High-contrast homogenization: Entry.Response.Cutoff.CutoffDualityBoundInputs #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The negative-Besov duality bound, assembled on the reference cube #
This file serves the cutoff estimate e.response.cutoff.estimate. On the adapted cell
U_t = q ⋄_t, it identifies the cutoff-weighted pairing of a harmonic optimizer's gradient and
flux defects, after integration by parts and the change of variables x = q y, with minus the
centred cube average of the potential against the flux defect contracted with the cutoff's
gradient (AK.HC Lemma A.1, (A.4)). It reads the generic negative-Besov duality bound of the
cutoff-product estimate on the reference cube, records the Besov scale weight of that cube at
every scale, and shows that coordinatewise square-integrability of a doubled field on the
adapted cell supplies the measurability and the two slotwise pullback bounds the duality
estimate needs. Instantiating these at the centred doubled optimizer field, it assembles the
resulting coefficient and seminorm bounds into the shape the final inequality consumes: a
dimension-only constant times the scale weight 3^{-t} times the square of the scale-average
seminorm.
The cutoff pairing as a centred cube average #
On the adapted cell U_t = q ⋄_t the cutoff-weighted pairing of the gradient defect of a harmonic
optimizer against its flux defect is, after integration by parts, the negative of the normalized
average of the centred potential against the flux defect contracted with the gradient of the
cutoff. Composing that identity with the change of variables x = q y and with the adjointness of
the pulled-back pairing rewrites it on the reference cube originCube d t, where the first slot of
the pairing is the pulled-back flux defect q⁻¹ (b ∇v - q₀) and the second is the gradient of the
pulled-back cutoff.
Because the flux defect is weakly divergence free, the pairing against the cutoff gradient has zero
average, so centring the potential changes nothing. The resulting identity is the exact shape in
which the negative-Besov duality bound e.response.cutoff.estimate (AK.HC Lemma A.1, (A.4)) is
stated, and it is the identity half of the pathwise cutoff bound: no inequality is involved.
The cutoff-weighted pairing of the gradient defect of a harmonic optimizer against its flux
defect, normalized over the adapted cell U_t = q ⋄_t, equals the negative of the normalized
average over the reference cube originCube d t of the Euclidean pairing of the pulled-back flux
defect q⁻¹ (b ∇v - q₀) against the centred pulled-back potential multiplied into the gradient of
the pulled-back cutoff. With q the selected grid and the potential centred at its cube average,
this is the identity form of the cutoff estimate e.response.cutoff.estimate (AK.HC Lemma A.1,
(A.4)) on which the negative-Besov duality bound is applied.
The negative-Besov duality bound at the pulled-back cutoff data #
The generic cutoff-product duality bound
abs_cubeAverage_vecDot_centered_scalar_cutoff_le_scaledWeakNormProduct in namespace
Book.Ch05.Section53.JUpperBoundWeakNorms
is the direct full-dual pairing form of AK.HC Lemma A.1, (A.4). This file reads it on the
reference cube after the linear change of variables of the cutoff argument: with q = respGrid jStar F
the centred potential is the pulled-back response respCenteredPullbackH1, the flux is the
pulled-back flux defect q⁻¹ (b ∇v - q0), and the weight field is the gradient of the pulled-back
cutoff scalarCutoffGradientField (φ ∘ q). Every analytic hypothesis of the generic bound — the
membrane and smoothness data of the weight field, the derivative bound, the square integrability of
the flux, and the two negative-Besov seminorm bounds — is carried through unchanged at the
pulled-back data.
The negative-Besov duality bound (AK.HC Lemma A.1, (A.4)) evaluated on the reference cube
openCubeSet (originCube d t) at the pulled-back cutoff data. With q = respGrid jStar F, the
potential is the centred pulled-back response respCenteredPullbackH1, the flux is the pulled-back
flux defect q⁻¹ (b ∇v - q0), and the weight field is the gradient of the pulled-back cutoff
φ ∘ q; the two exponents take the symmetric value s = t = 1/2.
The conclusion is the generic bound with its four coefficient lets expanded: the scaled gradient
and flux seminorms, the cutoff-product coefficient built from the derivative bound B and the
cutoff gradient, and the flux coefficient carrying the scale weights.
Constant arithmetic of the pathwise cutoff bound #
The negative-Besov duality bound of the cutoff argument yields, on a reference cube, the product
of two coefficient bounds and two pulled-back seminorm bounds. This file assembles those factors
into the single shape consumed by the annealed estimate: a dimensional constant times the scale
weight 3 ^ (-t) times the square of the scale-average seminorm.
The product of the two coefficient bounds and the two pulled-back seminorm bounds of the
pathwise cutoff estimate is bounded by the dimensional constant K * L * 3 * d ^ 2 times the
scale weight 3 ^ (-t) and the squared scale-average seminorm A ^ 2. The coefficient bounds
are gradCoeff ≤ K * 3 ^ (-t) and fluxCoeff ≤ L, the seminorm bounds are scaledGrad ≤ G * A
and scaledFlux ≤ f * A, and the two Frobenius factors satisfy G * f ≤ 3 * d ^ 2; all
quantities are nonnegative. This is the form in which the pathwise cutoff bound
e.response.cutoff.estimate feeds the annealed one.
The Besov scale weight of the reference cube #
The Besov scale weight cubeBesovScaleWeight s Q = (cubeScaleFactor Q) ^ (-s) assigns to a
triadic cube Q the geometric weight 3^{-s · scale Q}, and the scale factor is
cubeScaleFactor Q = 3 ^ (scale Q). On the reference cube originCube d t, whose scale is t,
both collapse to elementary powers of 3: the scale factor is 3 ^ t and the weight is
3^{-s t}. In particular the exponent -1/2 weight is 3^{t/2} and the exponent 0 weight is
1. These are the weight evaluations used when the cutoff support of the response is expanded in
the cube Besov scale.
The scale factor of the reference cube originCube d t is 3 ^ t, the integer power being
read as a real power: cubeScaleFactor (originCube d t) = 3 ^ (t : ℝ).
The exponent -1/2 Besov scale weight of the reference cube originCube d t is the positive
power 3^{t/2}: cubeBesovScaleWeight (-(1/2)) (originCube d t) = 3 ^ ((t : ℝ) / 2).
The exponent 0 Besov scale weight of the reference cube originCube d t is 1:
cubeBesovScaleWeight 0 (originCube d t) = 1.
The pulled-back cube seminorms of a square-integrable doubled field #
The slotwise pullback bounds of the cutoff argument, partialSeminorm_pullback_fst_le and
partialSeminorm_pullback_snd_le, take as side conditions the integrability of the squared
Euclidean length of the metric-scaled doubled field together with its a.e. strong measurability.
This file supplies both from the single hypothesis that all 2d coordinates of the field are
square integrable on the adapted cell.
Paper: the cutoff estimate e.response.cutoff.estimate and its pullback to the reference cube.
The metric scaling of a doubled field, by m^{1/2} in the first slot and m^{-1/2} in the
second, carries square integrable coordinates to square integrable coordinates.
The slotwise pullback bound for the gradient slot, with the square-integrability side conditions supplied coordinatewise.
The slotwise pullback bound for the flux slot, with the square-integrability side conditions supplied coordinatewise.
The two negative-Besov inputs of the duality bound #
The slotwise pullback bounds partialSeminorm_pullback_fst_le_of_coords and
partialSeminorm_pullback_snd_le_of_coords hold for an arbitrary doubled field X, subject to
coordinatewise square integrability on the adapted cell. This file instantiates them at the
centred doubled optimizer field X = (∇v, b∇v) − Y. The first slot is then the gradient of the
centred pulled-back potential, and the second the pulled-back flux defect, while the
square-integrability hypotheses are supplied by the elliptic representative of the coefficient.
Paper: the cutoff estimate e.response.cutoff.estimate and the scale-average seminorm bound of
e.response.weak.estimate.
The negative-Besov seminorm on the reference cube of the gradient of the centred pulled-back
potential is bounded by the scale-average seminorm of the metric-scaled doubled optimizer state,
with the metric factor qᵀ m^{-1/2}. This is the gradient slot of the duality bound, obtained by
applying partialSeminorm_pullback_fst_le_of_coords to the centred doubled optimizer field and
rewriting its first slot through the gradient of the pulled-back centred potential.
The negative-Besov seminorm on the reference cube of the pulled-back flux defect is bounded by
the scale-average seminorm of the metric-scaled doubled optimizer state, with the metric factor
q⁻¹ m^{1/2}. This is the flux slot of the duality bound, obtained by applying
partialSeminorm_pullback_snd_le_of_coords to the centred doubled optimizer field.