| import Mathlib |
| import OAI.Analysis.Conductivity.Geometry.TorusLocalizedMoments |
| import OAI.Analysis.Conductivity.Fourier.FourierMatchingInterpolation |
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| namespace OAI |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology Real MeasureTheory Matrix |
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| lemma flatTensorLaplacian_congr_nhds (s : Fin 3 → ℝ) {f g : Coord3 → ℝ} {x : Coord3} |
| (h : f=ᶠ[𝓝 x] g) : flatTensorLaplacian s f x=flatTensorLaplacian s g x := |
| congrArg (flatTensorLaplacianCLM s) ((h.iteratedFDeriv ℝ 2).self_of_nhds) |
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| lemma fourier_left_coeff_bound {k : ℤ} {b : (Fin 2 → ℤ) → ℝ} {B : ℝ} |
| (hb : ∀ h,|b h|≤B) (h : Fin 2 → ℤ) : |
| |(Pi.single ![0,k] (1:ℝ) : (Fin 2 → ℤ) → ℝ) h+b h|≤1+B := |
| (abs_add_le _ _).trans (add_le_add (single_fourier_coeff_bound _ _) (hb h)) |
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| lemma fourierMatchingPair_left_fourier {s : Fin 3 → ℝ} |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| (k : ℤ) {f g : ℝ×Coord3 → ℝ} {a b pa pb : (Fin 2 → ℤ) → ℝ} {B p : ℝ} |
| (hb : ∀ h,|b h|≤B) (hp : pb ![0,k]=0) |
| (hf : ∀ y : Coord3,y 0∈Icc 0 6 → f (p,y)=flatFourier s a pa y) |
| (hg : ∀ y : Coord3,y 0∈Icc 0 6 → g (p,y)=flatFourier s b pb y) |
| {x : Coord3} (hx : x 0∈Ioo 0 2) : |
| (fun y => fourierMatchingPair s k f g (p,y) 0)=ᶠ[𝓝 x] affineFourierField s a pa 1 0 ∧ |
| (fun y => fourierMatchingPair s k f g (p,y) 1)=ᶠ[𝓝 x] |
| flatFourier s (fun h => (Pi.single ![0,k] (1:ℝ) : (Fin 2 → ℤ) → ℝ) h+b h) pb := by |
| have hn : ∀ᶠ y in 𝓝 x,y 0∈Ioo 0 2 := |
| (isOpen_Ioo.preimage (continuous_apply 0)).mem_nhds hx |
| constructor |
| · filter_upwards [hn] with y hy |
| change y 0+fourierMatchingCutoff (y 0)*f (p,y)=_ |
| rw [fourierMatchingCutoff_left hy.2.le,one_mul,hf y ⟨hy.1.le,by linarith [hy.2]⟩] |
| simp only [affineFourierField,one_mul,add_zero] |
| · filter_upwards [hn] with y hy |
| change flatPhaseMode s ![0,k] 0 y+fourierMatchingCutoff (y 0)*g (p,y)=_ |
| rw [fourierMatchingCutoff_left hy.2.le,one_mul,hg y ⟨hy.1.le,by linarith [hy.2]⟩, |
| flatFourier_add hs (single_fourier_coeff_bound ![0,k]) hb hy.1, |
| flatFourier_single,hp,one_mul] |
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| lemma fourierMatchingPair_right_fourier (s : Fin 3 → ℝ) (k : ℤ) (f g : ℝ×Coord3 → ℝ) |
| (pa pb : (Fin 2 → ℤ) → ℝ) (hp : pb ![0,k]=0) (p : ℝ) {x : Coord3} (hx : 4<x 0) : |
| (fun y => fourierMatchingPair s k f g (p,y) 0)=ᶠ[𝓝 x] affineFourierField s (fun _ => 0) pa 1 0 ∧ |
| (fun y => fourierMatchingPair s k f g (p,y) 1)=ᶠ[𝓝 x] flatFourier s (Pi.single ![0,k] 1) pb := by |
| have hn : ∀ᶠ y in 𝓝 x,4<y 0 := (isOpen_lt continuous_const (continuous_apply 0)).mem_nhds hx |
| constructor |
| · filter_upwards [hn] with y hy |
| change y 0+fourierMatchingCutoff (y 0)*f (p,y)=_ |
| simp only [fourierMatchingCutoff_right hy.le,zero_mul,add_zero,affineFourierField, |
| flatFourier_zero,one_mul] |
| · filter_upwards [hn] with y hy |
| change flatPhaseMode s ![0,k] 0 y+fourierMatchingCutoff (y 0)*g (p,y)=_ |
| rw [fourierMatchingCutoff_right hy.le,zero_mul,add_zero,flatFourier_single,hp,one_mul] |
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| end ScalarConductivity |
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| namespace ScalarConductivity |
| open Set Filter Topology Real MeasureTheory Matrix |
| open scoped Matrix.Norms.Elementwise |
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| lemma periodicSourceMoment_neg (T : ℝ) (u : Coord3 → Fin 2 → ℝ) (r : Fin 2 → Coord3 → ℝ) : |
| periodicSourceMoment T u (fun j x => -r j x)=-periodicSourceMoment T u r := by |
| have hpoint (x : Coord3) : u x 1*(-r 0 x)-u x 0*(-r 1 x)= |
| -(u x 1*r 0 x-u x 0*r 1 x) := by ring |
| ext i |
| fin_cases i |
| · change (∫ t,∫ y in 0..T,∫ z in 0..T,-r 0 ![t,y,z])=-(∫ t,∫ y in 0..T,∫ z in 0..T,r 0 ![t,y,z]) |
| simp only [intervalIntegral.integral_neg,integral_neg] |
| · change (∫ t,∫ y in 0..T,∫ z in 0..T,-r 1 ![t,y,z])=-(∫ t,∫ y in 0..T,∫ z in 0..T,r 1 ![t,y,z]) |
| simp only [intervalIntegral.integral_neg,integral_neg] |
| · change (∫ t,∫ y in 0..T,∫ z in 0..T,u ![t,y,z] 1*(-r 0 ![t,y,z])-u ![t,y,z] 0*(-r 1 ![t,y,z]))= |
| -(∫ t,∫ y in 0..T,∫ z in 0..T,u ![t,y,z] 1*r 0 ![t,y,z]-u ![t,y,z] 0*r 1 ![t,y,z]) |
| simp only [hpoint,intervalIntegral.integral_neg,integral_neg] |
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| theorem fourier_matching_three_moments {s : Fin 3 → ℝ} |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| {a b : Fin 2 → (Fin 2 → ℤ) → ℝ} {pa pb : Fin 2 → (Fin 2 → ℤ) → ℝ} |
| {Ba Bb ga gb : Fin 2 → ℝ} {l d α : ℝ} {β : Fin 2 → ℝ} |
| (ha : ∀ e h,|a e h|≤Ba e) (hb : ∀ e h,|b e h|≤Bb e) |
| (hga : ∀ e,0<ga e) (hgb : ∀ e,0<gb e) |
| (hra : ∀ e h,a e h≠0 → ga e≤torusRate s h) |
| (hrb : ∀ e h,b e h≠0 → gb e≤torusRate s h) |
| (hl : 0<l) (hld : l≤d) {u : Coord3 → Fin 2 → ℝ} {r : Fin 2 → Coord3 → ℝ} |
| (hu : ContDiff ℝ (↑(⊤:ℕ∞)) u) (hup : AngularPeriodic (2*Real.pi) u) |
| (he0 : ∀ e x,x 0=![l,d] e → (fun y => u y 0)=ᶠ[𝓝 x] affineFourierField s (a e) (pa e) α (β e)) |
| (he1 : ∀ e x,x 0=![l,d] e → (fun y => u y 1)=ᶠ[𝓝 x] flatFourier s (b e) (pb e)) |
| (hrs : ∀ j,tsupport (r j)⊆{x | x 0∈Icc l d}) |
| (hre : ∀ j x,x 0∈Icc l d → r j x=symmetricSource (fun _ => flatBackgroundTensor s) u j x) : |
| periodicSourceMoment (2*Real.pi) u r=0 ∧ |
| periodicSourceMoment (2*Real.pi) u (fun j x => -r j x)=0 := by |
| have ht (e : Fin 2) : 0<![l,d] e := by fin_cases e; exact hl; exact hl.trans_le hld |
| have he (e : Fin 2) := fourier_pair_boundary_flux hs (ha e) (hb e) (hga e) (hgb e) |
| (hra e) (hrb e) α (β e) (ht e) (hu.differentiable (by simp)) (he0 e) (he1 e) |
| have heL := he 0 |
| have heR := he 1 |
| simp only [Matrix.cons_val_zero,Matrix.cons_val_one] at heL heR |
| have hm := periodic_localized_source_moment_formula (by positivity : 0≤2*Real.pi) hld |
| (H:=fun _ => flatBackgroundTensor s) contDiff_const hu (fun _ _ => rfl) hup |
| (fun _ => flatBackgroundTensor_symm s) hrs hre |
| have hz : periodicSourceMoment (2*Real.pi) u r=0 := by |
| rw [hm] |
| ext i |
| fin_cases i |
| · change _-_=0 |
| rw [heR.1,heL.1,sub_self] |
| · change _-_=0 |
| rw [heR.2.1,heL.2.1,sub_self] |
| · change _-_=0 |
| rw [heR.2.2,heL.2.2,sub_self] |
| exact ⟨hz,by rw [periodicSourceMoment_neg,hz,neg_zero]⟩ |
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| end ScalarConductivity |
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| end |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology Real MeasureTheory Matrix |
|
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| lemma fourierMatchingResidual_exact {s : Fin 3 → ℝ} |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| (k : ℤ) {f g : ℝ×Coord3 → ℝ} {a b pa pb : (Fin 2 → ℤ) → ℝ} {A B p : ℝ} |
| (hfs : ContDiff ℝ (↑(⊤:ℕ∞)) f) (hgs : ContDiff ℝ (↑(⊤:ℕ∞)) g) |
| (ha : ∀ h,|a h|≤A) (hb : ∀ h,|b h|≤B) (hp : pb ![0,k]=0) |
| (hf : ∀ y : Coord3,y 0∈Icc 0 6 → f (p,y)=flatFourier s a pa y) |
| (hg : ∀ y : Coord3,y 0∈Icc 0 6 → g (p,y)=flatFourier s b pb y) |
| (j : Fin 2) (x : Coord3) (hx : 0<x 0) : |
| fourierMatchingResidual s k f g j (p,x)= |
| -symmetricSource (fun _ => flatBackgroundTensor s) (fun y => fourierMatchingPair s k f g (p,y)) j x := by |
| have hu : ContDiff ℝ (↑(⊤:ℕ∞)) (fun y : Coord3 => fourierMatchingPair s k f g (p,y)) := |
| (fourierMatchingPair_smooth s k hfs hgs).comp ((contDiff_const (c:=p)).prodMk contDiff_id) |
| by_cases hmid : x 0∈Icc 2 4 |
| · simp only [fourierMatchingResidual,fourierResidualCutoff_one hmid,neg_mul,one_mul] |
| have hz : symmetricSource (fun _ => flatBackgroundTensor s) (fun y => fourierMatchingPair s k f g (p,y)) j x=0 := by |
| rw [flatBackgroundTensor_source hu] |
| by_cases hleft : x 0<2 |
| · have he := fourierMatchingPair_left_fourier hs k hb hp hf hg (x:=x) ⟨hx,hleft⟩ |
| fin_cases j |
| · change flatTensorLaplacian s (fun y => fourierMatchingPair s k f g (p,y) 0) x=0 |
| rw [flatTensorLaplacian_congr_nhds s he.1] |
| exact affineFourierField_harmonic hs ha 1 0 hx |
| · change flatTensorLaplacian s (fun y => fourierMatchingPair s k f g (p,y) 1) x=0 |
| rw [flatTensorLaplacian_congr_nhds s he.2] |
| exact flatFourier_harmonic hs (fourier_left_coeff_bound hb) hx |
| · have hright : 4<x 0 := by |
| by_contra hh |
| exact hmid ⟨le_of_not_gt hleft,le_of_not_gt hh⟩ |
| have he := fourierMatchingPair_right_fourier s k f g pa pb hp p hright |
| fin_cases j |
| · change flatTensorLaplacian s (fun y => fourierMatchingPair s k f g (p,y) 0) x=0 |
| rw [flatTensorLaplacian_congr_nhds s he.1] |
| exact affineFourierField_harmonic hs (B:=0) (by intro h; simp) 1 0 (by linarith) |
| · change flatTensorLaplacian s (fun y => fourierMatchingPair s k f g (p,y) 1) x=0 |
| rw [flatTensorLaplacian_congr_nhds s he.2] |
| exact flatFourier_harmonic hs (single_fourier_coeff_bound ![0,k]) (by linarith) |
| simp only [fourierMatchingResidual,hz,mul_zero,neg_zero] |
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| lemma fourier_single_plus_rate {s : Fin 3 → ℝ} {k : ℤ} {b : (Fin 2 → ℤ) → ℝ} |
| (hb : ∀ h,b h≠0 → torusRate s ![0,k]≤torusRate s h) : |
| ∀ h,(Pi.single ![0,k] (1:ℝ) : (Fin 2 → ℤ) → ℝ) h+b h≠0 → torusRate s ![0,k]≤torusRate s h := by |
| classical |
| intro h hh |
| by_cases he : h=![0,k] |
| · subst h; exact le_rfl |
| · apply hb h |
| simpa only [Pi.single_eq_of_ne he,zero_add] using hh |
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| lemma fourierMatchingResidual_moments {s : Fin 3 → ℝ} |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| (k : ℤ) {f g : ℝ×Coord3 → ℝ} {a b pa pb : (Fin 2 → ℤ) → ℝ} {A B p ga : ℝ} |
| (hfs : ContDiff ℝ (↑(⊤:ℕ∞)) f) (hgs : ContDiff ℝ (↑(⊤:ℕ∞)) g) |
| (hfp : ∀ p,AngularPeriodic (2*Real.pi) (fun x => f (p,x))) |
| (hgp : ∀ p,AngularPeriodic (2*Real.pi) (fun x => g (p,x))) |
| (ha : ∀ h,|a h|≤A) (hb : ∀ h,|b h|≤B) (hp : pb ![0,k]=0) |
| (hga : 0<ga) (hgb : 0<torusRate s ![0,k]) |
| (hra : ∀ h,a h≠0 → ga≤torusRate s h) |
| (hrb : ∀ h,b h≠0 → torusRate s ![0,k]≤torusRate s h) |
| (hf : ∀ y : Coord3,y 0∈Icc 0 6 → f (p,y)=flatFourier s a pa y) |
| (hg : ∀ y : Coord3,y 0∈Icc 0 6 → g (p,y)=flatFourier s b pb y) : |
| periodicSourceMoment (2*Real.pi) (fun y => fourierMatchingPair s k f g (p,y)) |
| (fun j y => fourierMatchingResidual s k f g j (p,y))=0 := by |
| classical |
| let aa : Fin 2 → (Fin 2 → ℤ) → ℝ := ![a,fun _ => 0] |
| let bb : Fin 2 → (Fin 2 → ℤ) → ℝ := ![fun h => (Pi.single ![0,k] (1:ℝ) : (Fin 2 → ℤ) → ℝ) h+b h,Pi.single ![0,k] 1] |
| have haa : ∀ e h,|aa e h|≤![A,0] e := by intro e h; fin_cases e; exact ha h; simp [aa] |
| have hbb : ∀ e h,|bb e h|≤![1+B,1] e := by |
| intro e h |
| fin_cases e |
| · exact fourier_left_coeff_bound hb h |
| · exact single_fourier_coeff_bound _ _ |
| have hraa : ∀ e h,aa e h≠0 → ga≤torusRate s h := by |
| intro e h hh |
| fin_cases e |
| · exact hra h hh |
| · exact False.elim (hh rfl) |
| have hrbb : ∀ e h,bb e h≠0 → torusRate s ![0,k]≤torusRate s h := by |
| intro e h hh |
| fin_cases e |
| · exact fourier_single_plus_rate hrb h hh |
| · by_cases he : h=![0,k] |
| · subst h; exact le_rfl |
| · exact False.elim (hh (Pi.single_eq_of_ne he 1)) |
| have he (e : Fin 2) (x : Coord3) (hx : x 0=![1,5] e) : |
| (fun y => fourierMatchingPair s k f g (p,y) 0)=ᶠ[𝓝 x] affineFourierField s (aa e) pa 1 0 ∧ |
| (fun y => fourierMatchingPair s k f g (p,y) 1)=ᶠ[𝓝 x] flatFourier s (bb e) pb := by |
| fin_cases e |
| · exact fourierMatchingPair_left_fourier hs k hb hp hf hg ⟨by simpa using hx ▸ (show (0:ℝ)<1 by norm_num),by simpa using hx ▸ (show (1:ℝ)<2 by norm_num)⟩ |
| · exact fourierMatchingPair_right_fourier s k f g pa pb hp p (by simpa using hx ▸ (show (4:ℝ)<5 by norm_num)) |
| have hrs (j : Fin 2) : tsupport (fun y => -fourierMatchingResidual s k f g j (p,y))⊆{x | x 0∈Icc 1 5} := by |
| rw [show (fun y => -fourierMatchingResidual s k f g j (p,y))= -(fun y => fourierMatchingResidual s k f g j (p,y)) from rfl,tsupport_neg] |
| exact fourierMatchingResidual_support s k f g p j |
| have hre (j : Fin 2) (x : Coord3) (hx : x 0∈Icc 1 5) : |
| -fourierMatchingResidual s k f g j (p,x)= symmetricSource (fun _ => flatBackgroundTensor s) |
| (fun y => fourierMatchingPair s k f g (p,y)) j x := by |
| rw [fourierMatchingResidual_exact hs k hfs hgs ha hb hp hf hg j x (by linarith [hx.1]),neg_neg] |
| have hu : ContDiff ℝ (↑(⊤:ℕ∞)) (fun y : Coord3 => fourierMatchingPair s k f g (p,y)) := |
| (fourierMatchingPair_smooth s k hfs hgs).comp ((contDiff_const (c:=p)).prodMk contDiff_id) |
| have hm := fourier_matching_three_moments hs haa hbb (fun _ => hga) (fun _ => hgb) |
| hraa hrbb (by norm_num : (0:ℝ)<1) (by norm_num : (1:ℝ)≤5) |
| hu |
| (fourierMatchingPair_periodic s k hfp hgp p) (fun e x hx => (he e x hx).1) (fun e x hx => (he e x hx).2) hrs hre |
| simpa only [neg_neg,Function.comp_apply] using hm.2 |
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| end ScalarConductivity |
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| end |
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| end OAI |
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