File size: 3,443 Bytes
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 76 | 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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