| import Mathlib | |
| import OAI.Analysis.Conductivity.Geometry.PhysicalBoxCorrection | |
| import OAI.Analysis.Conductivity.Variational.PhysicalOverlapMoments | |
| namespace OAI | |
| noncomputable section | |
| namespace ScalarConductivity | |
| open Set MeasureTheory Filter Topology Matrix | |
| open scoped Matrix.Norms.Elementwise | |
| lemma localPiolaSource_tsupport_image | |
| (X : OpenPartialHomeomorph Coord3 Coord3) (r : Coord3 → ℝ) | |
| (hc : HasCompactSupport r) (hs : tsupport r⊆X.source) : | |
| tsupport (localPiolaSource X r)⊆X '' tsupport r := by | |
| exact closure_minimal ((localPiolaSource_support X r).trans | |
| (image_mono (fun _ hx => subset_closure hx.1))) | |
| (hc.image_of_continuousOn (X.continuousOn.mono hs)).isClosed | |
| lemma localPiolaSource_inverse | |
| (X : OpenPartialHomeomorph Coord3 Coord3) | |
| (hX : DifferentiableOn ℝ X X.source) (hXi : DifferentiableOn ℝ X.symm X.target) | |
| (r : Coord3 → ℝ) (hs : Function.support r⊆X.target) : | |
| localPiolaSource X (localPiolaSource X.symm r)=r := by | |
| funext y | |
| by_cases hy : y∈X.target | |
| · have he := congrArg (fun D : Coord3 →L[ℝ] Coord3 => D.det) | |
| (local_fderiv_symm_comp X.symm hXi hX hy) | |
| simp only [OpenPartialHomeomorph.symm_symm,ContinuousLinearMap.det, | |
| ContinuousLinearMap.toLinearMap_comp,ContinuousLinearMap.coe_id, | |
| LinearMap.det_comp,LinearMap.det_id] at he | |
| have habs : |(fderiv ℝ X (X.symm y)).det| * |(fderiv ℝ X.symm y).det|=1 := by | |
| rw [←abs_mul] | |
| exact congrArg abs he |>.trans abs_one | |
| classical | |
| change (if y∈X.target then |(fderiv ℝ X (X.symm y)).det|⁻¹ * | |
| (if X.symm y∈X.source then |(fderiv ℝ X.symm (X (X.symm y))).det|⁻¹ * | |
| r (X (X.symm y)) else 0) else 0) = r y | |
| rw [ite_eq_left hy,ite_eq_left (X.map_target hy),X.right_inv hy] | |
| rw [←mul_assoc,←mul_inv,habs,inv_one,one_mul] | |
| · rw [localPiolaSource,ite_eq_right hy,Function.notMem_support.mp (fun h => hy (hs h))] | |
| def correctionBox (a b : Fin 3 → ℝ) : Set Coord3 := | |
| boxCoordinates ⁻¹' ((Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2)) | |
| lemma localPiolaSource_inverse_support | |
| (X : OpenPartialHomeomorph Coord3 Coord3) (r : Coord3 → ℝ) | |
| (hc : HasCompactSupport r) {V : Set Coord3} (hV : V⊆X.source) | |
| (hs : tsupport r⊆X '' V) : | |
| tsupport (localPiolaSource X.symm r)⊆V := by | |
| have ht : tsupport r⊆X.target := hs.trans (by rintro y ⟨x,hx,rfl⟩; exact X.map_source (hV hx)) | |
| intro x hx | |
| obtain ⟨y,hy,rfl⟩ := localPiolaSource_tsupport_image X.symm r hc ht hx | |
| obtain ⟨z,hz,rfl⟩ := hs hy | |
| simpa only [X.left_inv (hV hz)] using hz | |
| theorem local_physical_symmetric_correction | |
| (X : OpenPartialHomeomorph Coord3 Coord3) | |
| (hX : ContDiffOn ℝ (↑(⊤:ℕ∞)) X X.source) | |
| (hXi : ContDiffOn ℝ (↑(⊤:ℕ∞)) X.symm X.target) | |
| {a b : Fin 3 → ℝ} (hab : ∀ i,a i<b i) (hbox : correctionBox a b⊆X.source) | |
| (u : Coord3 → Fin 2 → ℝ) (hu : EqOn u (coordinatePair∘X.symm) X.target) | |
| (r : Fin 2 → Coord3 → ℝ) | |
| (hr : ∀ j,ContDiff ℝ (↑(⊤:ℕ∞)) (r j)) | |
| (hc : ∀ j,HasCompactSupport (r j)) | |
| (hs : ∀ j,tsupport (r j)⊆X '' correctionBox a b) | |
| (hz : ∀ j,(∫ y,r j y)=0) | |
| (ht : (∫ y,u y 1*r 0 y-u y 0*r 1 y)=0) : | |
| ∃ H : Coord3 → Mat3,ContDiff ℝ (↑(⊤:ℕ∞)) H ∧ HasCompactSupport H ∧ | |
| tsupport H⊆X.target ∧ (∀ y,(H y).IsSymm) ∧ | |
| ∀ j (ψ : Coord3 → ℝ),ContDiff ℝ (↑(⊤:ℕ∞)) ψ → | |
| (∫ y,fderiv ℝ ψ y ((H y*gradientColumns (fderiv ℝ u y)).col j))= | |
| -(∫ y,ψ y*r j y) := by | |
| let p : Fin 2 → Coord3 → ℝ := fun j => localPiolaSource X.symm (r j) | |
| have hst (j) : tsupport (r j)⊆X.target := | |
| (hs j).trans (by rintro y ⟨x,hx,rfl⟩; exact X.map_source (hbox hx)) | |
| have hp (j) : ContDiff ℝ (↑(⊤:ℕ∞)) (p j) := | |
| localPiolaSource_smooth X.symm hXi hX (r j) (hr j) (hc j) (hst j) | |
| have hpc (j) : HasCompactSupport (p j) := | |
| (localPiolaSource_compact X.symm (r j) (hc j) (hst j)).1 | |
| have hps (j) : tsupport (p j)⊆correctionBox a b := | |
| localPiolaSource_inverse_support X (r j) (hc j) hbox (hs j) | |
| let q : Fin 2 → Box3 → ℝ := fun j => p j∘boxCoordinates.symm | |
| have hq (j) : ContDiff ℝ (↑(⊤:ℕ∞)) (q j) := (hp j).comp boxCoordinates.symm.contDiff | |
| have hqc (j) : HasCompactSupport (q j) := (hpc j).comp_homeomorph boxCoordinates.symm.toHomeomorph | |
| have hqs (j) : tsupport (q j)⊆ | |
| (Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2) := by | |
| intro z hz | |
| have h := hps j ((tsupport_comp_subset_preimage (p j) boxCoordinates.symm.continuous) hz) | |
| simpa only [correctionBox,mem_preimage,boxCoordinates.apply_symm_apply] using h | |
| have hd := hX.differentiableOn (by simp) | |
| have hdi := hXi.differentiableOn (by simp) | |
| have hback (j) : localPiolaSource X (p j)=r j := | |
| localPiolaSource_inverse X hd hdi (r j) ((subset_tsupport _).trans (hst j)) | |
| have hqz (j) : (∫ z,q j z)=0 := by | |
| rw [←boxCoordinates_integral (q j)] | |
| simp only [q,Function.comp_apply,boxCoordinates.symm_apply_apply] | |
| have he := localPiolaSource_pairing volume X.symm hdi | |
| (fun _ hy => local_fderiv_det_ne_zero X.symm hdi hd hy) (r j) | |
| ((subset_tsupport _).trans (hst j)) (fun _ => 1) | |
| simpa only [one_mul] using he.trans (by simpa only [one_mul] using hz j) | |
| have hqt : (∫ z : Box3,z.1.2*q 0 z-z.1.1*q 1 z)=0 := by | |
| rw [←boxCoordinates_integral (fun z : Box3 => z.1.2*q 0 z-z.1.1*q 1 z)] | |
| have he := localPiolaSource_torque X hd hdi (p 0) (p 1) | |
| ((subset_tsupport _).trans ((hps 0).trans hbox)) | |
| ((subset_tsupport _).trans ((hps 1).trans hbox)) | |
| rw [hback 0,hback 1] at he | |
| simp only [q,Function.comp_apply,boxCoordinates.symm_apply_apply] | |
| change (∫ x,x 1*p 0 x-x 0*p 1 x)=0 | |
| rw [←he] | |
| refine (integral_congr_ae ?_).trans ht | |
| filter_upwards [] with y | |
| by_cases hy : y∈X.target | |
| · rw [hu hy]; rfl | |
| · rw [image_eq_zero_of_notMem_tsupport (fun h => hy (hst 0 h)), | |
| image_eq_zero_of_notMem_tsupport (fun h => hy (hst 1 h)),mul_zero,mul_zero,mul_zero,mul_zero] | |
| obtain ⟨H,hH,hHc,hHs,hsy,hweak⟩ := physical_box_symmetric_correction X hX hXi hab hbox | |
| (hq 0) (hq 1) (hqc 0) (hqc 1) (hqs 0) (hqs 1) (hqz 0) (hqz 1) hqt | |
| refine ⟨H,hH,hHc,hHs,hsy,?_⟩ | |
| intro j ψ hψ | |
| have he := hweak j ψ hψ | |
| have hpsource : (fun x => ![q 0 (boxCoordinates x),q 1 (boxCoordinates x)] j)=p j := by | |
| funext x | |
| fin_cases j <;> simp [q] | |
| rw [hpsource,hback j] at he | |
| refine (integral_congr_ae ?_).trans he | |
| filter_upwards [] with y | |
| by_cases hy : y∈X.target | |
| · have hf : u =ᶠ[𝓝 y] coordinatePair∘X.symm := | |
| Filter.eventuallyEq_of_mem (X.open_target.mem_nhds hy) (fun z hz => hu hz) | |
| rw [hf.fderiv_eq] | |
| · rw [image_eq_zero_of_notMem_tsupport (fun h => hy (hHs h)),Matrix.zero_mul,Matrix.zero_mul] | |
| end ScalarConductivity | |
| end | |
| end OAI | |