File size: 7,364 Bytes
59d1c80 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 | import Mathlib
import OAI.Geometry.IntegralFillings.Currents.TestBounds
namespace OAI
section
open Filter Set
open Set Filter MeasureTheory TopologicalSpace
open scoped Topology ENNReal
open Set MeasureTheory
open scoped RealInnerProductSpace
open Matrix
open scoped RealInnerProductSpace MatrixOrder
open Set MeasureTheory Measure Filter Module
open Set Filter MeasureTheory Measure ContinuousLinearMap
open scoped Topology Convolution NNReal
open Set Filter MeasureTheory Measure Metric
open scoped Topology ContDiff
open Set Filter Metric
open scoped Topology NNReal
open Set MeasureTheory Filter
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
open MeasureTheory Filter Set Metric
open scoped Topology Pointwise NNReal
namespace SharpIntegralFillings
open MeasureTheory Set Filter
open scoped Topology NNReal
namespace Foundations
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [CompactSpace X]
omit [MeasurableSpace X] [BorelSpace X] in
lemma eval_tendsto_of_uniform_modulus {d : ℕ}
(F : (Fin d → X → ℝ) → ℝ) (π : Fin d → X → ℝ)
(πs : ℕ → Fin d → X → ℝ)
(hLip : ∀ i, ∃ K : ℝ≥0, ∀ j, LipschitzWith K (πs j i))
(hpt : ∀ i x, Tendsto (fun j => πs j i x) atTop (𝓝 (π i x)))
(hmod : ∀ K : ℝ≥0, ∃ C : ℝ≥0, ∀ (ρ σ : Fin d → X → ℝ),
(∀ i, LipschitzWith K (ρ i)) → (∀ i, LipschitzWith K (σ i)) →
∀ ε : ℝ≥0, (∀ i x, |ρ i x - σ i x| ≤ ε) → |F ρ - F σ| ≤ C * ε) :
Tendsto (fun j => F (πs j)) atTop (𝓝 (F π)) := by
choose Ks hKs using hLip
let K : ℝ≥0 := ∑ i, Ks i
have hK (i : Fin d) : Ks i ≤ K :=
Finset.single_le_sum (fun _ _ => bot_le) (Finset.mem_univ i)
have hseq (j : ℕ) (i : Fin d) : LipschitzWith K (πs j i) := (hKs i j).weaken (hK i)
have hlimit (i : Fin d) : LipschitzWith K (π i) :=
lipschitz_of_pointwise_limit (fun j => hseq j i) (hpt i)
obtain ⟨C, hC⟩ := hmod K
apply Metric.tendsto_nhds.mpr
intro ε hε
let δ : ℝ := ε / ((C : ℝ) + 1)
have hδ : 0 < δ := div_pos hε (by positivity)
have hev (i : Fin d) : ∀ᶠ j in atTop, ∀ x, |πs j i x - π i x| ≤ δ := by
have hu := lipschitz_pointwise_uniform (fun j => hseq j i) (hpt i)
filter_upwards [Metric.tendstoUniformly_iff.mp hu δ hδ] with j hj x
exact le_of_lt (by simpa [Real.dist_eq, abs_sub_comm] using hj x)
filter_upwards [Filter.eventually_all.mpr hev] with j hj
rw [Real.dist_eq]
have h := hC (πs j) π (hseq j) hlimit ⟨δ, hδ.le⟩ hj
refine h.trans_lt ?_
change (C : ℝ) * δ < ε
calc (C : ℝ) * δ < ((C : ℝ) + 1) * δ := by nlinarith
_ = ε := by dsimp [δ]; exact mul_div_cancel₀ ε (by positivity : (C : ℝ) + 1 ≠ 0)
lemma metricCurrent_of_bounded_weak_limit
{ι : Type*} {l : Filter ι} [NeBot l] {k : ℕ}
{Ts : ι → Functional X (k + 1)} {T : Functional X (k + 1)}
(hTs : ∀ j, IntegerRectifiable (Ts j))
(hcur : ∀ j, IsMetricCurrent (Ts j))
(hbd : ∀ j, IsMetricCurrent (boundarySucc (Ts j)))
(M N : ℝ≥0) (hM : ∀ j, mass (Ts j) ≤ M)
(hN : ∀ j, mass (boundarySucc (Ts j)) ≤ N)
(hlim : ∀ b π, Tendsto (fun j => Ts j b π) l (𝓝 (T b π))) :
IsMetricCurrent T := by
obtain ⟨μ, hμ, hctrl, _⟩ := exists_controls_of_bounded_weak_limit M hcur hM
(fun b π _ => hlim b π)
refine {
offDomain := ?_
linearFirst := ?_
linearCoord := ?_
sequentialContinuity := ?_
locality := ?_
finiteMass := ⟨μ, hμ, hctrl⟩ }
· intro b π h
apply tendsto_nhds_unique (hlim b π)
have heq : (fun j => Ts j b π) = fun _ => 0 :=
funext fun j => (hcur j).offDomain b π h
rw [heq]
exact tendsto_const_nhds
· intro b₁ b₂ π a c hb₁ hb₂ hπ
apply tendsto_nhds_unique (hlim (fun x => a * b₁ x + c * b₂ x) π)
convert ((hlim b₁ π).const_mul a).add ((hlim b₂ π).const_mul c) using 1
funext j
exact (hcur j).linearFirst b₁ b₂ π a c hb₁ hb₂ hπ
· intro b π i f a c hadm hf
apply tendsto_nhds_unique (hlim b (Function.update π i (fun x => a * π i x + c * f x)))
convert ((hlim b π).const_mul a).add
((hlim b (Function.update π i f)).const_mul c) using 1
funext j
exact (hcur j).linearCoord b π i f a c hadm hf
· intro b π πs hb hLip hpt
obtain ⟨Kb, hKb⟩ := hb.1
obtain ⟨B₀, hB₀⟩ := hb.2
let B : ℝ≥0 := ⟨max B₀ 0, le_max_right _ _⟩
have hB (x : X) : |b x| ≤ B := (hB₀ x).trans (le_max_left _ _)
apply eval_tendsto_of_uniform_modulus (T b) π πs hLip hpt
intro K
let C : ℝ≥0 := (k + 1) * (K + K) ^ k * (B * N + Kb * M)
refine ⟨C, ?_⟩
intro ρ σ hρ hσ ε hε
have h := weak_limit_test_difference_bound hTs hcur hbd M N hM hN
(fun b π _ => hlim b π) hKb ρ σ K hρ hσ B ε hB hε
simpa [C] using h
· intro b π hadm hloc
apply tendsto_nhds_unique (hlim b π)
have heq : (fun j => Ts j b π) = fun _ => 0 :=
funext fun j => (hcur j).locality b π hadm hloc
rw [heq]
exact tendsto_const_nhds
omit [BorelSpace X] [CompactSpace X] in
lemma boundarySucc_boundarySucc {k : ℕ} {T : Functional X (k + 1 + 1)}
(hT : IsMetricCurrent T) : boundarySucc (boundarySucc T) = fun _ _ => 0 := by
funext b π
by_cases h : Admissible b π
· rw [boundarySucc, ite_eq_left h]
have hadm : Admissible (fun _ : X => (1 : ℝ)) (Matrix.vecCons b π) :=
admissible_vecCons (BoundedLip.const 1) h.1.1 h.2
rw [boundarySucc, ite_eq_left hadm]
apply hT.eq_zero_of_const_coord (admissible_vecCons (BoundedLip.const 1)
⟨0, LipschitzWith.const 1⟩ hadm.2) 0 1
intro x
rfl
· rw [boundarySucc, ite_eq_right h]
omit [MeasurableSpace X] [BorelSpace X] [CompactSpace X] in
lemma boundarySucc_weak_limit {ι : Type*} {l : Filter ι} {k : ℕ}
{Ts : ι → Functional X (k + 1)} {T : Functional X (k + 1)}
(hlim : ∀ b π, Tendsto (fun j => Ts j b π) l (𝓝 (T b π))) :
∀ b π, Tendsto (fun j => boundarySucc (Ts j) b π) l (𝓝 (boundarySucc T b π)) := by
intro b π
by_cases h : Admissible b π
· simpa only [boundarySucc, ite_eq_left h] using hlim (fun _ => 1) (Matrix.vecCons b π)
· simp only [boundarySucc, ite_eq_right h]
exact tendsto_const_nhds
lemma normal_of_bounded_integral_weak_limit
{ι : Type*} {l : Filter ι} [NeBot l] {k : ℕ}
{Ts : ι → Functional X (k + 1 + 1)} {T : Functional X (k + 1 + 1)}
(hTs : ∀ j, IsIntegral (k + 1 + 1) (Ts j))
(M N : ℝ≥0) (hM : ∀ j, mass (Ts j) ≤ M)
(hN : ∀ j, mass (boundarySucc (Ts j)) ≤ N)
(hlim : ∀ b π, Tendsto (fun j => Ts j b π) l (𝓝 (T b π))) :
IsMetricCurrent T ∧ IsMetricCurrent (boundarySucc T) := by
have hcur (j : ι) := (hTs j).1
have hint (j : ι) := (hTs j).2.1
have hbd (j : ι) := (hTs j).2.2.1
have hbdi (j : ι) := (hTs j).2.2.2
refine ⟨metricCurrent_of_bounded_weak_limit hint hcur hbd M N hM hN hlim, ?_⟩
apply metricCurrent_of_bounded_weak_limit hbdi hbd (fun j => ?_) N 0 hN
(fun j => ?_) (boundarySucc_weak_limit hlim)
· rw [boundarySucc_boundarySucc (hcur j)]
exact isMetricCurrent_zero k
· rw [boundarySucc_boundarySucc (hcur j), mass_zero]
rfl
end Foundations
end SharpIntegralFillings
end
end OAI
|