| import Mathlib |
| import OAI.Analysis.Conductivity.Fourier.PeriodicBandCompactness |
| import OAI.Analysis.Conductivity.Walls.EndingJoinCutoff |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology Real MeasureTheory Matrix |
| open scoped Matrix.Norms.Elementwise |
|
|
| def delayedInitialTensor (lam k J R : ℝ) (x : Coord3) : Mat3 := |
| fun i j => pureModeTensor (initialConnectorAngular (lam/k) (1/2) J (k*(x 0-R))) 1 |
| (angularAxisSwap i) (angularAxisSwap j) |
|
|
| lemma delayedInitialTensor_smooth (lam k J R : ℝ) : |
| ContDiff ℝ (↑(⊤:ℕ∞)) (delayedInitialTensor lam k J R) := by |
| have hq : ContDiff ℝ (↑(⊤:ℕ∞)) (fun x : Coord3 => initialConnectorAngular (lam/k) (1/2) J (k*(x 0-R))) := |
| (initialConnectorAngular_smooth _ _ _).comp (contDiff_const.mul ((contDiff_apply ℝ ℝ 0).sub contDiff_const)) |
| have hh := pureModeTensor_smooth hq (1:Fin 3) |
| apply contDiff_pi.mpr; intro i |
| apply contDiff_pi.mpr; intro j |
| exact contDiff_pi.mp (contDiff_pi.mp hh (angularAxisSwap i)) (angularAxisSwap j) |
|
|
| lemma delayedInitialTensor_symm (lam k J R : ℝ) (x : Coord3) : |
| (delayedInitialTensor lam k J R x).IsSymm := by |
| ext i j |
| exact congrFun (congrFun (pureModeTensor_symm _ _) (angularAxisSwap i)) (angularAxisSwap j) |
|
|
| lemma delayedEndingTensor_initial {lam k J L K R : ℝ} (x : Coord3) (hx : k*(x 0-R)<J+2) : |
| delayedEndingTensor lam k J L K R x=delayedInitialTensor lam k J R x := by |
| ext i j |
| change finiteEndingTensor (lam/k) J L K (permutedDilation angularAxisSwap k (x-Pi.single 0 R)) |
| (angularAxisSwap i) (angularAxisSwap j)=_ |
| have ht : (permutedDilation angularAxisSwap k (x-Pi.single 0 R)) 0=k*(x 0-R) := by |
| simp [permutedDilation,angularAxisSwap_apply] |
| rw [finiteEndingTensor,ht,ite_eq_left hx] |
| rfl |
|
|
| def endingJoinMiddleTensor (s : Fin 3 → ℝ) (lam k J : ℝ) (x : Coord3) : Symmetric3 := |
| ⟨(1-endingJoinCutoff (x 0)) • flatBackgroundTensor s+ |
| endingJoinCutoff (x 0) • delayedInitialTensor lam k J 7 x, |
| ((flatBackgroundTensor_symm s).smul _).add ((delayedInitialTensor_symm _ _ _ _ _).smul _)⟩ |
|
|
| lemma endingJoinMiddleTensor_smooth (s : Fin 3 → ℝ) (lam k J : ℝ) : |
| ContDiff ℝ (↑(⊤:ℕ∞)) (fun x => (endingJoinMiddleTensor s lam k J x).val) := by |
| have hχ := endingJoinCutoff_smooth.comp (contDiff_apply ℝ ℝ (0:Fin 3)) |
| exact ((contDiff_const.sub hχ).smul contDiff_const).add (hχ.smul (delayedInitialTensor_smooth _ _ _ _)) |
|
|
| lemma endingJoinMiddle_regular_ae (s : Fin 3 → ℝ) (lam J : ℝ) {k : ℝ} (hk : k≠0) : |
| ∀ᵐ x : Coord3,x∈regularRegion (wallCoordinatePair (pureWallMode 1 lam k)) |
| (endingJoinMiddleTensor s lam k J) univ := by |
| have hu : ContDiff ℝ (↑(⊤:ℕ∞)) (wallCoordinatePair (pureWallMode 1 lam k)) := |
| wallCoordinatePair_smooth (pureWallMode_smooth _ _ _) |
| have hh := regularRegion_ae_of_simple_minor hu (endingJoinMiddleTensor_smooth s lam k J) isOpen_univ |
| (fun x _ hx => (pureMode_pairMinor_noncritical (by norm_num : (1:ℝ)≠0) hk x).resolve_left (not_not.mpr hx)) |
| filter_upwards [hh] with x hx |
| exact hx (mem_univ _) |
|
|
| lemma wallCoordinatePair_pure (lam k : ℝ) (x : Coord3) : |
| wallCoordinatePair (pureWallMode 1 lam k) x=![x 0,exp (-lam*x 0)*cos (k*x 2)] := by |
| simp [wallCoordinatePair,pureWallMode,boxCoordinates_apply] |
|
|
| end ScalarConductivity |
|
|
| end |
| end |
|
|
| end OAI |
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|
