File size: 3,855 Bytes
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 | import OAI.Geometry.Perimeter.Regularity.CoupledConeLink
import OAI.Geometry.Perimeter.Regularity.ConeLinkSupport
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal ContDiff
namespace RPISingularHarnack
namespace SphericalFrankel
def fullLink {n : ℕ} (E : Set (Euc n)) : Set (Euc n) :=
perimeterSupport E ∩ Metric.sphere 0 1
lemma fullLink_compact {n : ℕ} (E : Set (Euc n)) : IsCompact (fullLink E) :=
(isCompact_sphere (0 : Euc n) 1).inter_left (isClosed_perimeterSupport E)
lemma fullLink_nonempty {n : ℕ} (hn : 1 < n) {E : Set (Euc n)}
(hE : IsLocallyPerimeterMinimizing E) (hc : IsCone E)
(hv : (0 : Euc n) ∈ perimeterSupport E) : (fullLink E).Nonempty := by
obtain ⟨x,hx,hx0⟩ := minimizing_support_nonzero_point hn hE hv
refine ⟨radialUnit x, (isCone_perimeterSupport hE.1 hc _
(inv_pos.mpr (norm_pos_iff.mpr hx0)) x).mpr hx, ?_⟩
simpa [Metric.mem_sphere] using radialUnit_norm hx0
lemma product_ne_zero_of_ne_zero {n : ℕ} {μ ν : Measure (Frame n)}
[IsFiniteMeasure μ] [IsFiniteMeasure ν] (hμ : μ ≠ 0) (hν : ν ≠ 0) : μ.prod ν ≠ 0 := by
intro hz
have hp : μ Set.univ * ν Set.univ = 0 := by
rw [← Measure.prod_prod, Set.univ_prod_univ, hz]
simp
rcases mul_eq_zero.mp hp with h | h
· exact hμ (Measure.measure_univ_eq_zero.mp h)
· exact hν (Measure.measure_univ_eq_zero.mp h)
theorem minimizing_cone_links_intersect {n : ℕ} (hn : 2 < n) {E F : Set (Euc n)}
(hE : IsLocallyPerimeterMinimizing E) (hF : IsLocallyPerimeterMinimizing F)
(hcE : IsCone E) (hcF : IsCone F)
(hvE : (0 : Euc n) ∈ perimeterSupport E)
(hvF : (0 : Euc n) ∈ perimeterSupport F) :
(fullLink E ∩ fullLink F).Nonempty := by
have hn1 : 1 < n := by omega
let μ := annularLink E
let ν := annularLink F
let : IsFiniteMeasure μ := annularLink_finite hE
let : IsFiniteMeasure ν := annularLink_finite hF
have hμ0 : μ ≠ 0 := annularLink_ne_zero hn1 hE hcE hvE
have hν0 : ν ≠ 0 := annularLink_ne_zero hn1 hF hcF hvF
have hμ := annularLink_frames hn1 hE hcE
have hν := annularLink_frames hn1 hF hcF
have hS := minimizing_cones_coupledStationarity hn1 hE hF hcE hcF
have hC : Continuous (fun p : Euc n × Euc n ↦ inner ℝ p.1 p.2) :=
continuous_fst.inner continuous_snd
obtain ⟨p, hp, hmax⟩ := ((fullLink_compact E).prod (fullLink_compact F)).exists_isMaxOn
((fullLink_nonempty hn1 hE hcE hvE).prod (fullLink_nonempty hn1 hF hcF hvF))
hC.continuousOn
have hpn : ‖p.1‖ = 1 := by simpa [Metric.mem_sphere] using hp.1.2
have hqn : ‖p.2‖ = 1 := by simpa [Metric.mem_sphere] using hp.2.2
have hle : inner ℝ p.1 p.2 ≤ 1 := by
exact le_trans (le_abs_self _) (by simpa [hpn,hqn] using abs_real_inner_le_norm p.1 p.2)
by_cases heq : p.1 = p.2
· exact ⟨p.1,hp.1,heq ▸ hp.2⟩
· have hlt : inner ℝ p.1 p.2 < 1 := lt_of_le_of_ne hle (by
intro he
exact heq ((inner_eq_one_iff_of_norm_eq_one hpn hqn).mp he))
have hbound : ∀ᵐ q ∂μ.prod ν, angle q.1 q.2 ≤ max 0 (inner ℝ p.1 p.2) := by
apply (Measure.ae_prod_iff_ae_ae
(isClosed_le continuous_angle continuous_const).measurableSet).mpr
filter_upwards [annularLink_radial_support hn1 hE hcE] with e he
filter_upwards [annularLink_radial_support hn1 hF hcF] with f hf
apply le_trans _ (le_max_right _ _)
change inner ℝ e.1 f.1 ≤ inner ℝ p.1 p.2
exact hmax (a := (e.1,f.1)) ⟨⟨he.1, by simpa [Metric.mem_sphere] using he.2⟩,
⟨hf.1, by simpa [Metric.mem_sphere] using hf.2⟩⟩
exact (no_positive_angle_gap hn (product_ne_zero_of_ne_zero hμ0 hν0)
(product_frames hμ hν) hS (le_max_left _ _) (max_lt (by norm_num) hlt) hbound).elim
end SphericalFrankel
end RPISingularHarnack
end
end OAI
|