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a458723 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | import Mathlib
import OAI.Analysis.Conductivity.Geometry.RegularPatch
import OAI.Analysis.Conductivity.Variational.FiniteEndPhysicalRegularity
import OAI.Analysis.Conductivity.Flux.FiniteEndPhysicalSmoothTensor
import OAI.Analysis.Conductivity.Flux.VariableCollarTensor
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set Filter Topology MeasureTheory Matrix
open scoped Matrix.Norms.Elementwise
def variableFaceField (u : Coord3 → Fin 2 → ℝ) (a b : ℝ) (i j : Fin 4) : Coord3 → Fin 2 → ℝ :=
fun y => u (flatEndCoordinates a b (sourceFaceAngles i j (sourceCollarInverse i j y)))
def variableCollarField (u : Coord3 → Fin 2 → ℝ) (a b : ℝ) : Coord3 → Fin 2 → ℝ :=
sourceFacePaste (variableFaceField u a b) 0
lemma variableCollarField_open (u : Coord3 → Fin 2 → ℝ) (a b : ℝ)
(i j : Fin 4) {x : Coord3} (hx : x∈sourceCollarOpenBox) :
variableCollarField u a b (sourceCollarPiece i j x)=
u (flatEndCoordinates a b (sourceFaceAngles i j x)) := by
rw [variableCollarField,sourceFacePaste_open _ _ i j hx]
simp only [variableFaceField,sourceCollarInverse_point i j hx]
lemma flatEndCoordinates_fderiv_surjective {a : ℝ} (ha : a≠0) (b : ℝ) (x : Coord3) :
Function.Surjective (fderiv ℝ (flatEndCoordinates a b) x) := by
change Function.Surjective (fderiv ℝ (endAxialAffine a b) x)
rw [(endAxialAffine_hasFDeriv a b x).fderiv]
exact (LinearMap.equivOfDetNeZero (Matrix.toLin' (endAxialMatrix a)) (by
simpa [LinearMap.det_toLin',endAxialMatrix,Matrix.det_diagonal,Fin.prod_univ_succ] using ha)).surjective
lemma sourceFaceAngles_fderiv_surjective (i j : Fin 4) {x : Coord3}
(hx : x∈sourceCollarOpenBox) : Function.Surjective (fderiv ℝ (sourceFaceAngles i j) x) := by
rw [(sourceFaceAngles_hasFDeriv i j x).fderiv]
exact (LinearMap.equivOfDetNeZero (Matrix.toLin' (sourceFaceAngleMatrix i j x)) (by
rw [LinearMap.det_toLin']; exact (sourceFaceAngleMatrix_det_pos i j (sourceCollarOpenBox_subset hx)).ne')).surjective
lemma variableCollarTensor_local_eq (K : Coord3 → Mat3) (a b : ℝ)
(i j : Fin 4) {x : Coord3} (hx : x∈sourceCollarOpenBox) :
variableCollarTensor K a b =ᶠ[𝓝 (sourceCollarPiece i j x)]
(attachedVariableTensor (K ∘ flatEndCoordinates a b ∘ sourceFaceAngles i j) a i j ∘ sourceCollarInverse i j) := by
change ∀ᶠ y in 𝓝 (sourceCollarPiece i j x), _ = _
rw [←sourceCollarPiece_map_nhds i j hx,eventually_map]
filter_upwards [isOpen_sourceCollarOpenBox.mem_nhds hx] with y hy
simpa only [Function.comp_def,sourceCollarInverse_point i j hy] using
variableCollarTensor_open K a b i j hy
lemma variableCollarTensor_smooth_at {K : Coord3 → Mat3} {a : ℝ} (ha : a≠0) (b : ℝ)
(i j : Fin 4) {x : Coord3} (hx : x∈sourceCollarOpenBox)
(hK : ContDiffAt ℝ (↑(⊤:ℕ∞)) K (flatEndCoordinates a b (sourceFaceAngles i j x))) :
ContDiffAt ℝ (↑(⊤:ℕ∞)) (variableCollarTensor K a b) (sourceCollarPiece i j x) := by
have hk : ContDiffAt ℝ (↑(⊤:ℕ∞)) (K ∘ flatEndCoordinates a b ∘ sourceFaceAngles i j) x :=
hK.comp x ((contDiff_flatEndCoordinates a b).contDiffAt.comp x
(contDiff_sourceFaceAngles i j).contDiffAt)
have ht : ContDiffAt ℝ (↑(⊤:ℕ∞))
(attachedVariableTensor (K ∘ flatEndCoordinates a b ∘ sourceFaceAngles i j) a i j)
(sourceCollarInverse i j (sourceCollarPiece i j x)) := by
rw [sourceCollarInverse_point i j hx]
exact contDiffAt_attachedVariableTensor hk ha i j hx
exact (ht.comp _ (sourceCollarInverse_contDiffAt i j hx)).congr_of_eventuallyEq
(variableCollarTensor_local_eq K a b i j hx)
theorem variableCollar_regular_at {u : Coord3 → Fin 2 → ℝ} {K : Coord3 → Mat3}
(hS : ∀ x,(K x).IsSymm) {a : ℝ} (ha : a≠0) (b : ℝ)
(i j : Fin 4) {x : Coord3} (hx : x∈sourceCollarOpenBox) {U : Set Coord3}
(hr : flatEndCoordinates a b (sourceFaceAngles i j x)∈
regularRegion u (fun y => ⟨K y,hS y⟩) U) :
sourceCollarPiece i j x∈regularRegion (variableCollarField u a b)
(fun y => ⟨variableCollarTensor K a b y,variableCollarTensor_symm hS a b y⟩)
(sourceCollarPiece i j '' sourceCollarOpenBox) := by
obtain ⟨O,hOU,hO,hxO⟩ := mem_regularRegion_iff.mp hr
let V := sourceCollarPiece i j '' sourceCollarOpenBox
let g : Coord3 → Coord3 := fun y => flatEndCoordinates a b (sourceFaceAngles i j (sourceCollarInverse i j y))
have hV : IsOpen V := isOpen_sourceCollarOpenImage i j
have hg : ContDiffOn ℝ (↑(⊤:ℕ∞)) g V :=
(contDiff_flatEndCoordinates a b).comp_contDiffOn
((contDiff_sourceFaceAngles i j).comp_contDiffOn (sourceCollarInverse_contDiffOn i j))
let W := V∩g ⁻¹' O
have hW : IsOpen W := hg.continuousOn.isOpen_inter_preimage hV hO.1
have hm : MapsTo g W O := fun _ hy => hy.2
have hd : ∀ y∈W,Function.Surjective (fderiv ℝ g y) := by
rintro y ⟨⟨z,hz,rfl⟩,_⟩
have hinv := (sourceCollarInverse_contDiffAt i j hz).differentiableAt (by simp)
have hang := (contDiff_sourceFaceAngles i j).differentiable (by simp)
have hflat := (contDiff_flatEndCoordinates a b).differentiable (by simp)
change Function.Surjective (fderiv ℝ (flatEndCoordinates a b ∘ (sourceFaceAngles i j ∘ sourceCollarInverse i j)) _)
rw [fderiv_comp _ (hflat _) ((hang _).comp _ hinv),fderiv_comp _ (hang _) hinv]
have hsur : Function.Surjective (fderiv ℝ (sourceFaceAngles i j)
(sourceCollarInverse i j (sourceCollarPiece i j z))) := by
simpa only [sourceCollarInverse_point i j hz] using sourceFaceAngles_fderiv_surjective i j hz
exact (flatEndCoordinates_fderiv_surjective ha b _).comp
(hsur.comp (sourceCollarInverse_fderiv_surjective i j ⟨z,hz,rfl⟩))
have hB : ContDiffOn ℝ (↑(⊤:ℕ∞)) (variableCollarTensor K a b) W := by
rintro y hy
obtain ⟨z,hz,rfl⟩ := hy.1
apply (variableCollarTensor_smooth_at ha b i j hz ?_).contDiffWithinAt
have hgz : g (sourceCollarPiece i j z)=flatEndCoordinates a b (sourceFaceAngles i j z) := by
simp only [g,sourceCollarInverse_point i j hz]
simpa only [hgz] using (hO.2.2.1 _ hy.2).contDiffAt (hO.1.mem_nhds hy.2)
have hp : RegularPatch (fun y => u (g y))
(fun y => ⟨variableCollarTensor K a b y,variableCollarTensor_symm hS a b y⟩) W :=
⟨hW,hO.2.1.comp (hg.mono inter_subset_left) hm,hB,
hO.2.2.2.smoothCoordinates hO.1 hW hO.2.1 (hg.mono inter_subset_left) hm hd⟩
have hc : EqOn (variableCollarField u a b) (fun y => u (g y)) W := by
rintro y ⟨⟨z,hz,rfl⟩,_⟩
simpa only [g,sourceCollarInverse_point i j hz] using variableCollarField_open u a b i j hz
refine mem_regularRegion_iff.mpr ⟨W,inter_subset_left,hp.congr hc (fun _ _ => rfl),?_⟩
refine ⟨⟨x,hx,rfl⟩,?_⟩
change g (sourceCollarPiece i j x)∈O
simpa only [g,sourceCollarInverse_point i j hx] using hxO
end ScalarConductivity
end
end
end OAI
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