| import Mathlib |
| import OAI.Analysis.Conductivity.Geometry.ControlledCorrectionBoxes |
| import OAI.Analysis.Conductivity.Variational.MetricControlledPiola |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set Matrix MeasureTheory Filter Topology |
| open scoped Matrix.Norms.Elementwise |
|
|
| theorem BoundedPhysicallyCorrectable.piola |
| (X : OpenPartialHomeomorph Coord3 Coord3) |
| (hX : ContDiffOn ℝ (↑(⊤:ℕ∞)) X X.source) |
| (hXi : ContDiffOn ℝ (↑(⊤:ℕ∞)) X.symm X.target) |
| {u v : Coord3 → Fin 2 → ℝ} (hu : ContDiff ℝ (↑(⊤:ℕ∞)) u) |
| (hv : ContDiff ℝ (↑(⊤:ℕ∞)) v) (hpot : EqOn u (v∘X.symm) X.target) |
| {U : Set Coord3} (hU : U⊆X.source) {C N : ℝ} (hC : 0≤C) |
| (hbound : ∀ x∈U,9*|(fderiv ℝ X x).det|⁻¹*‖operatorMatrix (fderiv ℝ X x)‖* |
| ‖operatorMatrix (fderiv ℝ X x)‖≤C) |
| {r : PhysicalSourcePair} (hr : CompactSmoothPair r) (hs : PairSupported r U) |
| (hc : BoundedPhysicallyCorrectable v U r N) : |
| BoundedPhysicallyCorrectable u X.target (fun j => localPiolaSource X (r j)) (C*N) := by |
| obtain ⟨B,hB,hBc,hBs,hsy,hsrc,hBb⟩ := hc |
| have hBs' := hBs.trans hU |
| have hsr (j) := (hs j).trans hU |
| have hweak (j) (φ : Coord3 → ℝ) (hφ : ContDiff ℝ (↑(⊤:ℕ∞)) φ) : |
| (∫ x,fderiv ℝ φ x ((B x*gradientColumns (fderiv ℝ v x)).col j))=-(∫ x,φ x*r j x) := by |
| have h := coordinateDivergence_weak (symmetricFlux_smooth hB hv j) (symmetricFlux_compact hBc v j) φ hφ |
| change (∫ x,fderiv ℝ φ x ((B x*gradientColumns (fderiv ℝ v x)).col j))= |
| -(∫ x,φ x*symmetricSource B v j x) at h |
| simpa only [hsrc] using h |
| let H := localPiolaTensor X B |
| have hH := localPiolaTensor_smooth X hX hXi B hB hBc hBs' |
| have hHc := (localPiolaTensor_compact X B hBc hBs').1 |
| have hHs := (localPiolaTensor_compact X B hBc hBs').2 |
| have hpr : CompactSmoothPair (fun j => localPiolaSource X (r j)) := fun j => |
| ⟨localPiolaSource_smooth X hX hXi (r j) (hr j).1 (hr j).2 (hsr j), |
| (localPiolaSource_compact X (r j) (hr j).2 (hsr j)).1⟩ |
| refine ⟨H,hH,hHc,hHs,localPiolaTensor_isSymm X B hsy,?_,?_⟩ |
| · apply symmetricSource_eq_of_weak hu hH hHc hpr |
| intro j ψ hψ |
| have he : (∫ y,fderiv ℝ ψ y ((H y*gradientColumns (fderiv ℝ (v∘X.symm) y)).col j))= |
| -(∫ y,ψ y*localPiolaSource X (r j) y) := by |
| simp_rw [H,localPiolaTensor_columns X (hX.differentiableOn (by simp)) |
| (hXi.differentiableOn (by simp)) B v (hv.differentiable (by simp)).differentiableOn] |
| have ht := tensorColumn_tsupport B (fun x => gradientColumns (fderiv ℝ v x)) j |
| exact localPiolaFlux_weak_source volume X hX hXi _ _ |
| (hBc.of_isClosed_subset isClosed_closure ht) (hr j).2 (ht.trans hBs') (hsr j) (hweak j) ψ hψ |
| refine (integral_congr_ae ?_).trans he |
| filter_upwards [] with y |
| by_cases hy : y∈X.target |
| · have hf : u =ᶠ[𝓝 y] v∘X.symm := |
| Filter.eventuallyEq_of_mem (X.open_target.mem_nhds hy) (fun z hz => hpot hz) |
| rw [hf.fderiv_eq] |
| · have hzero : H y=0 := image_eq_zero_of_notMem_tsupport (fun h => hy (hHs h)) |
| change (fderiv ℝ ψ y) ((H y*gradientColumns (fderiv ℝ u y)).col j)=_ |
| simp only [hzero,Matrix.zero_mul] |
| · intro y |
| exact (localPiolaTensor_bound_of_coefficient X hC hbound B ((subset_tsupport _).trans hBs) y).trans |
| (mul_le_mul_of_nonneg_left (hBb _) hC) |
|
|
| theorem physicalSourceMoment_piola |
| (X : OpenPartialHomeomorph Coord3 Coord3) |
| (hX : ContDiffOn ℝ (↑(⊤:ℕ∞)) X X.source) |
| (hXi : ContDiffOn ℝ (↑(⊤:ℕ∞)) X.symm X.target) |
| {u v : Coord3 → Fin 2 → ℝ} (hu : ContDiff ℝ (↑(⊤:ℕ∞)) u) |
| (hv : ContDiff ℝ (↑(⊤:ℕ∞)) v) (hpot : EqOn u (v∘X.symm) X.target) |
| {r : PhysicalSourcePair} (hr : CompactSmoothPair r) (hs : PairSupported r X.source) : |
| physicalSourceMoment u (fun j => localPiolaSource X (r j))=physicalSourceMoment v r := by |
| have hd := hX.differentiableOn (by simp) |
| have hdi := hXi.differentiableOn (by simp) |
| have hpr : CompactSmoothPair (fun j => localPiolaSource X (r j)) := fun j => |
| ⟨localPiolaSource_smooth X hX hXi (r j) (hr j).1 (hr j).2 (hs j), |
| (localPiolaSource_compact X (r j) (hr j).2 (hs j)).1⟩ |
| have he (j) (φ : Coord3 → ℝ) := localPiolaSource_pairing volume X hd |
| (fun _ hx => local_fderiv_det_ne_zero X hd hdi hx) (r j) ((subset_tsupport _).trans (hs j)) φ |
| have ht (i j : Fin 2) : (∫ y,u y i*localPiolaSource X (r j) y)=(∫ x,v x i*r j x) := by |
| rw [he] |
| apply integral_congr_ae |
| filter_upwards [] with x |
| by_cases hx : x∈X.source |
| · rw [hpot (X.map_source hx),Function.comp_apply,X.left_inv hx] |
| · rw [image_eq_zero_of_notMem_tsupport (fun h => hx (hs j h)),mul_zero,mul_zero] |
| ext i |
| fin_cases i |
| · change (∫ y,localPiolaSource X (r 0) y)=(∫ x,r 0 x) |
| simpa only [one_mul] using he 0 (fun _ => 1) |
| · change (∫ y,localPiolaSource X (r 1) y)=(∫ x,r 1 x) |
| simpa only [one_mul] using he 1 (fun _ => 1) |
| · change (∫ y,u y 1*localPiolaSource X (r 0) y-u y 0*localPiolaSource X (r 1) y)= |
| ∫ x,v x 1*r 0 x-v x 0*r 1 x |
| have hi (w : Coord3 → Fin 2 → ℝ) (hw : Continuous w) (s : PhysicalSourcePair) |
| (hs : CompactSmoothPair s) (i j : Fin 2) : Integrable (fun x => w x i*s j x) := |
| ((continuous_apply i|>.comp hw).mul (hs j).1.continuous).integrable_of_hasCompactSupport (hs j).2.mul_left |
| rw [integral_sub (hi u hu.continuous _ hpr 1 0) (hi u hu.continuous _ hpr 0 1),ht,ht, |
| integral_sub (hi v hv.continuous r hr 1 0) (hi v hv.continuous r hr 0 1)] |
|
|
| end ScalarConductivity |
|
|
| end |
| end |
|
|
| end OAI |
|
|
