| import Mathlib |
| import OAI.Analysis.Conductivity.Fourier.AngularPeriodization |
| import OAI.Analysis.Conductivity.Fourier.SmoothAngularCutoff |
|
|
| namespace OAI |
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| section |
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|
| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology Real MeasureTheory |
|
|
| def angularCutoff (T : ℝ) (x : Coord3) : ℝ := periodCutoff T (x 1)*periodCutoff T (x 2) |
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| lemma angularCutoff_smooth (T : ℝ) : ContDiff ℝ (↑(⊤:ℕ∞)) (angularCutoff T) := |
| ((periodCutoff_smooth T).comp (by fun_prop)).mul |
| ((periodCutoff_smooth T).comp (by fun_prop)) |
|
|
| lemma angularCutoff_support {T : ℝ} (hT : 0<T) : |
| tsupport (angularCutoff T)⊆{x | x 1∈Icc (-T) T ∧ x 2∈Icc (-T) T} := by |
| apply closure_minimal _ ((isClosed_Icc.preimage (continuous_apply 1)).inter |
| (isClosed_Icc.preimage (continuous_apply 2))) |
| intro x hx |
| have ha : periodCutoff T (x 1)≠0 := fun h => hx (by simp [angularCutoff,h]) |
| have hb : periodCutoff T (x 2)≠0 := fun h => hx (by simp [angularCutoff,h]) |
| exact ⟨periodCutoff_support hT (subset_tsupport _ ha),periodCutoff_support hT (subset_tsupport _ hb)⟩ |
|
|
| lemma angularCutoff_finsum {T : ℝ} (hT : 0<T) (x : Coord3) : |
| (∑ᶠ n : Fin 2 → ℤ,angularCutoff T (x+angularShift T n))=1 := by |
| let a : ℤ → ℝ := fun n => periodCutoff T (x 1+T*n) |
| let b : ℤ → ℝ := fun n => periodCutoff T (x 2+T*n) |
| have ha : (∑ᶠ n,a n)=1 := periodCutoff_finsum hT _ |
| have hb : (∑ᶠ n,b n)=1 := periodCutoff_finsum hT _ |
| have hf : Function.HasFiniteSupport (fun z : ℤ×ℤ => a z.1*b z.2) := by |
| apply ((hasFiniteSupport_of_finsum_eq_one ha).prod (hasFiniteSupport_of_finsum_eq_one hb)).subset |
| intro z hz |
| exact ⟨fun h => hz (by simp [h]),fun h => hz (by simp [h])⟩ |
| change (∑ᶠ n : Fin 2 → ℤ,a (n 0)*b (n 1))=1 |
| have he := finsum_comp_equiv (MeasurableEquiv.finTwoArrow (α:=ℤ)).symm.toEquiv |
| (f:=fun n : Fin 2 → ℤ => a (n 0)*b (n 1)) |
| change (∑ᶠ z : ℤ×ℤ,a z.1*b z.2)=(∑ᶠ n : Fin 2 → ℤ,a (n 0)*b (n 1)) at he |
| rw [←he,finsum_curry _ hf] |
| simp_rw [←mul_finsum,hb,mul_one] |
| exact ha |
|
|
| lemma angularCutoff_periodize {T : ℝ} (hT : 0<T) {f : Coord3 → ℝ} (hp : AngularPeriodic T f) : |
| angularPeriodize T (fun x => angularCutoff T x*f x)=f := by |
| funext x |
| unfold angularPeriodize |
| have he : (fun n : Fin 2 → ℤ => angularCutoff T (x+angularShift T n)*f (x+angularShift T n))= |
| (fun n => angularCutoff T (x+angularShift T n)*f x) := by |
| funext n; rw [hp n x] |
| rw [he] |
| rw [←finsum_mul,angularCutoff_finsum hT,one_mul] |
|
|
| lemma angularCutoff_compact_support {T a b : ℝ} (hT : 0<T) {f : Coord3 → ℝ} |
| (hf : tsupport f⊆{x | x 0∈Icc a b}) : |
| tsupport (fun x => angularCutoff T x*f x)⊆Icc ![a,-T,-T] ![b,T,T] := by |
| intro x hx |
| have h0 := hf ((tsupport_mul_subset_right (f:=angularCutoff T) (g:=f)) hx) |
| have hθ := angularCutoff_support hT ((tsupport_mul_subset_left (f:=angularCutoff T) (g:=f)) hx) |
| constructor <;> intro i <;> fin_cases i |
| · exact h0.1 |
| · exact hθ.1.1 |
| · exact hθ.2.1 |
| · exact h0.2 |
| · exact hθ.1.2 |
| · exact hθ.2.2 |
|
|
| lemma angularCutoff_compact {T a b : ℝ} (hT : 0<T) {f : Coord3 → ℝ} |
| (hf : tsupport f⊆{x | x 0∈Icc a b}) : |
| HasCompactSupport (fun x => angularCutoff T x*f x) := |
| isCompact_Icc.of_isClosed_subset (isClosed_tsupport _) (angularCutoff_compact_support hT hf) |
|
|
| lemma AngularPeriodic.coordinate_one {T : ℝ} {f : Coord3 → ℝ} (hp : AngularPeriodic T f) |
| (t z : ℝ) : Function.Periodic (fun y => f ![t,y,z]) T := by |
| intro y |
| have hh := hp ![1,0] ![t,y,z] |
| have he : ![t,y,z]+angularShift T ![1,0]=![t,y+T,z] := by |
| ext i; fin_cases i <;> simp [angularShift] |
| rw [he] at hh |
| exact hh |
|
|
| lemma AngularPeriodic.coordinate_two {T : ℝ} {f : Coord3 → ℝ} (hp : AngularPeriodic T f) |
| (t y : ℝ) : Function.Periodic (fun z => f ![t,y,z]) T := by |
| intro z |
| have hh := hp ![0,1] ![t,y,z] |
| have he : ![t,y,z]+angularShift T ![0,1]=![t,y,z+T] := by |
| ext i; fin_cases i <;> simp [angularShift] |
| rw [he] at hh |
| exact hh |
|
|
| end ScalarConductivity |
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| end |
| end |
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| end OAI |
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|
