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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 | import Mathlib
import OAI.Analysis.Conductivity.Flux.AssembleCompatibleTensors
import OAI.Analysis.Conductivity.Flux.OpenRegularTensorTarget
import OAI.Analysis.Conductivity.Geometry.RegularRegionFiniteGluing
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 ENNReal
structure CompactGlobalUpdate (μ : Measure Coord3) (U : Set Coord3)
(u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) where
du : Coord3 → Fin 2 → ℝ
dF : Coord3 → Matrix (Fin 3) (Fin 2) ℝ
tensor : Coord3 → Symmetric3
smooth_du : ContDiff ℝ (↑(⊤ : ℕ∞)) du
compact_du : HasCompactSupport du
support_du : tsupport du ⊆ U
smooth_dF : ContDiff ℝ (↑(⊤ : ℕ∞)) dF
compact_dF : HasCompactSupport dF
support_dF : tsupport dF ⊆ U
cauchy_dF : ∀ j (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ →
(∫ x, fderiv ℝ ψ x ((dF x).col j) ∂μ) = 0
measurable_tensor : Measurable tensor
constitutive : ∀ x, conductivityFlux (fun y => u y+du y) tensor x = conductivityFlux u A x+dF x
theorem global_regular_depth_reduction
(μ : Measure Coord3) [μ.IsAddHaarMeasure] [Measure.InnerRegularCompactLTTop μ]
(u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) (hAm : Measurable A)
{U : Set Coord3} (hUb : Bornology.IsBounded 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)
(hreg : μ (U \ regularRegion u A U) = 0)
{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)
(hgraph : ∀ᵐ x ∂μ, x ∈ U → A x ∈ matrixFiniteLaminate a b)
{n : ℕ} (hsc : (spectralExtension (depthClass T n)).Nonempty)
{G : Set Coord3} (hGm : MeasurableSet G)
(hG : G ⊆ regularRegion u A U ∩ A ⁻¹' spectralExtension (depthClass T (n+1)))
{ε : ℝ} (hε : 0 < ε) :
∃ 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) ∧
μ (U \ R.tensor ⁻¹' spectralExtension (depthClass T n)) ≤ μ (U \ G) + ENNReal.ofReal ε := by
classical
obtain ⟨N,O,P,hO,hd,hP,hPG,hloss⟩ := finite_regular_depth_reduction μ u A hUb hdiv
ha hT hsub hperm hsc hGm hG hε
obtain ⟨du,dF,B,hus,huc,hUs,hFs,hFc,hFS,hpair,hB,heq,hoff,hconst⟩ :=
assemble_compatible_tensors μ O (fun i => (hO i).1) hd u A hAm P
have hOU : (⋃ i, O i) ⊆ U := iUnion_subset (fun i => subset_closure.trans (hO i).2.2.1)
let R : CompactGlobalUpdate μ U u A := ⟨du,dF,B,hus,huc,hUs.trans hOU,
hFs,hFc,hFS.trans hOU,hpair,hB,hconst⟩
have hnew : ∀ i, RegularPatch (fun x => u x+du x) B (O i) := by
intro i
exact (hP i).congr (fun x hx => by rw [(heq i).1 hx]) (heq i).2.2
have huo : EqOn (fun x => u x+du x) u (⋃ i, O i)ᶜ := by
intro x hx
change u x+du x = u x
rw [image_eq_zero_of_notMem_tsupport (fun hs => hx (hUs hs)),add_zero]
refine ⟨R,regularRegion_finite_gluing_ae μ hreg O
(fun i => subset_closure.trans (hO i).2.2.1) hnew huo hoff (fun i => (hO i).2.2.2),?_,?_⟩
· filter_upwards [hgraph] with x hx
intro hxU
change B x ∈ matrixFiniteLaminate a b
by_cases hxin : x ∈ ⋃ i, O i
· obtain ⟨i,hi⟩ := mem_iUnion.mp hxin
rw [(heq i).2.2 hi]
exact hPG i x hi
· rw [hoff hxin]
exact hx hxU
· let bad : Fin N → Set Coord3 := fun i =>
{x | x ∈ O i ∧ (P i).tensor x ∉ spectralExtension (depthClass T n)}
have hset : U \ B ⁻¹' spectralExtension (depthClass T n) ⊆
(U \ G) ∪ ((G \ ⋃ i, O i) ∪ (⋃ i, bad i)) := by
intro x hx
by_cases hxG : x ∈ G
· right
by_cases hxO : x ∈ ⋃ i, O i
· right
obtain ⟨i,hi⟩ := mem_iUnion.mp hxO
exact mem_iUnion.mpr ⟨i,hi,by rw [← (heq i).2.2 hi]; exact hx.2⟩
· exact Or.inl ⟨hxG,hxO⟩
· exact Or.inl ⟨hx.1,hxG⟩
calc
μ (U \ R.tensor ⁻¹' spectralExtension (depthClass T n)) ≤
μ (U \ G) + (μ (G \ ⋃ i, O i) + μ (⋃ i, bad i)) :=
(measure_mono hset).trans ((measure_union_le _ _).trans
(add_le_add le_rfl (measure_union_le _ _)))
_ ≤ μ (U \ G) + (μ (G \ ⋃ i, O i) + ∑ i, μ (bad i)) :=
add_le_add le_rfl (add_le_add le_rfl (measure_iUnion_fintype_le μ bad))
_ ≤ μ (U \ G) + ENNReal.ofReal ε := add_le_add le_rfl hloss
end ScalarConductivity
end
end OAI
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