| import OAI.Geometry.Perimeter.Regularity.RawConeLink |
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| namespace OAI |
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| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal ContDiff |
| namespace RPISingularHarnack |
| namespace SphericalFrankel |
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| 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 |
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| 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⟩ |
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| 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) |
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| 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'] |
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| 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 |
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| 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 |
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| 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⟩ |
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| end SphericalFrankel |
| end RPISingularHarnack |
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| end |
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| end OAI |
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