High-contrast homogenization: Entry.Response.Rows.TerminalEnergyMeasurability #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The Terminal-Cell Energy Identity and Its Measurability #
At a response maximizer, this file shows the variation energy of the optimizer is exactly twice
the pathwise response, so the volume average of the energy density over the terminal cell equals
twice the recentred response, for both signs. Exact partition averaging spreads this over the
aligned subcells of the coarse scale, and the energy density is a.e. nonnegative on every
subcell, giving the same identity and nonnegativity for an a.e. elliptic representative. Since
two response maximizers for the same loads and coefficient share the same doubled optimizer field
a.e., every weighted average of the terminal optimizer's energy density is measurable in the
sample. It also records the Cauchy-Schwarz bound for a cutoff-weighted pairing of two
square-integrable fields, the bound |φ - 1| ≤ 1 for a cutoff of mean one, and that a
nonnegative summand of an integrable sum is itself integrable. The terminal-cell energy identity
and the sample measurability of the weighted optimizer energy are the first error row of
p.response.transfer at the level of the response carriers.
The terminal cell energy of the optimizer on the response carriers #
At a response maximizer the variation energy of the optimizer is exactly twice the response, so the
volume average of its energy density over the terminal cell is 2 J_t^∓. Exact partition averaging
then spreads that identity over the aligned subcells of the coarse scale, and the energy density is
nonnegative almost everywhere on every such subcell because the coefficient has an elliptic
representative on the terminal cell. These are the pathwise facts from which the sample-side
integrability of the subcell energies is obtained by domination, and they are the first error row of
p.response.transfer at the level of the carriers.
Integrability of the terminal optimizer energy, minus sign. On the terminal cell the
variation energy density of a terminal maximizer for a_- = a - g is integrable; the coefficient is
elliptic only almost everywhere, so the fact is transported from an elliptic representative.
The terminal energy identity, minus sign. At a terminal maximizer for a_- = a - g the
volume average of the optimizer energy density over the terminal cell is twice the terminal
response, the exact energy identity behind the first error row of p.response.transfer.
Nonnegativity of the subcell energy, minus sign. On every aligned subcell of the coarse scale the average of the terminal optimizer energy density is nonnegative, because the terminal coefficient is elliptic almost everywhere there.
Exact partition averaging of the terminal energy, minus sign. The normalized sum over the aligned subcells of the coarse scale of the optimizer energy averages equals the terminal energy average, since those subcells partition the terminal cell up to a null set.
Integrability of the terminal optimizer energy, plus sign. The twin of the minus statement
for a_+ = aᵗ + g.
The terminal energy identity, plus sign. The twin of the minus identity for
a_+ = aᵗ + g.
Nonnegativity of the subcell energy, plus sign. The twin of the minus statement for
a_+ = aᵗ + g.
Exact partition averaging of the terminal energy, plus sign. The twin of the minus
statement for a_+ = aᵗ + g.
The cutoff-weighted pairing bound on a cell #
This file records the crude Cauchy--Schwarz bound for a cutoff-weighted pairing of two
square-integrable vector fields on a cell, used in the cutoff argument of
e.response.cutoff.estimate (AK.HC Lemma A.1, (A.4)). If φ is a cutoff with 0 ≤ φ ≤ 2,
A and B are vector fields and vol denotes the normalized cell average, then
|vol (φ · (A · B))| ≤ 2 · √(vol |A|²) · √(vol |B|²).
The three ingredients are the pointwise Cauchy--Schwarz inequality for the Euclidean dot
product, the monotonicity of the normalized cell average under pointwise domination, and an
averaged Cauchy--Schwarz (Hölder with conjugate exponents 2 and 2) for nonnegative
functions.
Monotonicity of the normalized cell average: if |f| ≤ g pointwise on the measurable set
U and both are integrable on U, then |vol f| ≤ vol g. The prefactor (vol U)⁻¹ is
nonnegative and the set integral is monotone. This is the averaging step of the
cutoff-weighted pairing bound e.response.cutoff.estimate.
The weighted optimizer energy is measurable in the sample #
Two response maximizers for the same loads and the same coefficient have the same doubled optimizer
field almost everywhere on the response cell, by Chapter-2 almost-everywhere gradient uniqueness
(AK.HC (2.9)). Hence every weighted average of the energy density of the terminal optimizer of
p.response.transfer is computed by the canonical Chapter-2 selection, which depends measurably on
the sample. In particular the (φ - 1)-weighted terminal energy and the energy on each aligned
subcell are measurable functions of the sample, for an arbitrary family of maximizers.
The pointwise pairing of the doubled optimizer field with itself is the variation energy
∇v · (symmPart b) ∇v: the antisymmetric part of the coefficient contributes nothing to the
quadratic form, so the pairing may be read with either the full coefficient or its symmetric part.
The η-weighted terminal optimizer energy of the response functional for the recentred
coefficient a_-, for an arbitrary family of response maximizers, is measurable in the coefficient
sample. The energy is the pairing ⟨X.1, X.2⟩ = ∇v · a_- ∇v of the doubled optimizer field of the
maximizer; that field is determined almost everywhere by the measurable canonical Chapter-2
selection, so the weighted energy is a measurable function of the sample.
The η-weighted terminal optimizer energy of the response functional for the adjoint
coefficient a_+, for an arbitrary family of response maximizers, is measurable in the coefficient
sample. This is the adjoint twin of measurable_volumeAverage_weighted_energy_respCoeffMinus.
The fluctuation of a response cutoff #
A response cutoff φ of e.response.cutoff.estimate lies in [0, 2] and has volume average
one on its terminal cell, so its fluctuation φ - 1 is bounded by one pointwise and has mean
zero on that cell. Mean zero is what allows any constant to be subtracted from the factor paired
with the fluctuation without changing the weighted average: the centring step of the cutoff-mean
rows of the response-transfer estimate.
Integrability of one nonnegative term of an integrable sum #
The first error row of p.response.transfer needs the P-integrability of each subcell deficit
and of each subcell energy separately, while the exact partition identity supplies only the
integrability of their flat average. Both families are nonnegative, so each member is dominated
by the whole sum; with almost-everywhere strong measurability of the member, integrability
follows. This file records that elementary principle in the two forms the row consumes: a sum
bounded above by an integrable function, and a sum equal to one.
A single nonnegative term of a sum bounded by an integrable function is integrable. Let
f i be nonnegative for every i ∈ Z, let each f i be P-a.e.-strongly measurable, let g
be P-integrable, and suppose ∑ i ∈ Z, f i a ≤ g a for every a. Then each f i with
i ∈ Z is P-integrable. At every point the fixed term is bounded by the full sum through the
nonnegativity of the other terms and the sum is bounded by g; the norm in the domination
criterion is the term itself because the term is nonnegative.
A single nonnegative term of a sum equal to an integrable function is integrable. This
is the previous statement with the upper bound ∑ i ∈ Z, f i a ≤ g a strengthened to the
identity ∑ i ∈ Z, f i a = g a supplied by the exact partition identity of
p.response.transfer; the identity gives the bound by le_of_eq.