High-contrast homogenization: Entry.Response.Cutoff.DualityBoundHypotheses #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Hypotheses for the cutoff-pairing duality bound #
This file assembles analytic hypotheses feeding the negative-Besov duality bound behind the
cutoff-pairing estimate e.response.cutoff.estimate (AK.HC Lemma A.1, (A.4)). It records that the
adapted cell is the image of the reference cube under the linear map y ↦ q y, so a cell average
there identifies with a cube average after this change of variables, and shows the pulled-back
cutoff gradient field ξ is smooth coordinatewise, uniformly bounded to second order, and
measurable. It supplies the square-integrability of the pulled-back optimizer flux under an
almost-everywhere elliptic representative, the integrability of the cutoff-weighted pairing on a
cell, and the weak solenoidality of the flux defect of an A-harmonic function. It also shows
that the affine defect of an H¹ function is again H¹ with the expected gradient, and that the
integrand of the weak quantity W^\pm of e.response.weak.estimate is almost-everywhere
strongly measurable in the coefficient sample.
Measurability of the weak-quantity integrand of the response estimate #
The weak quantity W^\pm of e.response.weak.estimate is the sample expectation of the square of
the scale-average seminorm besovSeminorm of the recentred, canonically-metric-scaled cell averages
of the doubled optimizer state X_t^\pm. This file records that this integrand is
AEStronglyMeasurable in the coefficient sample.
The doubled optimizer state of an arbitrary response maximizer is determined almost everywhere on the response cell by the Chapter-2 canonical maximizer, by a.e. gradient uniqueness for response maximizers (AK.HC (2.9)); the canonical choice depends measurably on the sample by the Galerkin selection construction; and the scale-average seminorm is a countable sum of finite sums of measurable terms. The family is summed only over the triadic index box, so its entries outside the box may be replaced by a constant before applying the sum-measurability result.
The integrand a ↦ [M_0^{1/2} (X_t^\pm(a) - Y)]^2 of the weak quantity
e.response.weak.estimate is almost-everywhere strongly measurable in the coefficient sample, for
an arbitrary family of response maximizers u. The cell averages entering the doubled optimizer
state are the canonically selected ones up to a null set, and the canonical selection is measurable
in the sample.
The affine defect of an H¹ function #
Given an H¹ function u on a bounded open convex domain U, a vector P and a
constant c, the affine defect x ↦ u x - P · x - c is again H¹, with gradient
∇u - P. In the cutoff argument for e.response.cutoff.estimate the gradient
defect ∇v - P of an optimizer is paired against a flux defect, and integration
by parts requires that this defect be the gradient of an H¹ function: namely of
the affine defect of v, since the linear function x ↦ P · x and the constants
are H¹ on a bounded domain and H¹ is closed under subtraction.
The affine defect x ↦ u x - vecDot P x - c of an H¹ function u,
packaged as an H¹ function on U. The linear function x ↦ vecDot P x is
globally smooth, hence H¹ on the bounded domain U
(H1Function.ofContDiffOnIsOpenBoundedConvexDomain), and constants are H¹ on
a finite-measure domain (H1Function.const).
Equations
Instances For
The underlying function of the affine defect is x ↦ u x - vecDot P x - c.
The gradient of the affine defect is ∇u - P.
The change of variables from an adapted cell to the reference cube #
The adapted cell HighContrast.adaptedCell q t of generation t is the image of the reference open
cube openCubeSet (originCube d t) under the linear map y ↦ q y
(image_openCubeSet_originCube_eq_adaptedCell). This file records the two elementary facts that
let the normalized averages on the two sides be identified: the aligned cell at lattice index 0
is the centred adapted cell, and translating the reference cube by the zero lattice index fixes it.
Together with cubeAverage_comp_matVecMul, these transport a normalized average on the adapted
cell to the reference cube, which is the change of variables behind the adapted-to-Euclidean
comparison, HC Lemma 2.15, (2.127)--(2.128).
Translating the reference cube originCube d t by the zero lattice index leaves the cube
unchanged.
The normalized average of f over the adapted cell HighContrast.adaptedCell q t equals the
normalized average of the pulled-back function y ↦ f (matVecMul q y) over the reference cube
originCube d t. This is the change of variables x = q y for normalized averages, behind the
adapted-to-Euclidean comparison of HC Lemma 2.15, (2.127)--(2.128).
The pulled-back flux on the reference cube, with the elliptic datum packaged existentially #
The two inputs below restate the square-integrability of the optimizer flux and of its pullback to
the reference cube in the form in which the cutoff argument consumes them: the coefficient is only
assumed to agree almost everywhere on the adapted cell with a field uniformly elliptic there, and
the elliptic constants and representative are introduced by an existential rather than carried as
separate hypotheses. The a.e.-representative shape is forced by the coefficient carrier, whose
points are a.e. classes, and is enough for every MemLp consumer, all of which are a.e. invariant.
Paper: the cutoff estimate e.response.cutoff.estimate and its negative-Besov duality bound
(AK.HC Lemma A.1, (A.4)), whose flux slot is supplied here.
The pulled-back flux is L² on the reference cube, elliptic datum existentially
quantified. If the coefficient b agrees almost everywhere on the adapted cell
HighContrast.adaptedCell q t with a field f uniformly elliptic there, then the pulled-back
centred
flux defect y ↦ q⁻¹ ((optimizerField b u (q y) − Y).2) of an b-harmonic optimizer u is square
integrable for the normalized measure of the reference cube. This is the hflux slot of the
generic CG product bridge abs_cubeAverage_vecDot_centered_scalar_cutoff_le_scaledWeakNormProduct
used in the cutoff estimate e.response.cutoff.estimate.
The flux of an optimizer is L² on the cell, elliptic datum existentially quantified.
If the coefficient b agrees almost everywhere on the adapted cell HighContrast.adaptedCell q t
with
a field f uniformly elliptic there, then the flux x ↦ b x · u.toH1.grad x of a b-harmonic
optimizer u is square integrable on the cell. This is the hflux hypothesis of the
integration-by-parts step in the negative-Besov duality bound (AK.HC Lemma A.1, (A.4)) behind
the cutoff estimate e.response.cutoff.estimate.
Integrability of the cutoff-weighted pairing on a cell #
The integration-by-parts form of the cutoff pairing e.response.cutoff.estimate
(AK.HC Lemma A.1, (A.4)) integrates the product of a bounded cutoff with the Euclidean pairing of
two square-integrable vector fields against the cell measure. This file supplies the
integrability side conditions for that pairing, and the integrability of the squared Euclidean
length of a square-integrable vector field.
A bounded measurable weight times the Euclidean pairing of two coordinatewise square
integrable vector fields is integrable on a cell. This is the integrability side condition of
the cutoff-weighted pairing e.response.cutoff.estimate (AK.HC Lemma A.1, (A.4)).
The flux defect of an A-harmonic function is solenoidal #
For a set U ⊆ ℝ^d, a vector field is weakly solenoidal on U when its pairing with the
gradient of every compactly supported H¹₀(U) test function vanishes
(HCPolySupport.IsSolenoidalOn). The flux x ↦ b(x) ∇u(x) of an A-harmonic function is
solenoidal by the definition of A-harmonicity, and a constant field is solenoidal against
H¹₀(U) tests because the componentwise average of a zero-trace H¹ gradient vanishes.
Consequently the flux defect obtained by subtracting a constant vector from the flux is again
solenoidal; this is the field that the cutoff pairing of e.response.cutoff.estimate
(AK.HC Lemma A.1, (A.4)) pairs against.
The constant-field statement and the defect statement need integrability of the fields being
paired, which in this library is supplied by finiteness of the restricted volume measure (for the
constant) and by an explicit L²(U) witness (for the flux); the corresponding hypotheses appear
on the two refined statements below.
A constant vector field is weakly solenoidal on a finite-measure domain: every zero-trace
H¹ test gradient has vanishing componentwise integral, so pairing it with a constant vector
gives zero.
On a finite-measure domain, the flux defect x ↦ b(x) ∇u(x) - Q is weakly solenoidal
whenever the flux has an L²(U) representative: the flux is solenoidal, the constant field is
solenoidal, and both pairings are integrable, so their sum is solenoidal.
The pulled-back cutoff gradient as an admissible weight field #
For a cutoff φ and a matrix qq, the pulled-back cutoff is ψ(y) = φ (qq · y). Its gradient
field scalarCutoffGradientField ψ is the weight field ξ consumed by the negative-Besov duality
bound for the product term of the cutoff estimate (e.response.cutoff.estimate), which needs three
facts about ξ: it is smooth coordinatewise, each coordinate field has derivative bounded by the
second-derivative bound carried by the cutoff class, and ξ is bounded by the Lipschitz constant
of the pulled-back cutoff. The three declarations below record exactly those facts.
Coordinatewise smoothness of the gradient field of the pulled-back cutoff
(e.response.cutoff.estimate; the cutoff class of IsResponseCutoff).
Bound on the derivative of a coordinate of the gradient field of the pulled-back cutoff by the
second-derivative bound the cutoff class carries (e.response.cutoff.estimate; the cutoff class
of IsResponseCutoff).
Bound on the gradient field of the pulled-back cutoff by the Lipschitz constant of the
pullback (e.response.cutoff.estimate; the cutoff class of IsResponseCutoff).