File size: 3,241 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 | import Mathlib
import OAI.Analysis.Conductivity.Branching.AlignedEndingRegularity
namespace OAI
noncomputable section
namespace ScalarConductivity
open Set Filter Topology Real MeasureTheory Matrix
open scoped Matrix.Norms.Elementwise
def delayedEndingSymmetricTensor (lam k J L K R : ℝ) (x : Coord3) : Symmetric3 :=
alignedEndingSymmetricTensor lam k J L K (x-Pi.single 0 R)
def axialAmplitudeEquiv (a : ℝ) (ha : a≠0) : (Fin 2 → ℝ) ≃L[ℝ] (Fin 2 → ℝ) :=
ContinuousLinearEquiv.piCongrRight (fun i : Fin 2 =>
(LinearEquiv.smulOfNeZero ℝ ℝ (if i=0 then 1 else a) (by split_ifs <;> simp_all)).toContinuousLinearEquiv)
lemma axialAmplitudeEquiv_apply (a : ℝ) (ha : a≠0) (v : Fin 2 → ℝ) (i : Fin 2) :
axialAmplitudeEquiv a ha v i=(if i=0 then 1 else a)*v i := rfl
lemma RegularPatch.amplitude {u : Coord3 → Fin 2 → ℝ} {A : Coord3 → Symmetric3}
{O : Set Coord3} (h : RegularPatch u A O) (a : ℝ) (ha : a≠0) :
RegularPatch (fun x => axialAmplitudeEquiv a ha (u x)) A O := by
refine ⟨h.1,(axialAmplitudeEquiv a ha).contDiff.contDiffOn.comp h.2.1 (fun _ _ => mem_univ _),h.2.2.1,?_⟩
exact h.2.2.2.linearCoordinates h.1 h.2.1 (ContinuousLinearEquiv.refl ℝ Coord3) (axialAmplitudeEquiv a ha)
lemma delayedEnding_axialPair (lam k J L K R : ℝ) (x : Coord3) :
axialAmplitudeEquiv (exp (-lam*R)) (exp_pos _).ne'
(axialPair (alignedEndingValue lam k J L K) (x-Pi.single 0 R))+![R,0]=
axialPair (delayedEndingValue lam k J L K R) x := by
ext i
fin_cases i <;> simp [axialAmplitudeEquiv_apply,axialPair,delayedEndingValue]
lemma RegularPatch.delayedEnding {lam k J L K R : ℝ} {O : Set Coord3}
(h : RegularPatch (axialPair (alignedEndingValue lam k J L K))
(alignedEndingSymmetricTensor lam k J L K) O) :
RegularPatch (axialPair (delayedEndingValue lam k J L K R))
(delayedEndingSymmetricTensor lam k J L K R) ((fun x : Coord3 => x-Pi.single 0 R) ⁻¹' O) := by
have hh := (h.amplitude (exp (-lam*R)) (exp_pos _).ne').translate (Pi.single 0 R) ![R,0]
convert hh using 1
· funext x
exact (delayedEnding_axialPair _ _ _ _ _ _ _).symm
· rfl
lemma delayedEnding_regularRegion_ae {lam k J L K R : ℝ} (hk : k≠0) (hL : 0<L) (hK : 0<K)
{U : Set Coord3} (hU : IsOpen U) :
∀ᵐ x : Coord3,x∈U → x∈regularRegion (axialPair (delayedEndingValue lam k J L K R))
(delayedEndingSymmetricTensor lam k J L K R) U := by
have he := (measurePreserving_add_right (volume : Measure Coord3) (-Pi.single 0 R)).quasiMeasurePreserving.ae
(alignedEnding_regularRegion_ae (lam:=lam) (J:=J) hk hL hK isOpen_univ)
simp only [←sub_eq_add_neg] at he
filter_upwards [he] with x hx
intro hxU
obtain ⟨O,hOU,hO,hxO⟩ := mem_regularRegion_iff.mp (hx (mem_univ _))
have hh : x∈regularRegion (axialPair (delayedEndingValue lam k J L K R))
(delayedEndingSymmetricTensor lam k J L K R) univ := by
apply mem_regularRegion_iff.mpr
refine ⟨(fun y : Coord3 => y-Pi.single 0 R) ⁻¹' O,subset_univ _,?_,?_⟩
· exact hO.delayedEnding (R:=R)
· exact hxO
exact mem_regularRegion_congr_nhds hh (Filter.EventuallyEq.rfl) (Filter.EventuallyEq.rfl) hU hxU
end ScalarConductivity
end
end OAI
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