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a458723 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | import Mathlib
import OAI.Analysis.Conductivity.Scalarization.FineScalarSequence
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set Filter Topology
lemma dense_property_in_closure
{X : Type*} [TopologicalSpace X] (S : Set X) (P : X → Prop)
(h : ∀ x ∈ S, ∃ z : ℕ → X, Tendsto z atTop (𝓝 x) ∧
(∀ n, z n ∈ S) ∧ ∀ᶠ n in atTop, P (z n)) :
Dense {x : closure S | P x.val} := by
rw [Subtype.dense_iff]
apply closure_minimal _ isClosed_closure
intro x hx
obtain ⟨z,hz,hzS,hzP⟩ := h x hx
apply mem_closure_of_tendsto hz
filter_upwards [hzP] with n hn
exact ⟨⟨z n,subset_closure (hzS n)⟩,hn,rfl⟩
end ScalarConductivity
end
end
section
noncomputable section
namespace ScalarConductivity
open MeasureTheory Set Filter Topology TopologicalSpace
open scoped ENNReal
def finiteUpdatePairs
(μ : Measure Coord3) [μ.IsAddHaarMeasure]
(U : Set Coord3) (u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3)
(hE : MemLp (voltageGradient u) 2 (μ.restrict U))
(hF : MemLp (voltageFlux u A) 2 (μ.restrict U)) (a b : ℝ) :
Set (Lp FieldVector 2 (μ.restrict U) × Lp FieldVector 2 (μ.restrict U)) :=
{Z | ∃ R : CompactGlobalUpdate μ U u A,
μ (U \ regularRegion (fun x => u x+R.du x) R.tensor U) = 0 ∧
(∀ᵐ x ∂μ, x ∈ U → R.tensor x ∈ matrixFiniteLaminate a b) ∧
Z = R.fieldPair μ hE hF}
lemma finiteUpdatePairs_nonempty
(μ : Measure Coord3) [μ.IsAddHaarMeasure]
{U : Set Coord3} {u : Coord3 → Fin 2 → ℝ} {A : Coord3 → Symmetric3}
(hAm : Measurable A)
(hE : MemLp (voltageGradient u) 2 (μ.restrict U))
(hF : MemLp (voltageFlux u A) 2 (μ.restrict U)) {a b : ℝ}
(hreg : μ (U \ regularRegion u A U) = 0)
(hgraph : ∀ᵐ x ∂μ, x ∈ U → A x ∈ matrixFiniteLaminate a b) :
(finiteUpdatePairs μ U u A hE hF a b).Nonempty := by
refine ⟨(CompactGlobalUpdate.refl μ U u A hAm).fieldPair μ hE hF,
CompactGlobalUpdate.refl μ U u A hAm,?_,hgraph,rfl⟩
simpa only [CompactGlobalUpdate.refl,Pi.zero_apply,add_zero] using hreg
lemma dense_ApproxScalar_in_finiteUpdateClosure
(μ : Measure Coord3) [μ.IsAddHaarMeasure] [Measure.InnerRegularCompactLTTop μ]
{a b : ℝ} (ha : 0 < a) (hab : a < b)
{U : Set Coord3} (hUb : Bornology.IsBounded U) (hUm : MeasurableSet U)
[IsFiniteMeasure (μ.restrict U)] [(μ.restrict U).WeaklyRegular]
(u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3)
(hE : MemLp (voltageGradient u) 2 (μ.restrict U))
(hF : MemLp (voltageFlux u A) 2 (μ.restrict U))
(hint : SmoothFluxIntegrable μ U (conductivityFlux u A))
(hdiv : ∀ j (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → HasCompactSupport ψ →
tsupport ψ ⊆ U → (∫ x, fderiv ℝ ψ x ((conductivityFlux u A x).col j) ∂μ) = 0)
(hreg : μ (U \ regularRegion u A U) = 0) {ε : ℝ} (hε : 0 < ε) :
Dense {z : closure (toWeakSpace ℝ _ '' finiteUpdatePairs μ U u A hE hF a b) |
ApproxScalar (μ.restrict U) a b ε ((toWeakSpace ℝ _).symm z.val)} := by
apply dense_property_in_closure
rintro _ ⟨Z,⟨R,hregR,hgraphR,rfl⟩,rfl⟩
obtain ⟨hER,hFR⟩ := R.fields_memLp μ hUm hreg hE hF
obtain ⟨hintR,hdivR⟩ := R.preserves_harmonic μ hint hdiv
obtain ⟨S,hregS,hgraphS,hw,happrox⟩ := exists_fine_scalar_sequence μ ha hab hUb hUm
(fun x => u x+R.du x) R.tensor R.measurable_tensor hER hFR hintR hdivR hregR hgraphR
have he : ∀ n, (R.trans μ (S n)).fieldPair μ hE hF = (S n).fieldPair μ hER hFR :=
fun n => R.fieldPair_trans μ hUm (S n) hreg hregR hE hF hER hFR
refine ⟨fun n => toWeakSpace ℝ _ ((R.trans μ (S n)).fieldPair μ hE hF),?_,?_,?_⟩
· simpa only [he,R.fieldPair_eq μ hUm hreg hE hF hER hFR] using hw
· intro n
refine ⟨_,⟨R.trans μ (S n),?_,hgraphS n,rfl⟩,rfl⟩
simpa only [CompactGlobalUpdate.trans,Pi.add_apply,add_assoc] using hregS n
· filter_upwards [happrox ε hε] with n hn
change ApproxScalar (μ.restrict U) a b ε ((R.trans μ (S n)).fieldPair μ hE hF)
rwa [he n]
theorem exact_scalar_in_finiteUpdateClosure
(μ : Measure Coord3) [μ.IsAddHaarMeasure] [Measure.InnerRegularCompactLTTop μ]
{a b : ℝ} (ha : 0 < a) (hab : a < b)
{U : Set Coord3} (hUb : Bornology.IsBounded U) (hUm : MeasurableSet U)
[IsFiniteMeasure (μ.restrict U)] [(μ.restrict U).WeaklyRegular]
[SeparableSpace (Lp FieldVector 2 (μ.restrict U))]
(u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) (hAm : Measurable A)
(hE : MemLp (voltageGradient u) 2 (μ.restrict U))
(hF : MemLp (voltageFlux u A) 2 (μ.restrict U))
(hint : SmoothFluxIntegrable μ U (conductivityFlux u A))
(hdiv : ∀ j (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → HasCompactSupport ψ →
tsupport ψ ⊆ U → (∫ x, fderiv ℝ ψ x ((conductivityFlux u A x).col j) ∂μ) = 0)
(hreg : μ (U \ regularRegion u A U) = 0)
(hgraph : ∀ᵐ x ∂μ, x ∈ U → A x ∈ matrixFiniteLaminate a b) :
∃ (Z : Lp FieldVector 2 (μ.restrict U) × Lp FieldVector 2 (μ.restrict U))
(s : Coord3 → ℝ),
toWeakSpace ℝ _ Z ∈ closure (toWeakSpace ℝ _ '' finiteUpdatePairs μ U u A hE hF a b) ∧
Measurable s ∧ (∀ᵐ x ∂μ.restrict U, s x ∈ Icc a b ∧ Z.2 x = s x • Z.1 x) := by
apply exact_scalar_of_weak_density (μ.restrict U) hab.le
· apply isCompact_weak_closure_of_pair_bounds
rintro Z ⟨R,hregR,hgraphR,rfl⟩
exact R.fieldPair_energy_bound μ hUm hreg hE hF hint hdiv ha hab hgraphR
· exact ((finiteUpdatePairs_nonempty μ hAm hE hF hreg hgraph).image _).closure
· intro ε hε
exact dense_ApproxScalar_in_finiteUpdateClosure μ ha hab hUb hUm u A hE hF hint hdiv hreg hε
end ScalarConductivity
end
end
end OAI
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