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159357f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | 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
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