| import Mathlib |
| import OAI.Analysis.Conductivity.Flux.CollarCartesianDifferential |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology Matrix |
|
|
| def sourceCollarOpenBox : Set (Fin 3 → ℝ) := |
| {x | (-(1:ℝ)/100 < x 0 ∧ x 0 < 1/100) ∧ |x 1|<1 ∧ |x 2|<1} |
|
|
| lemma isOpen_sourceCollarOpenBox : IsOpen sourceCollarOpenBox := by |
| exact ((isOpen_lt continuous_const (continuous_apply 0)).inter |
| (isOpen_lt (continuous_apply 0) continuous_const)).inter |
| ((isOpen_lt (continuous_apply 1).abs continuous_const).inter |
| (isOpen_lt (continuous_apply 2).abs continuous_const)) |
|
|
| lemma sourceCollarOpenBox_subset : sourceCollarOpenBox ⊆ |
| sourceExtendedBox (-(1:ℝ)/100) (1/100) := by |
| intro x hx |
| exact mem_sourceExtendedBox.mpr ⟨⟨hx.1.1.le,hx.1.2.le⟩,hx.2.1.le,hx.2.2.le⟩ |
|
|
| def sourceCollarTangentEquiv (i j : Fin 4) (x : Fin 3 → ℝ) |
| (hx : x∈sourceExtendedBox (-(1:ℝ)/100) (1/100)) : |
| (Fin 3 → ℝ) ≃L[ℝ] (Fin 3 → ℝ) := |
| (LinearMap.equivOfDetNeZero (Matrix.toLin' (sourceCollarJacobian i j x)) |
| (by rw [LinearMap.det_toLin']; exact sourceCollarJacobian_extended_ne_zero i j hx)).toContinuousLinearEquiv |
|
|
| lemma sourceCollarTangentEquiv_apply (i j : Fin 4) (x : Fin 3 → ℝ) |
| (hx : x∈sourceExtendedBox (-(1:ℝ)/100) (1/100)) (v : Fin 3 → ℝ) : |
| sourceCollarTangentEquiv i j x hx v = sourceCollarJacobian i j x *ᵥ v := rfl |
|
|
| lemma sourceCollarPiece_hasStrictFDeriv (i j : Fin 4) (x : Fin 3 → ℝ) |
| (hx : x∈sourceExtendedBox (-(1:ℝ)/100) (1/100)) : |
| HasStrictFDerivAt (sourceCollarPiece i j) (sourceCollarTangentEquiv i j x hx).toContinuousLinearMap |
| x := by |
| have hf := (sourceCollarPiece_contDiff i j).hasStrictFDerivAt (x:=x) (by simp) |
| rw [(sourceCollarPiece_hasFDeriv i j x).fderiv] at hf |
| exact hf |
|
|
| lemma sourceCollarInverse_eventually_left (i j : Fin 4) {x : Fin 3 → ℝ} |
| (hx : x∈sourceCollarOpenBox) : |
| ∀ᶠ y in 𝓝 x,sourceCollarInverse i j (sourceCollarPiece i j y)=y := by |
| filter_upwards [isOpen_sourceCollarOpenBox.mem_nhds hx] with y hy |
| exact sourceCollarInverse_left_pos i j |
| (lt_of_lt_of_le (by norm_num : (0:ℝ)<1/50) |
| (sourceCollarRadius_extended j ⟨hy.1.1.le,hy.1.2.le⟩ hy.2.2.le).1) |
| (by linarith [hy.1.2]) hy.2.1.le hy.2.2.le |
|
|
| lemma sourceCollarInverse_hasStrictFDeriv (i j : Fin 4) {x : Fin 3 → ℝ} |
| (hx : x∈sourceCollarOpenBox) : |
| HasStrictFDerivAt (sourceCollarInverse i j) |
| (sourceCollarTangentEquiv i j x (sourceCollarOpenBox_subset hx)).symm.toContinuousLinearMap |
| (sourceCollarPiece i j x) := |
| (sourceCollarPiece_hasStrictFDeriv i j x (sourceCollarOpenBox_subset hx)).to_local_left_inverse |
| (sourceCollarInverse_eventually_left i j hx) |
|
|
| lemma sourceCollarPiece_map_nhds (i j : Fin 4) {x : Fin 3 → ℝ} |
| (hx : x∈sourceCollarOpenBox) : |
| Filter.map (sourceCollarPiece i j) (𝓝 x)=𝓝 (sourceCollarPiece i j x) := |
| (sourceCollarPiece_hasStrictFDeriv i j x (sourceCollarOpenBox_subset hx)).map_nhds_eq_of_equiv |
|
|
| lemma sourceCollarInverse_eventually_right (i j : Fin 4) {x : Fin 3 → ℝ} |
| (hx : x∈sourceCollarOpenBox) : |
| ∀ᶠ y in 𝓝 (sourceCollarPiece i j x),sourceCollarPiece i j (sourceCollarInverse i j y)=y := by |
| rw [←sourceCollarPiece_map_nhds i j hx,Filter.eventually_map] |
| filter_upwards [sourceCollarInverse_eventually_left i j hx] with y hy |
| rw [hy] |
|
|
| end ScalarConductivity |
|
|
| end |
|
|
| end OAI |
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|
