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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 | import Mathlib
import OAI.Analysis.Conductivity.Variational.ContDiffClmDet
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
open Set Filter
open scoped Topology
open Set Filter MeasureTheory
open scoped Topology Manifold ENNReal
open Set
namespace ScalarConductivity
open Set Filter Topology MeasureTheory
def inverseChartPiolaOperator {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(X : OpenPartialHomeomorph E E) (y : E) : E →L[ℝ] E :=
|(fderiv ℝ X.symm (X y)).det|⁻¹ • fderiv ℝ X.symm (X y)
lemma localPiolaFlux_inverse_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(X : OpenPartialHomeomorph E E) (F : E → E) {y : E} (hy : y ∈ X.source) :
localPiolaFlux X.symm F y = inverseChartPiolaOperator X y (F (X y)) := by
simp only [localPiolaFlux, X.symm_symm, show y ∈ X.symm.target from hy, ite_eq_left,
inverseChartPiolaOperator, _root_.smul_apply]
lemma inverseChartPiolaOperator_continuousAt
{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)
{a : E} (ha : a ∈ X.source) : ContinuousAt (inverseChartPiolaOperator X) a := by
have hca : ContDiffAt ℝ (↑(⊤ : ℕ∞)) (fderiv ℝ X.symm) (X a) :=
((hXi _ (X.map_source ha)).contDiffAt
(X.open_target.mem_nhds (X.map_source ha))).fderiv_right (by simp)
have hda := ((hX a ha).contDiffAt (X.open_source.mem_nhds ha)).continuousAt
have hd : ContinuousAt (fun y => fderiv ℝ X.symm (X y)) a := hca.continuousAt.comp hda
have hdet0 := local_fderiv_det_ne_zero X.symm (hXi.differentiableOn (by simp))
(hX.differentiableOn (by simp)) (X.map_source ha)
have hc : ContinuousAt (fun y => (fderiv ℝ X.symm (X y)).det) a :=
((contDiff_clm_det (E := E) (↑(⊤ : ℕ∞))).continuous.continuousAt).comp hd
exact (hc.abs.inv₀ (abs_ne_zero.mpr hdet0)).smul hd
lemma inverseChartPiolaOperator_eq_id
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(X : OpenPartialHomeomorph E E)
(hX : DifferentiableOn ℝ X X.source) (hXi : DifferentiableOn ℝ X.symm X.target)
{a : E} (ha : a ∈ X.source) (hDa : fderiv ℝ X a = ContinuousLinearMap.id ℝ E) :
inverseChartPiolaOperator X a = ContinuousLinearMap.id ℝ E := by
have hdi := local_fderiv_symm_comp X hX hXi ha
rw [hDa, ContinuousLinearMap.comp_id] at hdi
simp only [inverseChartPiolaOperator, hdi, ContinuousLinearMap.det,
ContinuousLinearMap.coe_id, LinearMap.det_id, abs_one, inv_one, one_smul]
theorem inverseChartPiola_freezing
{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)
{a : E} (ha : a ∈ X.source) (hDa : fderiv ℝ X a = ContinuousLinearMap.id ℝ E)
{ε : ℝ} (hε : 0 < ε) :
∃ V : Set E, IsOpen V ∧ a ∈ V ∧ V ⊆ X.source ∧
∀ (χ h : E → ℝ) (R : E) (y : E), y ∈ V →
‖localPiolaFlux X.symm (fun z => (χ (X.symm z) * h z) • R) y -
(χ y * h (X y)) • R‖ ≤ ε * (|χ y * h (X y)| * ‖R‖) := by
have heq := inverseChartPiolaOperator_eq_id X (hX.differentiableOn (by simp))
(hXi.differentiableOn (by simp)) ha hDa
have hc : ContinuousAt (fun y => inverseChartPiolaOperator X y -
ContinuousLinearMap.id ℝ E) a :=
(inverseChartPiolaOperator_continuousAt X hX hXi ha).sub continuousAt_const
have hevent : ∀ᶠ y in 𝓝 a,
‖inverseChartPiolaOperator X y - ContinuousLinearMap.id ℝ E‖ < ε := by
have ht := hc.norm.eventually (gt_mem_nhds (show
‖inverseChartPiolaOperator X a - ContinuousLinearMap.id ℝ E‖ < ε by
rw [heq, sub_self, norm_zero]; exact hε))
exact ht
obtain ⟨V, hVsub, hV, haV⟩ := mem_nhds_iff.mp (hevent.and (X.open_source.mem_nhds ha))
refine ⟨V, hV, haV, fun y hy => (hVsub hy).2, ?_⟩
intro χ h R y hy
rw [localPiolaFlux_inverse_eq X _ (hVsub hy).2, X.left_inv (hVsub hy).2]
rw [map_smul]
have hid : (χ y * h (X y)) • (inverseChartPiolaOperator X y R) -
(χ y * h (X y)) • R =
(χ y * h (X y)) • ((inverseChartPiolaOperator X y - ContinuousLinearMap.id ℝ E) R) := by
simp only [_root_.sub_apply, ContinuousLinearMap.id_apply, smul_sub]
rw [hid, norm_smul, Real.norm_eq_abs]
calc
|χ y * h (X y)| * ‖(inverseChartPiolaOperator X y - ContinuousLinearMap.id ℝ E) R‖ ≤
|χ y * h (X y)| * (ε * ‖R‖) :=
mul_le_mul_of_nonneg_left ((ContinuousLinearMap.le_opNorm _ _).trans
(mul_le_mul_of_nonneg_right (hVsub hy).1.le (norm_nonneg _))) (abs_nonneg _)
_ = ε * (|χ y * h (X y)| * ‖R‖) := by ring
end ScalarConductivity
end
end OAI
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