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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 | import Mathlib
import OAI.Geometry.IntegralFillings.Charts.Extensions
import OAI.Geometry.IntegralFillings.Charts.CurrentAxioms
namespace OAI
section
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 Set MeasureTheory Filter
open Filter Set
open scoped Topology NNReal
open Set Filter MeasureTheory TopologicalSpace
open scoped Topology ENNReal
open Set MeasureTheory
open scoped RealInnerProductSpace
open Matrix
open scoped RealInnerProductSpace MatrixOrder
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
open MeasureTheory Filter Set Metric
open scoped Topology Pointwise NNReal
namespace SharpIntegralFillings
open Set MeasureTheory Filter
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
(μ : Measure E) [Measure.IsAddHaarMeasure μ]
lemma LipschitzWith.ae_fderiv_zero_on_level {F : E → ℝ} {K : ℝ≥0}
(hF : LipschitzWith K F) (c : ℝ) :
∀ᵐ x ∂μ, F x = c → fderiv ℝ F x = 0 := by
have hset : MeasurableSet {x | F x = c} := isClosed_eq hF.continuous continuous_const |>.measurableSet
apply (ae_restrict_iff' hset).mp
have heq : EqOn F (fun _ => c) {x | F x = c} := fun _ hx => hx
filter_upwards [ae_uniqueDiffWithinAt μ {x | F x = c},ae_restrict_mem hset,
(hF.ae_differentiableAt (μ := μ)).filter_mono (ae_mono Measure.restrict_le_self)]
with x hx hxc hd
rw [← fderivWithin_eq_fderiv hx hd, fderivWithin_congr' heq hxc]
simpa only [Function.const_def,Pi.zero_apply] using
congrFun (fderivWithin_const (𝕜 := ℝ) (E := E) (s := {x | F x = c}) c) x
lemma ae_fderivWithin_zero_on_level {s : Set E} (hs : MeasurableSet s)
{f : E → ℝ} {K : ℝ≥0} (hf : LipschitzOnWith K f s) (c : ℝ) :
∀ᵐ x ∂μ.restrict s, f x = c → fderivWithin ℝ f s x = 0 := by
obtain ⟨F,hF,heq⟩ := hf.extend_real
filter_upwards [ae_fderivWithin_eq_fderiv_extension μ hs hF heq,
(LipschitzWith.ae_fderiv_zero_on_level μ hF c).filter_mono
(ae_mono Measure.restrict_le_self),ae_restrict_mem hs] with x hx hz hxs
intro hfc
rw [hx]
exact hz ((heq hxs).symm.trans hfc)
end SharpIntegralFillings
namespace SharpIntegralFillings.IntegerChart
variable {X : Type*} [MetricSpace X] [Nonempty X] {k : ℕ}
(C : IntegerChart X (k+1))
lemma boundary_action_locality (b : X → ℝ) (π : Fin k → X → ℝ) (h : Admissible b π)
(hloc : ∃ (i : Fin k) (U : Set X) (c : ℝ), IsOpen U ∧ Function.support b ⊆ U ∧
∀ x ∈ U, π i x = c) : C.action (fun _ => 1) (Matrix.vecCons b π) = 0 := by
obtain ⟨i,U,c,hU,hb,hπ⟩ := hloc
obtain ⟨L,J,hφ,_⟩ := C.bilipschitz
obtain ⟨K,hK⟩ := h.1.1
have hadm : Admissible (fun _ : X => (1:ℝ)) (Matrix.vecCons b π) :=
⟨BoundedLip.const 1,fun j => Fin.cases h.1.1 (fun l => h.2 l) j⟩
rw [action,ite_eq_left hadm]
apply integral_eq_zero_of_ae
filter_upwards [ae_restrict_mem C.borel,
ae_fderivWithin_zero_on_level volume C.borel (C.scalar_lipschitzOn hφ hK) 0]
with z hz hzero
have hdet : C.jacobian (Matrix.vecCons b π) z = 0 := by
by_cases hbz : C.scalar b z = 0
· apply Matrix.det_eq_zero_of_row_eq_zero 0
intro j
change fderivWithin ℝ (C.scalar b) C.domain z (EuclideanSpace.single j 1) = 0
rw [hzero hbz]; rfl
· have hext : C.paramExtended z = C.param ⟨z,hz⟩ := by simp [paramExtended,hz]
have hzU : C.paramExtended z ∈ U := by
rw [hext]
apply hb
simpa [Function.mem_support,C.scalar_eq hz] using hbz
have hh := (C.continuousOn_paramExtended z hz).preimage_mem_nhdsWithin (hU.mem_nhds hzU)
have heq : C.scalar (π i) =ᶠ[𝓝[C.domain] z] fun _ => c := by
filter_upwards [hh,eventually_mem_nhdsWithin] with y hy hys
rw [C.scalar_eq hys]
have hye : C.paramExtended y = C.param ⟨y,hys⟩ := by simp [paramExtended,hys]
exact hπ _ (hye ▸ hy)
have heqz : C.scalar (π i) z = c := by
rw [C.scalar_eq hz]
exact hπ _ (hext ▸ hzU)
have hd0 : fderivWithin ℝ (C.scalar (π i)) C.domain z = 0 :=
(heq.fderivWithin_eq (𝕜 := ℝ) heqz).trans (by
simpa only [Function.const_def,Pi.zero_apply] using
congrFun (fderivWithin_const (𝕜 := ℝ) (E := Euc (k+1)) (s := C.domain) c) z)
apply Matrix.det_eq_zero_of_row_eq_zero i.succ
intro j
change fderivWithin ℝ (C.scalar (π i)) C.domain z (EuclideanSpace.single j 1) = 0
rw [hd0]; rfl
rw [hdet,mul_zero]; rfl
end SharpIntegralFillings.IntegerChart
namespace SharpIntegralFillings
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] {k : ℕ}
omit [BorelSpace X] in
lemma IntegerRectifiable.boundary_locality {T : Functional X (k+1)}
(hT : IntegerRectifiable T) (b : X → ℝ) (π : Fin k → X → ℝ) (hab : Admissible b π)
(hloc : ∃ (i : Fin k) (U : Set X) (c : ℝ), IsOpen U ∧ Function.support b ⊆ U ∧
∀ x ∈ U, π i x = c) : boundarySucc T b π = 0 := by
classical
rw [boundarySucc,ite_eq_left hab]
obtain ⟨C,hd,hC,hs,heq⟩ := hT
rw [heq]
suffices hall : ∀ i, (C i).action (fun _ => 1) (Matrix.vecCons b π) = 0 by
simp only [hall,tsum_zero]
intro i
rcases isEmpty_or_nonempty X with hX | hX
· have hb0 : (fun _ : X => (1:ℝ)) = fun _ => 0 := funext (fun x => isEmptyElim x)
rw [hb0]
simpa using (hC i).linearFirst (fun _ => 0) (fun _ => 0) (Matrix.vecCons b π) 0 0
(BoundedLip.const 0) (BoundedLip.const 0) (fun j => Fin.cases hab.1.1 (fun l => hab.2 l) j)
· exact (C i).boundary_action_locality b π hab hloc
end SharpIntegralFillings
end
end OAI
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