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8e4d533 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 | import Mathlib
import OAI.Geometry.CAT0Fillings.Gradient.ClosedLinear
import OAI.Geometry.CAT0Fillings.Gradient.Multiplier
import OAI.Geometry.CAT0Fillings.Calculus.ClosedChain
namespace OAI
section
open Set Filter
open scoped Topology NNReal
namespace CAT0Fillings.ClosedCalculus
noncomputable def levelApprox (a ε t : ℝ) := ε*Real.arctan ((t-a)/ε)
noncomputable def levelApproxDeriv (a ε t : ℝ) := ε^2/(ε^2+(t-a)^2)
lemma levelApprox_hasDerivAt {ε : ℝ} (hε : 0 < ε) (a t : ℝ) :
HasDerivAt (levelApprox a ε) (levelApproxDeriv a ε t) t := by
have hd := ((((hasDerivAt_id t).sub_const a).div_const ε).arctan).const_mul ε
convert hd using 1 <;> try rfl
dsimp [levelApproxDeriv]
field_simp [hε.ne']
lemma levelApproxDeriv_bound {ε : ℝ} (hε : 0 < ε) (a t : ℝ) : |levelApproxDeriv a ε t| ≤ 1 := by
have hp := add_pos_of_pos_of_nonneg (sq_pos_of_pos hε) (sq_nonneg (t-a))
dsimp [levelApproxDeriv]
rw [abs_of_nonneg (div_nonneg (sq_nonneg _) hp.le)]
exact (div_le_one hp).mpr (le_add_of_nonneg_right (sq_nonneg _))
lemma levelApprox_lipschitz {ε : ℝ} (hε : 0 < ε) (a : ℝ) :
LipschitzWith 1 (levelApprox a ε) := by
apply lipschitzWith_of_nnnorm_deriv_le (fun t => (levelApprox_hasDerivAt hε a t).differentiableAt)
intro t
rw [←NNReal.coe_le_coe,coe_nnnorm,NNReal.coe_one,(levelApprox_hasDerivAt hε a t).deriv,Real.norm_eq_abs]
exact levelApproxDeriv_bound hε a t
lemma continuous_levelApproxDeriv {ε : ℝ} (hε : 0 < ε) (a : ℝ) :
Continuous (levelApproxDeriv a ε) := by
change Continuous (fun t : ℝ => ε^2/(ε^2+(t-a)^2))
exact continuous_const.div (continuous_const.add ((continuous_id.sub continuous_const).pow 2))
(fun t => (add_pos_of_pos_of_nonneg (sq_pos_of_pos hε) (sq_nonneg (t-a))).ne')
lemma levelApprox_error {ε : ℝ} (hε : 0 ≤ ε) (a t : ℝ) :
|levelApprox a ε t| ≤ ε*(Real.pi/2) := by
dsimp [levelApprox]
rw [abs_mul,abs_of_nonneg hε]
exact mul_le_mul_of_nonneg_left (abs_le.mpr
⟨(Real.neg_pi_div_two_lt_arctan _).le,(Real.arctan_lt_pi_div_two _).le⟩) hε
lemma levelApproxDeriv_tendsto {e : ℕ → ℝ} (he : Tendsto e atTop (𝓝 0))
(hep : ∀ j, 0 < e j) (a t : ℝ) :
Tendsto (fun j => levelApproxDeriv a (e j) t) atTop (𝓝 (if t = a then 1 else 0)) := by
by_cases ht : t = a
· subst t
simp only [levelApproxDeriv,sub_self,zero_pow (by decide : 2 ≠ 0),add_zero,
div_self (pow_ne_zero 2 (hep _).ne'),ite_true]
exact tendsto_const_nhds
· rw [ite_eq_right ht]
have hta : t-a ≠ 0 := sub_ne_zero.mpr ht
have hh := (he.pow 2).div ((he.pow 2).add_const ((t-a)^2))
(by simpa using pow_ne_zero 2 hta)
convert hh using 1 <;> simp only [levelApproxDeriv,zero_pow (by decide : 2 ≠ 0),zero_add,zero_div]
rfl
end CAT0Fillings.ClosedCalculus
end
section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
namespace CAT0Fillings.ChartGeometry
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X (k+1)}
{hT : IsMetricCurrent T} (q : ChartGeometry hT)
lemma closed_chain_uniform {F D : ℝ → ℝ} {J : ℝ≥0} (hF : LipschitzWith J F)
(hD : Measurable D) {C : ℝ} (hC : 0 ≤ C) (hbD : ∀ t, |D t| ≤ C)
(Fj Dj : ℕ → ℝ → ℝ) (K : ℕ → ℝ≥0)
(hFj : ∀ j, LipschitzWith (K j) (Fj j))
(hdj : ∀ j t, HasDerivAt (Fj j) (Dj j t) t)
(hcj : ∀ j, Continuous (Dj j)) (hbj : ∀ j t, |Dj j t| ≤ C)
(hdlim : ∀ t, Tendsto (fun j => Dj j t) atTop (𝓝 (D t)))
(a : ℕ → ℝ) (ha : Tendsto a atTop (𝓝 0))
(he : ∀ j t, |Fj j t-F t| ≤ a j) (P : q.Sobolev) :
∃ Q : q.Sobolev,
(q.inclusion Q : X → ℝ) =ᵐ[MassMeasure.currentMassMeasure hT]
(fun x => F ((q.inclusion P) x)) ∧
(q.closedGradient Q : (ℕ × Euc (k+1)) → Euc (k+1)) =ᵐ[q.atlasMeasure]
(fun w => D ((q.inclusion P) (q.atlasParam w)) • (q.closedGradient P) w) := by
choose Q hQ hG using (fun j => q.closed_chain (hFj j) (hdj j) (hcj j) hC (hbj j) P)
let b (j : ℕ) (w : ℕ × Euc (k+1)) := Dj j ((q.inclusion P) (q.atlasParam w))
let c (w : ℕ × Euc (k+1)) := D ((q.inclusion P) (q.atlasParam w))
have hpm := (Lp.aestronglyMeasurable (q.inclusion P)).comp_measurePreserving q.atlas_preserving
have hb (j : ℕ) : AEStronglyMeasurable (b j) q.atlasMeasure := (hcj j).comp_aestronglyMeasurable hpm
have hc : AEStronglyMeasurable c q.atlasMeasure := (hD.comp_aemeasurable hpm.aemeasurable).aestronglyMeasurable
have hbC (j : ℕ) : ∀ᵐ w ∂q.atlasMeasure, |b j w| ≤ C := Eventually.of_forall fun _ => hbj j _
have hcC : ∀ᵐ w ∂q.atlasMeasure, |c w| ≤ C := Eventually.of_forall fun _ => hbD _
have hgl := ClosedCalculus.mulLp_tendsto hb hc hC hbC hcC
(Eventually.of_forall fun point => hdlim ((q.inclusion P) (q.atlasParam point)))
(tendsto_const_nhds (x := q.closedGradient P))
have heG (j : ℕ) : ClosedCalculus.mulLp (hb j) (hbC j) (q.closedGradient P) = q.closedGradient (Q j) := by
apply Lp.ext
exact (ClosedCalculus.mulLp_ae (hb j) (hbC j) _).trans (hG j).symm
simp only [heG] at hgl
let V := q.composeValue hF (q.inclusion P)
have hv : Tendsto (fun j => q.inclusion (Q j)) atTop (𝓝 V) := by
apply tendsto_iff_norm_sub_tendsto_zero.mpr
apply squeeze_zero (fun _ => norm_nonneg _)
(g := fun j => a j*‖value (hT := hT) (LipschitzWith.const (1:ℝ))‖)
· intro j
apply Lp.norm_le_mul_norm_of_ae_le_mul
filter_upwards [Lp.coeFn_sub (q.inclusion (Q j)) V,hQ j,q.composeValue_ae hF (q.inclusion P),
((Foundations.boundedLip_of_lipschitz (LipschitzWith.const (1:ℝ))).memLp
(μ := MassMeasure.currentMassMeasure hT) 2).coeFn_toLp] with x hs hq hv h1
change (value (hT := hT) (LipschitzWith.const (1:ℝ)) : X → ℝ) x = 1 at h1
rw [hs,Pi.sub_apply,hq,hv,h1,Real.norm_eq_abs,norm_one,mul_one]
exact he j _
· simpa only [zero_mul] using ha.mul_const ‖value (hT := hT) (LipschitzWith.const (1:ℝ))‖
obtain ⟨R,hR,hRG⟩ := q.closedGradient_closed V (ClosedCalculus.mulLp hc hcC (q.closedGradient P)) Q hv hgl
refine ⟨R,?_,?_⟩
· rw [hR]
exact q.composeValue_ae hF (q.inclusion P)
· rw [hRG]
exact ClosedCalculus.mulLp_ae hc hcC _
end CAT0Fillings.ChartGeometry
end
section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
namespace CAT0Fillings.ChartGeometry
open ClosedCalculus
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X (k+1)}
{hT : IsMetricCurrent T} (q : ChartGeometry hT)
lemma closedGradient_level_zero (hz : IsCycle T) (P : q.Sobolev) (a : ℝ) :
∀ᵐ w ∂q.atlasMeasure, (q.inclusion P) (q.atlasParam w) = a →
(q.closedGradient P) w = 0 := by
let e (j : ℕ) : ℝ := 1/((j:ℝ)+1)
have hep (j : ℕ) : 0 < e j := by dsimp [e]; positivity
have he : Tendsto e atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
have hD : Measurable (fun t : ℝ => if t = a then (1:ℝ) else 0) := by
convert (show Measurable (({a} : Set ℝ).indicator (fun _ : ℝ => (1:ℝ))) from
measurable_const.indicator (measurableSet_singleton a)) using 1
funext t
simp only [Set.indicator_apply,Set.mem_singleton_iff]
have hb (t : ℝ) : |if t = a then (1:ℝ) else 0| ≤ 1 := by split_ifs <;> norm_num
obtain ⟨R,hR,hG⟩ := q.closed_chain_uniform (F := fun _ => (0:ℝ))
(D := fun t => if t = a then (1:ℝ) else 0) (LipschitzWith.const (0:ℝ))
hD (by norm_num : (0:ℝ) ≤ 1) hb
(fun j => levelApprox a (e j)) (fun j => levelApproxDeriv a (e j)) (fun _ => 1)
(fun j => levelApprox_lipschitz (hep j) a) (fun j => levelApprox_hasDerivAt (hep j) a)
(fun j => continuous_levelApproxDeriv (hep j) a) (fun j => levelApproxDeriv_bound (hep j) a)
(levelApproxDeriv_tendsto he hep a) (fun j => e j*(Real.pi/2))
(by simpa only [zero_mul] using he.mul_const (Real.pi/2))
(fun j t => by simpa only [sub_zero] using levelApprox_error (hep j).le a t) P
have hR0 : q.inclusion R = 0 := Lp.ext (hR.trans (Lp.coeFn_zero ℝ 2 (MassMeasure.currentMassMeasure hT)).symm)
have hRz : R = 0 := q.inclusion_injective hz (hR0.trans (map_zero q.inclusion).symm)
rw [hRz,map_zero] at hG
filter_upwards [hG,(Lp.coeFn_zero (Euc (k+1)) 2 q.atlasMeasure)] with w hw hz
intro ha
simpa only [hz,ha,ite_true,one_smul,Pi.zero_apply] using hw.symm
end CAT0Fillings.ChartGeometry
end
end OAI
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