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2fca478 | 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 | import OAI.Geometry.Perimeter.Regularity.RawConeLink
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal ContDiff
namespace RPISingularHarnack
namespace SphericalFrankel
lemma minimizing_perimeter_singleton {n : ℕ} (hn : 1 < n) {E : Set (Euc n)}
(hE : IsLocallyPerimeterMinimizing E) (x : Euc n) : perimeterMeasure E {x} = 0 := by
obtain ⟨C, D, hC0, hCt, hD0, hDt, hCD⟩ := minimizing_perimeter_hausdorff_comparison hn
have hd : (0 : ℝ) < (n - 1 : ℕ) := by exact_mod_cast (by omega : 0 < n-1)
let := Measure.nullSingletonClass_hausdorff (Euc n) hd
apply le_antisymm _ zero_le
calc
perimeterMeasure E {x} ≤ (C • (Measure.hausdorffMeasure (n-1 : ℕ)).restrict
(perimeterSupport E)) {x} := (hCD E hE).1 {x}
_ = 0 := by simp
lemma minimizing_support_nonzero_point {n : ℕ} (hn : 1 < n) {E : Set (Euc n)}
(hE : IsLocallyPerimeterMinimizing E) (hv : (0 : Euc n) ∈ perimeterSupport E) :
∃ x ∈ perimeterSupport E, x ≠ 0 := by
have hv' : (0 : Euc n) ∈ (perimeterMeasure E).support := by
rwa [perimeterMeasure_support]
have hp := Metric.nhds_basis_ball.mem_measureSupport.mp hv' 1 (by norm_num : (0 : ℝ)<1)
have hdiff : 0 < perimeterMeasure E (Metric.ball 0 1 \ {(0 : Euc n)}) := by
rwa [measure_sdiff_null (minimizing_perimeter_singleton hn hE 0)]
obtain ⟨x, hx, hs⟩ := (perimeterMeasure E).nonempty_inter_support_of_pos hdiff
exact ⟨x, by rwa [perimeterMeasure_support] at hs, hx.2⟩
lemma annularWeight_pos {n : ℕ} {x : Euc n} (h1 : 1 < ‖x‖) (h2 : ‖x‖ < 2) :
0 < annularWeight x := by
unfold annularWeight annularCut annularProfile
apply mul_pos
· apply mul_pos <;> apply expNegInvGlue.pos_of_pos
· nlinarith [norm_nonneg x]
· nlinarith [norm_nonneg x]
· exact inv_pos.mpr (lt_trans (by norm_num) h1)
lemma cone_support_annular_point {n : ℕ} (hn : 1 < n) {E : Set (Euc n)}
(hE : IsLocallyPerimeterMinimizing E) (hc : IsCone E)
(hv : (0 : Euc n) ∈ perimeterSupport E) :
∃ y ∈ perimeterSupport E, ‖y‖ = (3 : ℝ)/2 := by
obtain ⟨x, hx, hx0⟩ := minimizing_support_nonzero_point hn hE hv
have hn0 : 0 < ‖x‖ := norm_pos_iff.mpr hx0
refine ⟨((3:ℝ)/2/‖x‖) • x,
(isCone_perimeterSupport hE.1 hc _ (div_pos (by norm_num) hn0) x).mpr hx, ?_⟩
rw [norm_smul, Real.norm_of_nonneg (div_nonneg (by norm_num) hn0.le), div_mul_cancel₀ _ hn0.ne']
lemma annularFrameMeasure_ne_zero {n : ℕ} (hn : 1 < n) {E : Set (Euc n)}
(hE : IsLocallyPerimeterMinimizing E) (hc : IsCone E)
(hv : (0 : Euc n) ∈ perimeterSupport E) : annularFrameMeasure E ≠ 0 := by
obtain ⟨y, hy, hyn⟩ := cone_support_annular_point hn hE hc hv
have hyp : 0 < annularWeight y := annularWeight_pos (by rw [hyn]; norm_num) (by rw [hyn]; norm_num)
have hyb : y ∈ Metric.ball (0 : Euc n) 3 := by simp [Metric.mem_ball, hyn]
let U := {x : Euc n | 0 < annularWeight x}
have hU : IsOpen U := isOpen_lt continuous_const annularWeight_continuous
have hy' : y ∈ (perimeterMeasure E).support := by rwa [perimeterMeasure_support]
have hp : 0 < perimeterMeasure E (U ∩ Metric.ball 0 3) :=
(Measure.mem_support_iff_forall y).mp hy' _ ((hU.inter Metric.isOpen_ball).mem_nhds ⟨hyp,hyb⟩)
intro hz
have hzae : (fun x : Euc n ↦ (annularWeightNN x : ℝ≥0∞)) =ᵐ[
(perimeterMeasure E).restrict (Metric.ball 0 3)] 0 :=
(withDensity_eq_zero_iff annularWeightNN_measurable.coe_nnreal_ennreal.aemeasurable).mp hz
have hU0 : (perimeterMeasure E).restrict (Metric.ball 0 3) U = 0 := by
apply measure_eq_zero_iff_ae_notMem.mpr
filter_upwards [hzae] with x hx
intro hxp
have hw0 : annularWeight x = 0 := by
have hh : annularWeightNN x = 0 := ENNReal.coe_eq_zero.mp hx
exact congrArg (fun z : ℝ≥0 ↦ (z : ℝ)) hh
exact (ne_of_gt hxp) hw0
rw [Measure.restrict_apply hU.measurableSet] at hU0
exact hp.ne' hU0
lemma annularLink_ne_zero {n : ℕ} (hn : 1 < n) {E : Set (Euc n)}
(hE : IsLocallyPerimeterMinimizing E) (hc : IsCone E)
(hv : (0 : Euc n) ∈ perimeterSupport E) : annularLink E ≠ 0 := by
intro hz
apply annularFrameMeasure_ne_zero hn hE hc hv
exact (Measure.map_eq_zero_iff (frameOf_measurable hE.1).aemeasurable).mp hz
lemma annularLink_radial_support {n : ℕ} (hn : 1 < n) {E : Set (Euc n)}
(hE : IsLocallyPerimeterMinimizing E) (hc : IsCone E) :
∀ᵐ e ∂annularLink E, e.1 ∈ perimeterSupport E ∧ ‖e.1‖ = 1 := by
have hs : ∀ᵐ x ∂perimeterMeasure E, x ∈ perimeterSupport E := by
rw [← perimeterMeasure_support]
exact (perimeterMeasure E).support_mem_ae
have hf := annularLink_frames hn hE hc
have hr : ∀ᵐ e ∂annularLink E, e.1 ∈ perimeterSupport E := by
rw [annularLink, ae_map_iff (frameOf_measurable hE.1).aemeasurable
((isClosed_perimeterSupport E).preimage continuous_fst).measurableSet,
annularFrameMeasure, ae_withDensity_iff annularWeightNN_measurable.coe_nnreal_ennreal]
filter_upwards [ae_restrict_of_ae hs] with x hx hρ
have hx0 : x ≠ 0 := by
intro hh
apply hρ
apply ENNReal.coe_eq_zero.mpr
apply Subtype.ext
simp [hh, annularWeightNN, annularWeight]
exact (isCone_perimeterSupport hE.1 hc _ (inv_pos.mpr (norm_pos_iff.mpr hx0)) x).mpr hx
filter_upwards [hr,hf] with e he hframe
exact ⟨he,hframe.1⟩
end SphericalFrankel
end RPISingularHarnack
end
end OAI
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