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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | import Mathlib
import OAI.Analysis.Conductivity.Sobolev.LocalPiolaFlux
namespace OAI
noncomputable section
open MeasureTheory
open scoped ENNReal
open Matrix Filter Topology
open Set MeasureTheory Filter Topology
open scoped BigOperators
open Set MeasureTheory Filter Topology
open scoped Manifold
namespace ScalarConductivity
lemma contDiff_clm_det {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E] (n : WithTop ℕ∞) :
ContDiff ℝ n (fun L : E →L[ℝ] E => L.det) := by
classical
let b := Module.finBasis ℝ E
simp_rw [ContinuousLinearMap.det, ← LinearMap.det_toMatrix b, Matrix.det_apply',
LinearMap.toMatrix_apply]
apply ContDiff.sum
intro σ _
apply contDiff_const.mul
apply contDiff_prod
intro i _
change ContDiff ℝ n (fun L : E →L[ℝ] E => (b.coord (σ i)).toContinuousLinearMap (L (b i)))
exact (b.coord (σ i)).toContinuousLinearMap.contDiff.comp ((ContinuousLinearMap.apply ℝ E (b i)).contDiff)
lemma contDiff_of_contDiffOn_tsupport {E V : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup V] [NormedSpace ℝ V] {n : WithTop ℕ∞} {f : E → V} {s : Set E}
(hs : IsOpen s) (hf : ContDiffOn ℝ n f s) (hfS : tsupport f ⊆ s) : ContDiff ℝ n f := by
rw [contDiff_iff_contDiffAt]
intro x
by_cases hx : x ∈ tsupport f
· exact hf.contDiffAt (hs.mem_nhds (hfS hx))
· exact contDiffAt_const.congr_of_eventuallyEq (notMem_tsupport_iff_eventuallyEq.mp hx)
lemma localPiolaFlux_smooth {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E]
(X : OpenPartialHomeomorph E E) (hX : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X X.source)
(hXi : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X.symm X.target)
(F : E → E) (hF : ContDiff ℝ (↑(⊤ : ℕ∞)) F) (hc : HasCompactSupport F)
(hs : tsupport F ⊆ X.source) : ContDiff ℝ (↑(⊤ : ℕ∞)) (localPiolaFlux X F) := by
apply contDiff_of_contDiffOn_tsupport X.open_target _ (localPiolaFlux_tsupport X F hc hs)
apply X.open_target.contDiffOn_iff.mpr
intro y hy
have hxi := hXi.contDiffAt (X.open_target.mem_nhds hy)
have hx := hX.contDiffAt (X.open_source.mem_nhds (X.map_target hy))
have hd : ContDiffAt ℝ (↑(⊤ : ℕ∞)) (fun z => fderiv ℝ X (X.symm z)) y :=
(hx.fderiv_right (by simp)).comp y hxi
have hd0 : (fderiv ℝ X (X.symm y)).det ≠ 0 :=
local_fderiv_det_ne_zero X (hX.differentiableOn (by simp))
(hXi.differentiableOn (by simp)) (X.map_target hy)
have hdet := ((contDiff_clm_det (E := E) (↑(⊤ : ℕ∞))).contDiffAt.comp y hd).abs hd0
have hformula := (hdet.inv (abs_ne_zero.mpr hd0)).smul
(hd.clm_apply (hF.contDiffAt.comp y hxi))
apply hformula.congr_of_eventuallyEq
filter_upwards [X.open_target.mem_nhds hy] with z hz
simp [localPiolaFlux, hz]
lemma exists_smooth_extension_near_compact {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E] {U K : Set E} (hU : IsOpen U) (hK : IsCompact K) (hKU : K ⊆ U)
(φ : E → ℝ) (hφ : ContDiffOn ℝ (↑(⊤ : ℕ∞)) φ U) :
∃ g : E → ℝ, ContDiff ℝ (↑(⊤ : ℕ∞)) g ∧ ∀ x ∈ K, g =ᶠ[𝓝 x] φ := by
obtain ⟨χ, hχ₀, hχ₁, _⟩ := exists_contMDiffMap_zero_one_nhds_of_isClosed
(𝓘(ℝ, E)) hU.isClosed_compl hK.isClosed
(disjoint_left.mpr (fun x hx hxK => hx (hKU hxK))) (n := ⊤)
have hχ : ContDiff ℝ (↑(⊤ : ℕ∞)) (χ : E → ℝ) := contMDiff_iff_contDiff.mp χ.contMDiff
refine ⟨fun x => χ x * φ x, ?_, ?_⟩
· rw [contDiff_iff_contDiffAt]
intro x
by_cases hx : x ∈ U
· exact hχ.contDiffAt.mul (hφ.contDiffAt (hU.mem_nhds hx))
· have hzero : ∀ᶠ z in 𝓝 x, χ z = 0 := hχ₀.filter_mono (nhds_le_nhdsSet hx)
apply (contDiffAt_const : ContDiffAt ℝ (↑(⊤ : ℕ∞)) (fun _ : E => (0 : ℝ)) x).congr_of_eventuallyEq
filter_upwards [hzero] with z hz
simp [hz]
· intro x hx
have hone : ∀ᶠ z in 𝓝 x, χ z = 1 := hχ₁.filter_mono (nhds_le_nhdsSet hx)
filter_upwards [hone] with z hz
simp [hz]
lemma compactFlux_local_test
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E]
(μ : Measure E) {U : Set E} (hU : IsOpen U)
(F : E → E) (hc : HasCompactSupport F) (hs : tsupport F ⊆ U)
(hF : ∀ ψ : E → ℝ, ContDiff ℝ (↑(⊤ : ℕ∞)) ψ →
(∫ x, fderiv ℝ ψ x (F x) ∂μ) = 0)
(φ : E → ℝ) (hφ : ContDiffOn ℝ (↑(⊤ : ℕ∞)) φ U) :
(∫ x, fderiv ℝ φ x (F x) ∂μ) = 0 := by
obtain ⟨g, hg, hge⟩ := exists_smooth_extension_near_compact hU hc hs φ hφ
rw [← hF g hg]
apply integral_congr_ae
filter_upwards [] with x
by_cases hx : x ∈ tsupport F
· rw [(hge x hx).fderiv_eq]
· rw [image_eq_zero_of_notMem_tsupport hx, map_zero, map_zero]
lemma localPiolaFlux_zero_pairing
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
(μ : Measure E) [μ.IsAddHaarMeasure]
(X : OpenPartialHomeomorph E E) (hX : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X X.source)
(hXi : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X.symm X.target)
(F : E → E) (hc : HasCompactSupport F) (hs : tsupport F ⊆ X.source)
(hF : ∀ ψ : E → ℝ, ContDiff ℝ (↑(⊤ : ℕ∞)) ψ →
(∫ x, fderiv ℝ ψ x (F x) ∂μ) = 0)
(ψ : E → ℝ) (hψ : ContDiff ℝ (↑(⊤ : ℕ∞)) ψ) :
(∫ y, fderiv ℝ ψ y (localPiolaFlux X F y) ∂μ) = 0 := by
rw [localPiolaFlux_pairing μ X (hX.differentiableOn (by simp))
(fun _ hx => local_fderiv_det_ne_zero X (hX.differentiableOn (by simp))
(hXi.differentiableOn (by simp)) hx)
F (subset_closure.trans hs) ψ (hψ.differentiable (by simp))]
exact compactFlux_local_test μ X.open_source F hc hs hF _ (hψ.comp_contDiffOn hX)
end ScalarConductivity
end
end OAI
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