| import Mathlib |
| import OAI.Analysis.Conductivity.Branching.FiniteEndingRegularity |
| import OAI.Analysis.Conductivity.Branching.AlignedEndingData |
| import OAI.Analysis.Conductivity.Geometry.RegularPatchLinear |
|
|
| namespace OAI |
|
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|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Real Set Filter Topology MeasureTheory Matrix |
| open scoped Matrix.Norms.Elementwise |
|
|
| def permutedDilationEquiv (e : Equiv.Perm (Fin 3)) {k : ℝ} (hk : k≠0) : Coord3 ≃L[ℝ] Coord3 := |
| (ContinuousLinearEquiv.piCongrLeft ℝ (fun _ : Fin 3 => ℝ) e.symm).trans |
| (LinearEquiv.smulOfNeZero ℝ Coord3 k hk).toContinuousLinearEquiv |
|
|
| lemma permutedDilationEquiv_apply (e : Equiv.Perm (Fin 3)) {k : ℝ} (hk : k≠0) (x : Coord3) : |
| permutedDilationEquiv e hk x=permutedDilation e k x := by |
| ext i |
| simp [permutedDilationEquiv,permutedDilation,ContinuousLinearEquiv.piCongrLeft, |
| Homeomorph.piCongrLeft,Equiv.piCongrLeft] |
|
|
| def axialPairScaling {k : ℝ} (hk : k≠0) : (Fin 2 → ℝ) ≃L[ℝ] (Fin 2 → ℝ) := |
| ContinuousLinearEquiv.piCongrRight (fun i : Fin 2 => |
| (LinearEquiv.smulOfNeZero ℝ ℝ (if i=0 then k⁻¹ else 1) (by split_ifs <;> simp_all)).toContinuousLinearEquiv) |
|
|
| lemma axialPairScaling_apply {k : ℝ} (hk : k≠0) (v : Fin 2 → ℝ) (i : Fin 2) : |
| axialPairScaling hk v i=(if i=0 then k⁻¹ else 1)*v i := rfl |
|
|
| lemma alignedEnding_axialPair {lam k J L K : ℝ} (hk : k≠0) (x : Coord3) : |
| axialPairScaling hk (axialPair (finiteEndingValue (lam/k) J L K) |
| (permutedDilationEquiv angularAxisSwap hk x))=axialPair (alignedEndingValue lam k J L K) x := by |
| ext i |
| fin_cases i <;> simp [axialPairScaling_apply,permutedDilationEquiv_apply,axialPair, |
| permutedDilation,angularAxisSwap_apply,alignedEndingValue,hk] |
|
|
| def alignedEndingSymmetricTensor (lam k J L K : ℝ) (x : Coord3) : Symmetric3 := |
| ⟨alignedEndingTensor lam k J L K x,alignedEndingTensor_symm lam k J L K x⟩ |
|
|
| lemma RegularPatch.alignedEnding {lam k J L K : ℝ} (hk : k≠0) {O : Set Coord3} |
| (h : RegularPatch (axialPair (finiteEndingValue (lam/k) J L K)) |
| (finiteEndingSymmetricTensor (lam/k) J L K) O) : |
| RegularPatch (axialPair (alignedEndingValue lam k J L K)) |
| (alignedEndingSymmetricTensor lam k J L K) |
| ((permutedDilationEquiv angularAxisSwap hk) ⁻¹' O) := by |
| let P := permutedDilationEquiv angularAxisSwap hk |
| let Q := axialPairScaling hk |
| have he : (fun x => Q (axialPair (finiteEndingValue (lam/k) J L K) (P x)))= |
| axialPair (alignedEndingValue lam k J L K) := funext (alignedEnding_axialPair hk) |
| refine ⟨h.1.preimage P.continuous,?_,?_,?_⟩ |
| · rw [←he] |
| exact Q.contDiff.contDiffOn.comp (h.2.1.comp P.contDiff.contDiffOn (fun _ hx => hx)) |
| (fun _ _ => mem_univ _) |
| · apply contDiffOn_pi.mpr; intro i |
| apply contDiffOn_pi.mpr; intro j |
| have hh := h.2.2.1.comp P.contDiff.contDiffOn (fun _ hx => hx) |
| have hi := (contDiff_apply ℝ (Fin 3 → ℝ) (angularAxisSwap i)).contDiffOn.comp hh (fun _ _ => mem_univ _) |
| have hj := (contDiff_apply ℝ ℝ (angularAxisSwap j)).contDiffOn.comp hi (fun _ _ => mem_univ _) |
| change ContDiffOn ℝ (↑(⊤:ℕ∞)) (fun x => |
| finiteEndingTensor (lam/k) J L K (P x) (angularAxisSwap i) (angularAxisSwap j)) (P ⁻¹' O) at hj |
| convert hj using 1 |
| funext x |
| simp only [alignedEndingSymmetricTensor,alignedEndingTensor, |
| P,permutedDilationEquiv_apply] |
| · rw [←he] |
| exact TwoFieldRankRegular.linearCoordinates h.1 h.2.1 h.2.2.2 P Q |
|
|
| theorem alignedEnding_regularRegion_ae {lam k J L K : ℝ} (hk : k≠0) (hL : 0<L) (hK : 0<K) |
| {U : Set Coord3} (hU : IsOpen U) : |
| ∀ᵐ x : Coord3,x∈U → x∈regularRegion (axialPair (alignedEndingValue lam k J L K)) |
| (alignedEndingSymmetricTensor lam k J L K) U := by |
| have he := (permutedDilation_quasiMeasurePreserving angularAxisSwap hk).ae |
| (finiteEnding_regularRegion_ae (a:=lam/k) (J:=J) hL hK isOpen_univ) |
| 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 (alignedEndingValue lam k J L K)) |
| (alignedEndingSymmetricTensor lam k J L K) univ := by |
| apply mem_regularRegion_iff.mpr |
| refine ⟨(permutedDilationEquiv angularAxisSwap hk) ⁻¹' O,subset_univ _,hO.alignedEnding hk,?_⟩ |
| simpa only [mem_preimage,permutedDilationEquiv_apply] using hxO |
| exact mem_regularRegion_congr_nhds hh (Filter.EventuallyEq.rfl) (Filter.EventuallyEq.rfl) hU hxU |
|
|
| lemma alignedEnding_regularRegion_conull {lam k J L K : ℝ} (hk : k≠0) (hL : 0<L) (hK : 0<K) |
| {U : Set Coord3} (hU : IsOpen U) : |
| volume (U \ regularRegion (axialPair (alignedEndingValue lam k J L K)) |
| (alignedEndingSymmetricTensor lam k J L K) U)=0 := by |
| classical |
| have hh := ae_iff.mp (alignedEnding_regularRegion_ae (lam:=lam) (J:=J) hk hL hK hU) |
| convert hh using 2 |
| ext x |
| simp only [Set.mem_sdiff,mem_ofPred_eq] |
| tauto |
|
|
| end ScalarConductivity |
|
|
| end |
|
|
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology Real MeasureTheory Matrix |
|
|
| def delayedEndingValue (lam k J L K R : ℝ) (x : Coord3) : ℝ := |
| exp (-lam*R)*alignedEndingValue lam k J L K (x-Pi.single 0 R) |
|
|
| lemma delayedEndingValue_C2 {lam k J L K R : ℝ} (hJ : 0<J) (hL : 1≤L) (hK : 0<K) : |
| ContDiff ℝ 2 (delayedEndingValue lam k J L K R) := |
| contDiff_const.mul ((alignedEndingValue_C2 hJ hL hK).comp (contDiff_id.sub contDiff_const)) |
|
|
| lemma delayedEndingValue_initial_of_bound {lam k J L K R : ℝ} (hk : 0<k) (hJ : 0<J) |
| (x : Coord3) (hx : k*(x 0-R)≤1) : |
| delayedEndingValue lam k J L K R x=exp (-lam*x 0)*cos (k*x 2) := by |
| rw [delayedEndingValue,alignedEnding_initial hk hJ _ (by simpa using hx)] |
| simp only [Pi.sub_apply,Pi.single_eq_same,Pi.single_eq_of_ne (show (2:Fin 3)≠0 by decide),sub_zero] |
| rw [←mul_assoc,←exp_add] |
| congr 2 |
| ring |
|
|
| lemma delayedEndingValue_initial {lam k J L K R : ℝ} (hk : 0<k) (hJ : 0<J) |
| (x : Coord3) (hx : x 0≤R) : |
| delayedEndingValue lam k J L K R x=exp (-lam*x 0)*cos (k*x 2) := by |
| apply delayedEndingValue_initial_of_bound hk hJ |
| have hh := mul_nonpos_of_nonneg_of_nonpos hk.le (sub_nonpos.mpr hx) |
| linarith |
|
|
| lemma delayedEndingValue_terminal {lam k J L K R : ℝ} (hJ : 0<J) (hL : 1≤L) (hK : 0<K) |
| (x : Coord3) (hx : J+2+2*cascadeLength L K≤k*(x 0-R)) : |
| delayedEndingValue lam k J L K R x=0 := by |
| rw [delayedEndingValue,alignedEnding_terminal hJ hL hK _ (by simpa using hx),mul_zero] |
|
|
| lemma delayedEndingValue_periodic (lam J L K R : ℝ) (k : ℤ) : |
| AngularPeriodic (2*Real.pi) (delayedEndingValue lam k J L K R) := by |
| intro n x |
| change exp (-lam*R)*alignedEndingValue lam k J L K (x+angularShift (2*Real.pi) n-Pi.single 0 R)=_ |
| rw [add_sub_right_comm,alignedEndingValue_angularPeriodic] |
| rfl |
|
|
| def matchedEndingPair (u : Coord3 → Fin 2 → ℝ) (lam k J L K : ℝ) (x : Coord3) : Fin 2 → ℝ := |
| if x 0<13/2 then u x else ![x 0,delayedEndingValue lam k J L K 7 x] |
|
|
| lemma matchedEndingPair_C2 {u : Coord3 → Fin 2 → ℝ} (hu : ContDiff ℝ 2 u) |
| {lam k J L K : ℝ} (hk : 0<k) (hJ : 0<J) (hL : 1≤L) (hK : 0<K) |
| (he : ∀ x : Coord3,4≤x 0 → u x=![x 0,exp (-lam*x 0)*cos (k*x 2)]) : |
| ContDiff ℝ 2 (matchedEndingPair u lam k J L K) := by |
| have hv : ContDiff ℝ 2 (fun x : Coord3 => ![x 0,delayedEndingValue lam k J L K 7 x]) := by |
| apply contDiff_pi.mpr |
| intro i |
| fin_cases i |
| · exact contDiff_apply ℝ ℝ 0 |
| · exact delayedEndingValue_C2 hJ hL hK |
| apply contDiff_axial_paste hu hv (continuous_apply 0) (δ:=1/4) (by norm_num) |
| intro x hx |
| rw [he x (by linarith [hx.1]),delayedEndingValue_initial hk hJ x (by linarith [hx.2])] |
|
|
| lemma matchedEndingPair_periodic {u : Coord3 → Fin 2 → ℝ} |
| (hu : AngularPeriodic (2*Real.pi) u) (lam J L K : ℝ) (k : ℤ) : |
| AngularPeriodic (2*Real.pi) (matchedEndingPair u lam k J L K) := by |
| intro n x |
| have ht : (x+angularShift (2*Real.pi) n) 0=x 0 := by simp [angularShift] |
| simp only [matchedEndingPair,ht,hu n x,delayedEndingValue_periodic _ _ _ _ _ _ n x] |
|
|
| lemma matchedEndingPair_terminal {u : Coord3 → Fin 2 → ℝ} {lam k J L K : ℝ} |
| (hJ : 0<J) (hL : 1≤L) (hK : 0<K) (x : Coord3) |
| (hx : 13/2≤x 0) (hterm : J+2+2*cascadeLength L K≤k*(x 0-7)) : |
| matchedEndingPair u lam k J L K x=![x 0,0] := by |
| rw [matchedEndingPair,ite_eq_right (not_lt.mpr hx),delayedEndingValue_terminal hJ hL hK _ hterm] |
|
|
| end ScalarConductivity |
|
|
| end |
|
|
| end OAI |
|
|
