| import Mathlib |
| import OAI.Analysis.Conductivity.Variational.PhysicalRepresentativeGluing |
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| namespace OAI |
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| section |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology Matrix |
| open scoped Matrix.Norms.Elementwise |
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| theorem physical_region_glue {ι : Type*} [Countable ι] |
| (O : ι → Set Coord3) (hO : ∀ i,IsOpen (O i)) |
| {U : Set Coord3} (hUm : MeasurableSet U) (hOU : ∀ i,O i⊆U) |
| (hUb : ∀ y∈U,WithLp.toLp 2 y∈ball) |
| (hcover : ∀ᵐ y : Coord3,y∈U → y∈⋃ i,O i) |
| (w : Fin 2 → H1) (g : ι → Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) |
| (hreg : ∀ i,∀ᵐ y : Coord3,y∈O i → y∈regularRegion (g i) A (O i)) |
| (he : ∀ i,(fun y j => weakValue (w j) (WithLp.toLp 2 y))=ᵐ[volume.restrict (O i)] g i) : |
| ∃ v : Coord3 → Fin 2 → ℝ, |
| (∀ᵐ y : Coord3,y∈U → v y=fun j => weakValue (w j) (WithLp.toLp 2 y)) ∧ |
| (∀ᵐ y : Coord3,y∈U → ∀ i : Fin 3,∀ j : Fin 2, |
| fderiv ℝ (fun x => v x j) y (Pi.single i 1)=weakGradient (w j) (WithLp.toLp 2 y) i) ∧ |
| volume (U \ regularRegion v A U)=0 := by |
| let I := Σ i : ι,{V : Set Coord3 // V⊆O i ∧ RegularPatch (g i) A V} |
| let V : I → Set Coord3 := fun i => i.2.val |
| let f : I → Coord3 → Fin 2 → ℝ := fun i => g i.1 |
| have hV : ∀ i,IsOpen (V i) := fun i => i.2.property.2.1 |
| have hVU : ∀ i,V i⊆U := fun i => i.2.property.1.trans (hOU i.1) |
| have hVR : ∀ i,RegularPatch (f i) A (V i) := fun i => i.2.property.2 |
| have hc : ∀ᵐ y : Coord3,y∈U → y∈⋃ i,V i := by |
| filter_upwards [hcover,ae_all_iff.mpr hreg] with y hc hr hy |
| obtain ⟨i,hi⟩ := mem_iUnion.mp (hc hy) |
| obtain ⟨W,hWO,hW,hyW⟩ := mem_regularRegion_iff.mp (hr i hi) |
| exact mem_iUnion.mpr ⟨⟨i,⟨W,hWO,hW⟩⟩,hyW⟩ |
| have he' : ∀ i,(fun y j => weakValue (w j) (WithLp.toLp 2 y))=ᵐ[volume.restrict (V i)] f i := by |
| intro index |
| apply (ae_restrict_iff' (hV index).measurableSet).mpr |
| filter_upwards [(ae_restrict_iff' (hO index.1).measurableSet).mp (he index.1)] with point equality membership |
| exact equality (index.2.property.1 membership) |
| exact physical_representative_glue V hV hUm hVU hUb hc w f A hVR he' |
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| end ScalarConductivity |
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| end |
| end |
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| end OAI |
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