| 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 angle_swap {n : ℕ} (e f : Frame n) : angle f e = angle e f := real_inner_comm _ _ | |
| lemma cross_swap {n : ℕ} (e f : Frame n) : cross f e = cross e f := real_inner_comm _ _ | |
| lemma traceProduct_swap {n : ℕ} (e f : Frame n) : traceProduct f e = traceProduct e f := by | |
| simp only [traceProduct, angle_swap, real_inner_comm f.1 e.2, | |
| real_inner_comm f.2 e.1, real_inner_comm f.2 e.2] | |
| ring | |
| lemma continuous_defect_reverse {n : ℕ} : | |
| Continuous (fun p : Frame n × Frame n ↦ defect p.2 p.1) := by | |
| unfold defect proj | |
| fun_prop | |
| lemma product_frames {n : ℕ} {μ ν : Measure (Frame n)} [IsFiniteMeasure μ] [IsFiniteMeasure ν] | |
| (hμ : ∀ᵐ e ∂μ, IsFrame e) (hν : ∀ᵐ e ∂ν, IsFrame e) : | |
| ∀ᵐ p ∂μ.prod ν, IsFrame p.1 ∧ IsFrame p.2 := by | |
| apply (Measure.ae_prod_iff_ae_ae | |
| ((isClosed_frames n).prod (isClosed_frames n)).measurableSet).mpr | |
| filter_upwards [hμ] with e he | |
| filter_upwards [hν] with f hf | |
| exact ⟨he, hf⟩ | |
| lemma stationary_product_integral {n : ℕ} {μ ν : Measure (Frame n)} | |
| [IsFiniteMeasure μ] [IsFiniteMeasure ν] | |
| (hμ : ∀ᵐ e ∂μ, IsFrame e) (hν : ∀ᵐ e ∂ν, IsFrame e) | |
| (hS : LinkStationary μ) (W : Frame n → Euc n → Euc n) | |
| (hW : ∀ f, ContDiff ℝ ∞ (W f)) | |
| (G : Frame n × Frame n → ℝ) (hG : Continuous G) | |
| (heq : ∀ e f, IsFrame e → IsFrame f → G (e,f) = linkDiv e (W f)) : | |
| (∫ p, G p ∂μ.prod ν) = 0 := by | |
| rw [integral_prod_symm G (integrable_pairFrameFunction (product_frames hμ hν) hG)] | |
| calc | |
| (∫ f, ∫ e, G (e, f) ∂μ ∂ν) = ∫ f, (0 : ℝ) ∂ν := by | |
| apply integral_congr_ae | |
| filter_upwards [hν] with f hf | |
| calc | |
| (∫ e, G (e,f) ∂μ) = ∫ e, linkDiv e (W f) ∂μ := by | |
| apply integral_congr_ae | |
| filter_upwards [hμ] with e he | |
| exact heq e f he hf | |
| _ = 0 := hS.firstVariation (W f) (hW f) | |
| _ = 0 := by simp | |
| lemma stationary_gradient_one {n : ℕ} {μ ν : Measure (Frame n)} | |
| [IsFiniteMeasure μ] [IsFiniteMeasure ν] | |
| (hμ : ∀ᵐ e ∂μ, IsFrame e) (hν : ∀ᵐ e ∂ν, IsFrame e) | |
| (hS : LinkStationary μ) {φ : ℝ → ℝ} (hφ : ContDiff ℝ ∞ φ) : | |
| (∫ p, deriv φ (angle p.1 p.2) * defect p.1 p.2 ∂μ.prod ν) = | |
| ((n : ℝ)-2) * ∫ p, φ (angle p.1 p.2) * angle p.1 p.2 ∂μ.prod ν := by | |
| have hd := hφ.continuous_deriv (by simp) | |
| have h0 := stationary_product_integral hμ hν hS | |
| (fun f z ↦ φ (inner ℝ z f.1) • f.1) | |
| (fun f ↦ (hφ.comp (contDiff_id.inner ℝ contDiff_const)).smul contDiff_const) | |
| (fun p ↦ deriv φ (angle p.1 p.2) * defect p.1 p.2 - | |
| ((n : ℝ)-2) * (φ (angle p.1 p.2) * angle p.1 p.2)) | |
| (((hd.comp continuous_angle).mul continuous_defect).sub | |
| (continuous_const.mul ((hφ.continuous.comp continuous_angle).mul continuous_angle))) | |
| (fun e f he hf ↦ by | |
| rw [linkDiv_scalar_constant he (hφ.differentiable (by simp))] | |
| ring) | |
| have hi1 : Integrable (fun p : Frame n × Frame n ↦ deriv φ (angle p.1 p.2) * defect p.1 p.2) (μ.prod ν) := | |
| integrable_pairFrameFunction (product_frames hμ hν) | |
| ((hd.comp continuous_angle).mul continuous_defect) | |
| have hi2 : Integrable (fun p : Frame n × Frame n ↦ φ (angle p.1 p.2) * angle p.1 p.2) (μ.prod ν) := | |
| integrable_pairFrameFunction (product_frames hμ hν) | |
| ((hφ.continuous.comp continuous_angle).mul continuous_angle) | |
| rw [integral_sub hi1 (hi2.const_mul ((n : ℝ)-2)), integral_const_mul] at h0 | |
| exact sub_eq_zero.mp h0 | |
| lemma stationary_mixed_one {n : ℕ} {μ ν : Measure (Frame n)} | |
| [IsFiniteMeasure μ] [IsFiniteMeasure ν] | |
| (hμ : ∀ᵐ e ∂μ, IsFrame e) (hν : ∀ᵐ e ∂ν, IsFrame e) | |
| (hS : LinkStationary μ) {φ : ℝ → ℝ} (hφ : ContDiff ℝ ∞ φ) : | |
| (∫ p, deriv φ (angle p.1 p.2) * cross p.1 p.2 ∂μ.prod ν) + | |
| (∫ p, φ (angle p.1 p.2) * traceProduct p.1 p.2 ∂μ.prod ν) - | |
| ((n : ℝ)-2) * ∫ p, φ (angle p.1 p.2) * defect p.2 p.1 ∂μ.prod ν = 0 := by | |
| have hd := hφ.continuous_deriv (by simp) | |
| have hswap : Continuous (fun p : Frame n × Frame n ↦ defect p.2 p.1) := | |
| continuous_defect_reverse | |
| have hC : Continuous (fun p : Frame n × Frame n ↦ deriv φ (angle p.1 p.2) * cross p.1 p.2) := | |
| (hd.comp continuous_angle).mul continuous_cross | |
| have hT : Continuous (fun p : Frame n × Frame n ↦ φ (angle p.1 p.2) * traceProduct p.1 p.2) := | |
| (hφ.continuous.comp continuous_angle).mul continuous_traceProduct | |
| have hD : Continuous (fun p : Frame n × Frame n ↦ φ (angle p.1 p.2) * defect p.2 p.1) := | |
| (hφ.continuous.comp continuous_angle).mul hswap | |
| have h0 := stationary_product_integral hμ hν hS | |
| (fun f z ↦ φ (inner ℝ z f.1) • projMap f z) | |
| (fun f ↦ (hφ.comp (contDiff_id.inner ℝ contDiff_const)).smul (projMap f).contDiff) | |
| (fun p ↦ deriv φ (angle p.1 p.2) * cross p.1 p.2 + | |
| φ (angle p.1 p.2) * traceProduct p.1 p.2 - | |
| ((n : ℝ)-2) * (φ (angle p.1 p.2) * defect p.2 p.1)) | |
| ((hC.add hT).sub (continuous_const.mul hD)) | |
| (fun e f he hf ↦ by | |
| rw [linkDiv_scalar_proj he hf (hφ.differentiable (by simp))] | |
| ring) | |
| have hiC := integrable_pairFrameFunction (product_frames hμ hν) hC | |
| have hiT := integrable_pairFrameFunction (product_frames hμ hν) hT | |
| have hiD := integrable_pairFrameFunction (product_frames hμ hν) hD | |
| have hiCT : Integrable (fun p : Frame n × Frame n ↦ | |
| deriv φ (angle p.1 p.2) * cross p.1 p.2 + | |
| φ (angle p.1 p.2) * traceProduct p.1 p.2) (μ.prod ν) := hiC.add hiT | |
| rwa [integral_sub hiCT (hiD.const_mul ((n : ℝ)-2)), integral_add hiC hiT, | |
| integral_const_mul] at h0 | |
| lemma coupledStationarity_product {n : ℕ} {μ ν : Measure (Frame n)} | |
| [IsFiniteMeasure μ] [IsFiniteMeasure ν] | |
| (hμ : ∀ᵐ e ∂μ, IsFrame e) (hν : ∀ᵐ e ∂ν, IsFrame e) | |
| (hSμ : LinkStationary μ) (hSν : LinkStationary ν) : CoupledStationarity (μ.prod ν) := by | |
| constructor | |
| · intro φ hφ | |
| have hd := hφ.continuous_deriv (by simp) | |
| have hi1 : Integrable (fun p : Frame n × Frame n ↦ deriv φ (angle p.1 p.2) * defect p.1 p.2) (μ.prod ν) := | |
| integrable_pairFrameFunction (product_frames hμ hν) | |
| ((hd.comp continuous_angle).mul continuous_defect) | |
| have hi2 : Integrable (fun p : Frame n × Frame n ↦ deriv φ (angle p.1 p.2) * defect p.2 p.1) (μ.prod ν) := | |
| integrable_pairFrameFunction (product_frames hμ hν) | |
| ((hd.comp continuous_angle).mul (continuous_defect_reverse)) | |
| have h1 := stationary_gradient_one hμ hν hSμ hφ | |
| have h2 := stationary_gradient_one hν hμ hSν hφ | |
| rw [← integral_prod_swap (fun p : Frame n × Frame n ↦ deriv φ (angle p.1 p.2) * defect p.1 p.2), | |
| ← integral_prod_swap (fun p : Frame n × Frame n ↦ φ (angle p.1 p.2) * angle p.1 p.2)] at h2 | |
| simp only [Prod.fst_swap, Prod.snd_swap, angle_swap] at h2 | |
| simp_rw [mul_add] | |
| rw [integral_add hi1 hi2, h1, h2] | |
| ring | |
| · intro φ hφ | |
| have h1 := stationary_mixed_one hμ hν hSμ hφ | |
| have h2 := stationary_mixed_one hν hμ hSν hφ | |
| rw [← integral_prod_swap (fun p : Frame n × Frame n ↦ deriv φ (angle p.1 p.2) * cross p.1 p.2), | |
| ← integral_prod_swap (fun p : Frame n × Frame n ↦ φ (angle p.1 p.2) * traceProduct p.1 p.2), | |
| ← integral_prod_swap (fun p : Frame n × Frame n ↦ φ (angle p.1 p.2) * defect p.2 p.1)] at h2 | |
| simp only [Prod.fst_swap, Prod.snd_swap, angle_swap, cross_swap, traceProduct_swap] at h2 | |
| have hi1 : Integrable (fun p : Frame n × Frame n ↦ φ (angle p.1 p.2) * defect p.1 p.2) (μ.prod ν) := | |
| integrable_pairFrameFunction (product_frames hμ hν) | |
| ((hφ.continuous.comp continuous_angle).mul continuous_defect) | |
| have hi2 : Integrable (fun p : Frame n × Frame n ↦ φ (angle p.1 p.2) * defect p.2 p.1) (μ.prod ν) := | |
| integrable_pairFrameFunction (product_frames hμ hν) | |
| ((hφ.continuous.comp continuous_angle).mul (continuous_defect_reverse)) | |
| simp_rw [mul_add] | |
| rw [integral_add hi1 hi2] | |
| linarith | |
| lemma minimizing_cones_coupledStationarity {n : ℕ} (hn : 1 < n) {E F : Set (Euc n)} | |
| (hE : IsLocallyPerimeterMinimizing E) (hF : IsLocallyPerimeterMinimizing F) | |
| (hcE : IsCone E) (hcF : IsCone F) : | |
| CoupledStationarity ((annularLink E).prod (annularLink F)) := by | |
| let := annularLink_finite hE | |
| let := annularLink_finite hF | |
| exact coupledStationarity_product (annularLink_frames hn hE hcE) (annularLink_frames hn hF hcF) | |
| (annularLink_stationary hn hE hcE) (annularLink_stationary hn hF hcF) | |
| end SphericalFrankel | |
| end RPISingularHarnack | |
| end | |
| end OAI | |