| import Mathlib |
| import OAI.Analysis.Conductivity.Geometry.CoordinateWeakCorrection |
| import OAI.Analysis.Conductivity.Sources.LocalPiola |
|
|
| namespace OAI |
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|
| noncomputable section |
| namespace ScalarConductivity |
| open Set Matrix MeasureTheory |
| open scoped Matrix.Norms.Elementwise |
|
|
| def coordinatePairCLM : Coord3 →L[ℝ] (Fin 2 → ℝ) := |
| ContinuousLinearMap.pi (fun j => ContinuousLinearMap.proj (j.castLE (by decide : 2≤3))) |
|
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| def coordinatePair : Coord3 → Fin 2 → ℝ := coordinatePairCLM |
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| lemma coordinatePair_smooth : ContDiff ℝ (↑(⊤ : ℕ∞)) coordinatePair := |
| coordinatePairCLM.contDiff |
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| @[simp] lemma coordinatePair_zero (x : Coord3) : coordinatePair x 0=x 0 := rfl |
| @[simp] lemma coordinatePair_one (x : Coord3) : coordinatePair x 1=x 1 := rfl |
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| lemma coordinatePair_columns (B : Mat3) (x : Coord3) (j : Fin 2) : |
| (B*gradientColumns (fderiv ℝ coordinatePair x)).col j=(B.col (j.castLE (by decide : 2≤3))) := by |
| have hd : fderiv ℝ coordinatePair x=coordinatePairCLM := coordinatePairCLM.hasFDerivAt.fderiv |
| rw [hd] |
| ext i |
| fin_cases j |
| all_goals simp [Matrix.mul_apply,gradientColumns,LinearMap.toMatrix', |
| coordinatePairCLM,Matrix.transpose_apply,Matrix.col,Pi.single_apply] |
|
|
| theorem physical_box_symmetric_correction |
| (X : OpenPartialHomeomorph Coord3 Coord3) |
| (hX : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X X.source) |
| (hXi : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X.symm X.target) |
| {a b : Fin 3 → ℝ} (hab : ∀ i,a i<b i) |
| (hbox : boxCoordinates ⁻¹' ((Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2))⊆X.source) |
| {r₁ r₂ : Box3 → ℝ} |
| (hr₁ : ContDiff ℝ (↑(⊤ : ℕ∞)) r₁) (hr₂ : ContDiff ℝ (↑(⊤ : ℕ∞)) r₂) |
| (hs₁ : HasCompactSupport r₁) (hs₂ : HasCompactSupport r₂) |
| (hv₁ : tsupport r₁⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2)) |
| (hv₂ : tsupport r₂⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2)) |
| (hz₁ : (∫ p,r₁ p)=0) (hz₂ : (∫ p,r₂ p)=0) |
| (ht : (∫ p,p.1.2*r₁ p-p.1.1*r₂ p)=0) : |
| ∃ H : Coord3 → Mat3, |
| ContDiff ℝ (↑(⊤ : ℕ∞)) H ∧ HasCompactSupport H ∧ tsupport H⊆X.target ∧ |
| (∀ y,(H y).IsSymm) ∧ |
| ∀ j (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → |
| (∫ y,fderiv ℝ ψ y ((H y*gradientColumns (fderiv ℝ (coordinatePair∘X.symm) y)).col j))= |
| -(∫ y,ψ y*localPiolaSource X (fun x => ![r₁ (boxCoordinates x),r₂ (boxCoordinates x)] j) y) := by |
| obtain ⟨B,hB,hcB,htB,hsy,hw₁,hw₂⟩ := |
| compact_coordinate_box_correction hab hr₁ hr₂ hs₁ hs₂ hv₁ hv₂ hz₁ hz₂ ht |
| apply local_symmetric_source_transport X hX hXi B hB hcB (htB.trans hbox) hsy |
| coordinatePair coordinatePair_smooth.contDiffOn |
| · intro j |
| fin_cases j |
| · exact hs₁.comp_homeomorph boxCoordinates.toHomeomorph |
| · exact hs₂.comp_homeomorph boxCoordinates.toHomeomorph |
| · intro j |
| fin_cases j |
| · exact ((tsupport_comp_subset_preimage r₁ boxCoordinates.continuous).trans (preimage_mono hv₁)).trans hbox |
| · exact ((tsupport_comp_subset_preimage r₂ boxCoordinates.continuous).trans (preimage_mono hv₂)).trans hbox |
| · intro j ψ hψ |
| simp_rw [coordinatePair_columns] |
| fin_cases j |
| · exact hw₁ ψ hψ |
| · exact hw₂ ψ hψ |
|
|
| end ScalarConductivity |
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| end |
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| end OAI |
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