File size: 4,221 Bytes
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 | import Mathlib
import OAI.Analysis.Conductivity.Model
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
namespace ScalarConductivity
def completedPotential {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
(u : E → F) (D : E →L[ℝ] F) (Q : F →L[ℝ] E) (y : E) : E :=
Q (u y) + y - Q (D y)
lemma completedPotential_pairing {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
(u : E → F) (D : E →L[ℝ] F) (Q : F →L[ℝ] E)
(hQ : Function.RightInverse Q D) (y : E) :
D (completedPotential u D Q y) = u y := by
simp only [completedPotential, map_sub, map_add, hQ (u y), hQ (D y)]
abel
lemma completedPotential_smooth {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
{u : E → F} (hu : ContDiff ℝ (↑(⊤ : ℕ∞)) u)
(D : E →L[ℝ] F) (Q : F →L[ℝ] E) :
ContDiff ℝ (↑(⊤ : ℕ∞)) (completedPotential u D Q) :=
((Q.contDiff.comp hu).add contDiff_id).sub (Q.contDiff.comp D.contDiff)
lemma completedPotential_derivative {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
{u : E → F} {D : E →L[ℝ] F} {a : E} (hu : HasFDerivAt u D a)
(Q : F →L[ℝ] E) :
HasFDerivAt (completedPotential u D Q) (ContinuousLinearMap.id ℝ E) a := by
convert ((Q.hasFDerivAt.comp a hu).add (hasFDerivAt_id a)).sub
(Q.hasFDerivAt.comp a D.hasFDerivAt) using 1
· rfl
· abel
theorem exists_potential_chart
{E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
[NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F]
{u : E → F} (hu : ContDiff ℝ (↑(⊤ : ℕ∞)) u) (a : E)
(hD : Function.Surjective (fderiv ℝ u a)) {U : Set E} (hU : IsOpen U) (ha : a ∈ U) :
∃ X : OpenPartialHomeomorph E E,
a ∈ X.source ∧ X.source ⊆ U ∧
ContDiff ℝ (↑(⊤ : ℕ∞)) X ∧
ContDiffOn ℝ (↑(⊤ : ℕ∞)) X.symm X.target ∧
(∀ y, (fderiv ℝ u a) (X y) = u y) ∧
fderiv ℝ X a = ContinuousLinearMap.id ℝ E := by
let D := fderiv ℝ u a
obtain ⟨Q, hQ⟩ := ContinuousLinearMap.HasRightInverse.of_surjective_of_finiteDimensional hD
let f := completedPotential u D Q
have hf := completedPotential_smooth hu D Q
have hfa : HasFDerivAt f (ContinuousLinearEquiv.refl ℝ E : E →L[ℝ] E) a :=
completedPotential_derivative ((hu.differentiable (by simp) a).hasFDerivAt) Q
let X₀ := hf.contDiffAt.toOpenPartialHomeomorph f hfa (by simp)
let V := U ∩ {y | IsUnit (fderiv ℝ f y)}
have hV : IsOpen V := hU.inter (Units.isOpen.preimage (hf.continuous_fderiv (by simp)))
have haV : a ∈ V := ⟨ha, by change IsUnit (fderiv ℝ f a); rw [hfa.fderiv]; exact isUnit_one⟩
let X := X₀.restr V
have hXeq : (X : E → E) = f := rfl
have hsource : X.source = X₀.source ∩ V := by
simp only [X, OpenPartialHomeomorph.restr_source, hV.interior_eq]
have hinv : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X.symm X.target := by
intro y hy
have hx : X.symm y ∈ V := (hsource ▸ X.map_target hy).2
obtain ⟨v, hv⟩ := hx.2
let e : E ≃L[ℝ] E := ContinuousLinearEquiv.ofUnit v
have he : (e : E →L[ℝ] E) = fderiv ℝ X (X.symm y) := by
change (v : E →L[ℝ] E) = _
simpa only [hXeq] using hv
have hd : HasFDerivAt X (e : E →L[ℝ] E) (X.symm y) := by
rw [he, hXeq]
exact (hf.differentiable (by simp) _).hasFDerivAt
exact (X.contDiffAt_symm hy hd (hXeq ▸ hf.contDiffAt)).contDiffWithinAt
refine ⟨X, ?_, ?_, ?_, hinv, ?_, ?_⟩
· rw [hsource]
exact ⟨hf.contDiffAt.mem_toOpenPartialHomeomorph_source hfa (by simp), haV⟩
· rw [hsource]
exact fun _ hx => hx.2.1
· rw [hXeq]
exact hf
· intro y
exact completedPotential_pairing u D Q hQ y
· exact hfa.fderiv
end ScalarConductivity
end
end OAI
|