical optimizer state
paired against a fixed vector is measurable. This is the flux linear term in the expansion of the
cutoff pairing of e.response.cutoff.estimate (AK.HC Lemma A.1, (A.4)).
The two recentred response coefficient families #
The flux component of the canonical minus optimizer state is square integrable on the adapted cell; the a.e. elliptic representative of the recentred coefficient controls it.
The flux component of the canonical plus optimizer state is square integrable on the adapted cell.
The weighted coordinate volume averages of the canonical minus optimizer state are measurable,
so the potential linear term of the minus cutoff pairing of e.response.cutoff.estimate
(AK.HC Lemma A.1, (A.4)) is measurable.
The potential linear term of the plus cutoff pairing of e.response.cutoff.estimate
(AK.HC Lemma A.1, (A.4)) is measurable.
The flux linear term of the minus cutoff pairing of e.response.cutoff.estimate
(AK.HC Lemma A.1, (A.4)) is measurable.
The flux linear term of the plus cutoff pairing of e.response.cutoff.estimate
(AK.HC Lemma A.1, (A.4)) is measurable.
Reduction of the full pairing to its quadratic term #
If the cutoff-weighted volume average of the quadratic self-pairing of the canonical optimizer
state is almost-everywhere strongly measurable, then so is the full canonical cutoff pairing
itself. The integrand expands into that quadratic term and the three linear readouts whose
measurability is recorded above; the expansion of e.response.cutoff.estimate
(AK.HC Lemma A.1, (A.4)) is applied pointwise and the four integrands are integrable, so the volume
average of the sum is the corresponding sum of volume averages.
The absolute value of the full canonical cutoff pairing of e.response.cutoff.estimate
(AK.HC Lemma A.1, (A.4)) is almost-everywhere strongly measurable as soon as the cutoff-weighted
quadratic self-pairing of the canonical optimizer state is: the modulus is a continuous function of
the pairing recorded by aestronglyMeasurable_pairing_canonical_of_quadratic.
The canonical minus cutoff pairing of e.response.cutoff.estimate is almost-everywhere
strongly measurable, provided the cutoff-weighted volume average of the quadratic self-pairing of
the canonical minus optimizer state is. This is the hcanon_pairing minus conjunct with the
quadratic term isolated as an explicit hypothesis.
The canonical plus cutoff pairing of e.response.cutoff.estimate is almost-everywhere
strongly measurable, provided the cutoff-weighted volume average of the quadratic self-pairing of
the canonical plus optimizer state is.
Cell averages of an arbitrary response maximizer and the canonical selection #
For a coefficient field b that agrees almost everywhere on a Chapter-2 domain U with a field
f that is uniformly elliptic there, every response maximizer for the loads p, q has the
subcell averages of the doubled optimizer state of the Chapter-2 canonical maximizer for f.
The argument is that a response maximizer may be normalized to have mean zero without changing its
gradient, so it is a scalar canonical maximizer; Chapter-2 a.e. gradient uniqueness for response
maximizers (AK.HC (2.9)) then identifies its doubled optimizer state with that of the canonical
choice. This is the step that makes the doubled optimizer state X_t^\pm entering the weak
quantity W^\pm of e.response.weak.estimate a function of the sample through the measurable
canonical selection alone, independently of the choice of maximizer.
If b agrees almost everywhere on a Chapter-2 domain U with a field f that is uniformly
elliptic on U, then the cell average over any V ⊆ U of the doubled optimizer state of an
arbitrary response maximizer of the response functional for loads p, q equals the cell average
of the doubled optimizer state of the Chapter-2 canonical maximizer for f.
A response maximizer may be normalized to have mean zero without changing its gradient, so it
becomes a scalar canonical maximizer; Chapter-2 a.e. gradient uniqueness for response maximizers
(AK.HC (2.9)) then identifies its doubled optimizer state with that of the canonical choice. This
is the form in which the doubled optimizer state X_t^\pm entering the weak quantity W^\pm of
e.response.weak.estimate is seen to be determined by the measurable canonical selection,
independently of the choice of maximizer.