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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 | import Mathlib
import OAI.Analysis.Conductivity.Geometry.CoordinateWeakCorrection
import OAI.Analysis.Conductivity.Sources.LocalPiola
namespace OAI
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)))
def coordinatePair : Coord3 → Fin 2 → ℝ := coordinatePairCLM
lemma coordinatePair_smooth : ContDiff ℝ (↑(⊤ : ℕ∞)) coordinatePair :=
coordinatePairCLM.contDiff
@[simp] lemma coordinatePair_zero (x : Coord3) : coordinatePair x 0=x 0 := rfl
@[simp] lemma coordinatePair_one (x : Coord3) : coordinatePair x 1=x 1 := rfl
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
end
end OAI
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