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159357f | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | import Mathlib
import OAI.Analysis.Conductivity.Variational.TsupportFiniteSumSubset
import OAI.Analysis.Conductivity.Variational.Unchanged
namespace OAI
noncomputable section
open MeasureTheory
open scoped ENNReal
open Matrix Filter Topology
open Set MeasureTheory Filter Topology
open scoped BigOperators
open Set MeasureTheory Filter Topology
open scoped Manifold
open Set Filter
open scoped Topology
open Set Filter MeasureTheory
open scoped Topology Manifold ENNReal
open Set
namespace ScalarConductivity
open Matrix Set MeasureTheory Filter Topology
open scoped Matrix.Norms.Elementwise
theorem exists_compact_depth_reduction_regular
(μ : Measure Coord3) [μ.IsAddHaarMeasure]
(u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3)
{U : Set Coord3} (hU : IsOpen U) (hUb : Bornology.IsBounded U)
(hu : ContDiffOn ℝ (↑(⊤ : ℕ∞)) u U)
(hA : ContDiffOn ℝ (↑(⊤ : ℕ∞)) (fun x => (A x).val) U)
(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)
(hr : TwoFieldRankRegular u U)
{a b : ℝ} (ha : 0 < a) {T : Set DiagonalTriple}
(hT : IsOpen T) (hsub : ∀ d ∈ T, IsFiniteLaminate a b d)
(hperm : ∀ d ∈ T, ∀ e : Equiv.Perm (Fin 3), d ∘ e ∈ T)
{p : Coord3} (hpU : p ∈ U) {n : ℕ}
(hp : A p ∈ spectralExtension (depthClass T (n+1)))
(hsc : (spectralExtension (depthClass T n)).Nonempty) :
∃ W : Set Coord3, IsOpen W ∧ p ∈ W ∧ closure W ⊆ U ∧ IsCompact (closure W) ∧
∀ O : Set Coord3, IsOpen O → O ⊆ W → ∀ ε : ℝ, 0 < ε →
∃ R : CompactCompatibleReplacement μ O u A,
TwoFieldRankRegular (fun x => u x+R.du x) O ∧
(∀ x ∈ O, R.tensor x ∈ matrixFiniteLaminate a b) ∧
μ {x | x ∈ O ∧ R.tensor x ∉ spectralExtension (depthClass T n)} ≤ ENNReal.ofReal ε := by
classical
by_cases hn : A p ∈ spectralExtension (depthClass T n)
· have hTo := isOpen_spectralExtension (isOpen_depthClass ha hT hsub n) (depthClass_permute hperm n)
have hAc : ContinuousAt A p := Topology.IsInducing.subtypeVal.continuousAt_iff.mpr
(hA.continuousOn.continuousAt (hU.mem_nhds hpU))
have hev : ∀ᶠ x in 𝓝 p, x ∈ U ∧ A x ∈ spectralExtension (depthClass T n) :=
Filter.Eventually.and (hU.mem_nhds hpU) (hAc.preimage_mem_nhds (hTo.mem_nhds hn))
obtain ⟨V, hVs, hV, hpV⟩ := mem_nhds_iff.mp hev
obtain ⟨W, hW, hpW, hWV, hWc⟩ := exists_precompact_open_inside hV hpV
have hVU : V ⊆ U := fun x hx => (hVs hx).1
refine ⟨W, hW, hpW, hWV.trans hVU, hWc, ?_⟩
intro O hO hOW ε _
have hOV := hOW.trans (subset_closure.trans hWV)
have hOU := hOV.trans hVU
let R := CompactCompatibleReplacement.unchanged μ O u A (hA.mono hOU)
refine ⟨R, ?_, fun x hx => spectral_depth_subset hsub n (hVs (hOV hx)).2, ?_⟩
· simpa only [R, CompactCompatibleReplacement.unchanged, Pi.zero_apply, add_zero] using hr.mono hOU
· have he : {x | x ∈ O ∧ R.tensor x ∉ spectralExtension (depthClass T n)} = ∅ := by
ext x
simp only [mem_ofPred_eq, mem_empty_iff_false, iff_false, not_and, not_not]
intro hx
exact (hVs (hOV hx)).2
rw [he, measure_empty]
exact bot_le
· exact exists_compact_depth_reduction_regular_nontrivial μ u A hU hUb hu hA hdiv hr
ha hT hsub hperm hpU hp hn hsc
end ScalarConductivity
namespace ScalarConductivity
open Matrix Set MeasureTheory Filter Topology
open scoped Matrix.Norms.Elementwise
theorem assemble_compatible_replacements
(μ : Measure Coord3) [μ.IsAddHaarMeasure]
{I : Type*} [Fintype I] [DecidableEq I]
(O : I → Set Coord3) (hO : ∀ i, IsOpen (O i))
(hd : Pairwise (fun i j => Disjoint (O i) (O j)))
(u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3)
(R : ∀ i, CompactCompatibleReplacement μ (O i) u A) :
∃ (du : Coord3 → Fin 2 → ℝ) (dF : Coord3 → Matrix (Fin 3) (Fin 2) ℝ),
ContDiff ℝ (↑(⊤ : ℕ∞)) du ∧ HasCompactSupport du ∧ tsupport du ⊆ ⋃ i, O i ∧
ContDiff ℝ (↑(⊤ : ℕ∞)) dF ∧ HasCompactSupport dF ∧ tsupport dF ⊆ ⋃ i, O i ∧
(∀ j (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ →
(∫ x, fderiv ℝ ψ x ((dF x).col j) ∂μ) = 0) ∧
(∀ i, EqOn du (R i).du (O i) ∧ EqOn dF (R i).dF (O i) ∧
∀ x ∈ O i,
fderiv ℝ (fun y => u y+du y) x = fderiv ℝ (fun y => u y+(R i).du y) x) ∧
(∀ x ∉ ⋃ i, O i, du x = 0 ∧ dF x = 0) := by
let du : Coord3 → Fin 2 → ℝ := fun x => ∑ i, (R i).du x
let dF : Coord3 → Matrix (Fin 3) (Fin 2) ℝ := fun x => ∑ i, (R i).dF x
have heu : ∀ i, EqOn du (R i).du (O i) :=
supported_sum_eq_on O hd (fun i => (R i).du) (fun i => (R i).support_du)
have heF : ∀ i, EqOn dF (R i).dF (O i) :=
supported_sum_eq_on O hd (fun i => (R i).dF) (fun i => (R i).support_dF)
have hcdu : HasCompactSupport du := by
simpa only [du, ← Finset.sum_apply] using
(HasCompactSupport.finset_sum (s := Finset.univ) (fun i _ => (R i).compact_du))
have hcdF : HasCompactSupport dF := by
simpa only [dF, ← Finset.sum_apply] using
(HasCompactSupport.finset_sum (s := Finset.univ) (fun i _ => (R i).compact_dF))
refine ⟨du, dF, ContDiff.sum (fun i _ => (R i).smooth_du),
hcdu,
(tsupport_finite_sum_subset _).trans (iUnion_mono (fun i => (R i).support_du)),
ContDiff.sum (fun i _ => (R i).smooth_dF),
hcdF,
(tsupport_finite_sum_subset _).trans (iUnion_mono (fun i => (R i).support_dF)), ?_, ?_, ?_⟩
· exact finite_sum_cauchy_preserving μ (fun i => (R i).dF)
(fun i => (R i).smooth_dF) (fun i => (R i).compact_dF) (fun i => (R i).cauchy_dF)
· intro i
refine ⟨heu i, heF i, ?_⟩
intro x hx
apply Filter.EventuallyEq.fderiv_eq
filter_upwards [(hO i).mem_nhds hx] with y hy
rw [heu i hy]
· intro x hx
exact ⟨supported_sum_eq_zero_off O _ (fun i => (R i).support_du) hx,
supported_sum_eq_zero_off O _ (fun i => (R i).support_dF) hx⟩
end ScalarConductivity
end
end OAI
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