| import Mathlib |
| import OAI.Geometry.CubicTorus.Regularity.CoverEssentialBoundary |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace symmDiff |
| namespace CubicTorus |
|
|
| theorem isMinimizerAt_primary_cone_C1_offvertex {V : ℝ} (hV : V ∈ Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| (hcenter : quotientMap cubeCenter ∈ frontier (openRepresentative E)) |
| (r : ℕ → ℝ) (hr : ∀ i, 0 < r i) (hrlim : Tendsto r atTop (𝓝 0)) : |
| ∃ (A : Set Cover) (σ : FiniteMeasure BlowupNormalBundle) (θ : ℝ) (φ : ℕ → ℕ), |
| MeasurableSet A ∧ 0 < θ ∧ StrictMono φ ∧ |
| Tendsto (fun i ↦ ∫ x in Metric.closedBall (0 : Cover) 2, |
| |blowupCharacteristic E cubeCenter (r (φ i)) x-A.indicator (fun _ ↦ (1:ℝ)) x|) |
| atTop (𝓝 0) ∧ |
| (∀ b : ℝ, 0 < b → b ≤ 1 → |
| coverLocalPerimeter A (Metric.closedBall (0 : Cover) b) = ENNReal.ofReal (θ*b^2)) ∧ |
| (∀ B : Set Cover, MeasurableSet B → ∀ a b : ℝ, 0 < a → a < b → b ≤ 1 → |
| A ∆ B ⊆ Metric.ball (0 : Cover) a → |
| coverLocalPerimeter A (Metric.closedBall (0 : Cover) b) ≤ |
| coverLocalPerimeter B (Metric.closedBall (0 : Cover) b)) ∧ |
| (∀ t : ℝ, 1 ≤ t → ∀ᵐ x : Cover ∂volume, x ∈ Metric.ball (0 : Cover) 1 → |
| (t⁻¹ • x ∈ A ↔ x ∈ A)) ∧ |
| (∀ F : Cover → Cover, ContDiff ℝ (⊤ : ℕ∞) F → |
| tsupport F ⊆ Metric.closedBall (0 : Cover) 1 → |
| (∫ z, tangentialDivergence F z.1 z.2 ∂(σ : Measure BlowupNormalBundle)) = 0 ∧ |
| (∫ z, ⟪(z.2 : Cover),F z.1⟫ ∂(σ : Measure BlowupNormalBundle)) = ∫ x in A, divergence F x) ∧ |
| ∀ p : Cover, ‖p‖ < 1 → p ≠ 0 → p ∈ coverEssentialBoundary A → CoverC1BoundaryAt A p := by |
| obtain ⟨μ,hμ⟩ := exists_normalMeasure hE hmin.1.2.ne |
| obtain ⟨H,hH⟩ := isMinimizerAt_centered_multiplier hV hE hmin hcenter hμ |
| obtain ⟨A,σ,θ,φ,hA,hθ,hφ,hconv,hper,hcomp,hcone,hstat⟩ := |
| isMinimizerAt_minimizing_cone_blowup hV hE hmin hμ hH hcenter r hr hrlim |
| refine ⟨A,σ,θ,φ,hA,hθ,hφ,hconv,hper,hcomp,hcone,hstat,?_⟩ |
| intro p hp hp0 hboundary |
| exact actual_primary_offvertex_C1 hV hE hmin hcenter hμ hH |
| (fun i ↦ r (φ i)) (fun i ↦ hr _) (hrlim.comp hφ.tendsto_atTop) |
| hA hconv hcone hstat p hp hp0 hboundary |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| def graphMollifier (i : ℕ) : ContDiffBump (0 : Plane) where |
| rIn := ((i:ℝ)+1)⁻¹/2 |
| rOut := ((i:ℝ)+1)⁻¹ |
| rIn_pos := by positivity |
| rIn_lt_rOut := by |
| have hh : 0 < ((i:ℝ)+1)⁻¹ := by positivity |
| linarith |
|
|
| lemma graphMollifier_radius_tendsto : |
| Tendsto (fun i ↦ (graphMollifier i).rOut) atTop (𝓝 0) := by |
| simpa only [graphMollifier,one_div] using |
| (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)) |
|
|
| lemma graphSmooth_tendsto {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) (y : Plane) : |
| Tendsto (fun i ↦ graphSmooth (graphMollifier i) g y) atTop (𝓝 (g y)) := by |
| apply tendsto_iff_norm_sub_tendsto_zero.mpr |
| apply squeeze_zero (fun _ ↦ norm_nonneg _) (fun i ↦ ?_) |
| (by simpa only [mul_zero] using graphMollifier_radius_tendsto.const_mul (K:ℝ)) |
| exact graphSmooth_error _ hg y |
|
|
| theorem graphSmooth_fderiv_tendsto {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) |
| {y : Plane} (hc : ContinuousAt (fderiv ℝ g) y) : |
| Tendsto (fun i ↦ fderiv ℝ (graphSmooth (graphMollifier i) g) y) |
| atTop (𝓝 (fderiv ℝ g y)) := by |
| rw [Metric.tendsto_nhds] |
| intro ε hε |
| obtain ⟨δ,hδ,hclose⟩ := Metric.continuousAt_iff.mp hc (ε/2) (by positivity) |
| filter_upwards [graphMollifier_radius_tendsto.eventually (Iio_mem_nhds hδ)] with i hi |
| have ha := lipschitz_probability_average_hasFDerivAt (graphSmoothingMeasure_ac (graphMollifier i)) |
| hg (graphMollifier i).rOut_pos.le (graphSmoothingMeasure_radius (graphMollifier i)) y |
| change dist (fderiv ℝ (fun z ↦ ∫ w, g (z-w) ∂graphSmoothingMeasure (graphMollifier i)) y) _ < ε |
| rw [ha.2.fderiv,dist_eq_norm] |
| have he : (∫ _w : Plane, fderiv ℝ g y ∂graphSmoothingMeasure (graphMollifier i)) = fderiv ℝ g y := by simp |
| rw [← he,← integral_sub ha.1 (integrable_const _)] |
| apply lt_of_le_of_lt (norm_integral_le_integral_norm _) ?_ |
| have hb : (∫ w, ‖fderiv ℝ g (y-w)-fderiv ℝ g y‖ ∂graphSmoothingMeasure (graphMollifier i)) ≤ ε/2 := by |
| calc |
| _ ≤ ∫ _w : Plane, ε/2 ∂graphSmoothingMeasure (graphMollifier i) := by |
| apply integral_mono_ae (ha.1.sub (integrable_const _)).norm (integrable_const _) |
| filter_upwards [graphSmoothingMeasure_radius (graphMollifier i)] with w hw |
| have hh : dist (y-w) y < δ := by simpa only [dist_eq_norm,sub_sub_cancel_left,norm_neg] using hw.trans_lt hi |
| simpa only [dist_eq_norm,Pi.sub_apply] using (hclose hh).le |
| _ = _ := by simp |
| exact hb.trans_lt (by linarith) |
|
|
| lemma graphSmooth_gradient_tendsto {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) |
| {y : Plane} (hc : ContinuousAt (fderiv ℝ g) y) : |
| Tendsto (fun i ↦ gradient (graphSmooth (graphMollifier i) g) y) atTop (𝓝 (gradient g y)) := by |
| exact ((InnerProductSpace.toDual ℝ Plane).symm.continuous.tendsto _).comp |
| (graphSmooth_fderiv_tendsto hg hc) |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma graph_trace_volume_zero (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| {g : Plane → ℝ} (hg : Continuous g) : |
| volume {x : Cover | ⟪n,x-a⟫ = graphHeightExtension a n g x} = 0 := by |
| have hz : volume {x : Cover | ⟪n,x-a⟫ = 0} = 0 := by |
| simpa only [Set.preimage_ofPred_eq,sub_eq_add_neg] using |
| (measurePreserving_add_right volume (-a)).quasiMeasurePreserving.preimage_null |
| (hyperplane_volume_zero hn) |
| have hh := (graphShear_measurePreserving a hn hg.neg.measurable).quasiMeasurePreserving.preimage_null hz |
| convert hh using 2 |
| ext x |
| simp only [Set.mem_ofPred_eq,Set.mem_preimage,graphShear,normalMove,graphHeightExtension, |
| add_sub_right_comm,inner_add_right,inner_smul_right,real_inner_self_eq_norm_sq,hn, |
| one_pow,mul_one,Pi.neg_apply] |
| constructor <;> intro h <;> linarith |
|
|
| lemma measurable_graphSublevelIndicator (a n : Cover) {g : Plane → ℝ} (hg : Measurable g) : |
| Measurable (graphSublevelIndicator a n g) := by |
| exact measurable_const.indicator (measurableSet_lt (by fun_prop) |
| (hg.comp ((normalPlaneCoordinates n).measurable.comp (measurable_id.sub measurable_const)))) |
|
|
| lemma graphSublevelIndicator_bound (a n : Cover) (g : Plane → ℝ) (x : Cover) : |
| |graphSublevelIndicator a n g x| ≤ 1 := by |
| classical |
| by_cases hx : ⟪n,x-a⟫ < graphHeightExtension a n g x <;> |
| simp [graphSublevelIndicator,hx] |
|
|
| lemma graphSublevelIndicator_tendsto {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) |
| (a n : Cover) {x : Cover} (hx : ⟪n,x-a⟫ ≠ graphHeightExtension a n g x) : |
| Tendsto (fun i ↦ graphSublevelIndicator a n (graphSmooth (graphMollifier i) g) x) |
| atTop (𝓝 (graphSublevelIndicator a n g x)) := by |
| have ht := graphSmooth_tendsto hg (normalPlaneCoordinates n (x-a)) |
| have he : ∀ᶠ i in atTop, |
| graphSublevelIndicator a n (graphSmooth (graphMollifier i) g) x = graphSublevelIndicator a n g x := by |
| by_cases hl : ⟪n,x-a⟫ < graphHeightExtension a n g x |
| · filter_upwards [ht.eventually (Ioi_mem_nhds hl)] with i hi |
| change (if ⟪n,x-a⟫ < graphSmooth (graphMollifier i) g (normalPlaneCoordinates n (x-a)) then (1:ℝ) else 0) = _ |
| rw [ite_eq_left hi] |
| exact (indicator_of_mem hl (fun _ ↦ (1:ℝ))).symm |
| · have hh : graphHeightExtension a n g x < ⟪n,x-a⟫ := lt_of_le_of_ne (not_lt.mp hl) hx.symm |
| filter_upwards [ht.eventually (Iio_mem_nhds hh)] with i hi |
| have hi' : ¬ ⟪n,x-a⟫ < graphHeightExtension a n (graphSmooth (graphMollifier i) g) x := not_lt.mpr hi.le |
| simp [graphSublevelIndicator,hl,hi'] |
| apply tendsto_const_nhds.congr' |
| filter_upwards [he] with i hi |
| exact hi.symm |
|
|
| lemma graphSublevel_flux_mollification_tendsto (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) |
| {F : Cover → Cover} (hF : ContDiff ℝ (⊤ : ℕ∞) F) (hs : HasCompactSupport F) : |
| Tendsto (fun i ↦ ∫ x : Cover, graphSublevelIndicator a n (graphSmooth (graphMollifier i) g) x * divergence F x) |
| atTop (𝓝 (∫ x : Cover, graphSublevelIndicator a n g x * divergence F x)) := by |
| have hi : Integrable (divergence F) := (continuous_divergence hF).integrable_of_hasCompactSupport |
| (hs.of_isClosed_subset isClosed_closure (divergence_tsupport_subset F)) |
| apply tendsto_integral_of_dominated_convergence (fun x ↦ ‖divergence F x‖) |
| (fun i ↦ ((measurable_graphSublevelIndicator a n (graphSmooth_contDiff _ hg).continuous.measurable).mul |
| (continuous_divergence hF).measurable).aestronglyMeasurable) hi.norm |
| · intro i |
| filter_upwards [] with x |
| simp only [Pi.mul_apply,norm_mul] |
| exact mul_le_of_le_one_left (norm_nonneg _) (graphSublevelIndicator_bound a n _ x) |
| · have hae : ∀ᵐ x : Cover ∂volume, ⟪n,x-a⟫ ≠ graphHeightExtension a n g x := by |
| simpa only [ae_iff,not_not] using graph_trace_volume_zero a hn hg.continuous |
| filter_upwards [hae] with x hx |
| exact (graphSublevelIndicator_tendsto hg a n hx).mul_const _ |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma graphAreaVector_norm_linear {n : Cover} (hn : ‖n‖ = 1) |
| {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) (y : Plane) : |
| ‖graphAreaVector n g y‖ ≤ 1+(K:ℝ) := by |
| calc |
| _ ≤ ‖n‖+‖normalPlaneEmbedding n (gradient g y)‖ := norm_sub_le _ _ |
| _ = 1+‖fderiv ℝ g y‖ := by rw [hn,normalPlaneEmbedding_norm,plane_gradient_norm] |
| _ ≤ _ := add_le_add le_rfl (norm_fderiv_le_of_lipschitz ℝ hg) |
|
|
| lemma graph_trace_zero_outside {X : Type*} [Zero X] {a n : Cover} (hn : ‖n‖ = 1) |
| {F : Cover → X} {R : ℝ} (hs : tsupport F ⊆ Metric.closedBall a R) |
| (g : Plane → ℝ) {y : Plane} (hy : y ∉ Metric.closedBall (0 : Plane) R) : |
| F (a+normalGraph n g y) = 0 := by |
| apply image_eq_zero_of_notMem_tsupport |
| intro h |
| have hh := hs h |
| rw [Metric.mem_closedBall,dist_eq_norm,add_sub_cancel_left] at hh |
| exact hy (by simpa only [Metric.mem_closedBall,dist_zero_right] using |
| (normalGraph_norm_ge hn g y).trans hh) |
|
|
| theorem integral_C1graphSublevel_divergence (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) |
| {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y) |
| {F : Cover → Cover} (hF : ContDiff ℝ (⊤ : ℕ∞) F) |
| (hs : tsupport F ⊆ Metric.closedBall a R) : |
| (∫ x : Cover, graphSublevelIndicator a n g x * divergence F x) = |
| ∫ y : Plane, ⟪graphAreaVector n g y, F (a+normalGraph n g y)⟫ := by |
| have hFs : HasCompactSupport F := (isCompact_closedBall a R).of_isClosed_subset isClosed_closure hs |
| obtain ⟨M,hM⟩ := hF.continuous.norm.bddAbove_range_of_hasCompactSupport hFs.norm |
| have hMb (x : Cover) : ‖F x‖ ≤ M := hM (mem_range_self x) |
| have hM0 : 0 ≤ M := (norm_nonneg (F 0)).trans (hMb 0) |
| let B : Plane → ℝ := (Metric.closedBall (0 : Plane) R).indicator (fun _ ↦ (1+(K:ℝ))*M) |
| have hBi : Integrable B := (integrableOn_const (isCompact_closedBall (0 : Plane) R).measure_lt_top.ne).integrable_indicator Metric.isClosed_closedBall.measurableSet |
| have ht : Tendsto (fun i ↦ ∫ y : Plane, |
| ⟪graphAreaVector n (graphSmooth (graphMollifier i) g) y, |
| F (a+normalGraph n (graphSmooth (graphMollifier i) g) y)⟫) atTop |
| (𝓝 (∫ y : Plane, ⟪graphAreaVector n g y,F (a+normalGraph n g y)⟫)) := by |
| apply tendsto_integral_of_dominated_convergence B |
| (fun i ↦ ((graphAreaVector_continuous n (graphSmooth_contDiff _ hg)).inner |
| (graph_trace_continuous a n (graphSmooth_contDiff _ hg).continuous hF.continuous)).aestronglyMeasurable) hBi |
| · intro i |
| filter_upwards [] with y |
| by_cases hy : y ∈ Metric.closedBall (0 : Plane) R |
| · rw [show B y = (1+(K:ℝ))*M from indicator_of_mem hy _] |
| exact (norm_inner_le_norm _ _).trans |
| (mul_le_mul (graphAreaVector_norm_linear hn (graphSmooth_lipschitz _ hg) y) |
| (hMb _) (norm_nonneg _) (by positivity)) |
| · rw [graph_trace_zero_outside hn hs _ hy,inner_zero_right,norm_zero, |
| show B y = 0 from indicator_of_notMem hy _] |
| · filter_upwards [] with y |
| by_cases hy : y ∈ Metric.closedBall (0 : Plane) R |
| · have hd := graphSmooth_gradient_tendsto hg (hc y hy) |
| have hnG : Tendsto (fun i ↦ normalGraph n (graphSmooth (graphMollifier i) g) y) |
| atTop (𝓝 (normalGraph n g y)) := |
| tendsto_const_nhds.add ((graphSmooth_tendsto hg y).smul_const n) |
| have hnA : Tendsto (fun i ↦ graphAreaVector n (graphSmooth (graphMollifier i) g) y) |
| atTop (𝓝 (graphAreaVector n g y)) := |
| tendsto_const_nhds.sub (((normalPlaneLinearIsometry n).continuous.tendsto _).comp hd) |
| exact hnA.inner ((hF.continuous.tendsto _).comp (tendsto_const_nhds.add hnG)) |
| · simp only [graph_trace_zero_outside hn hs _ hy,inner_zero_right] |
| exact tendsto_const_nhds |
| apply tendsto_nhds_unique (graphSublevel_flux_mollification_tendsto a hn hg hF hFs) |
| simpa only [integral_graphSublevel_divergence a hn (graphSmooth_contDiff _ hg) hF hFs] using ht |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma graphAreaVector_continuousAt (n : Cover) {g : Plane → ℝ} {y : Plane} |
| (hc : ContinuousAt (fderiv ℝ g) y) : ContinuousAt (graphAreaVector n g) y := by |
| exact continuousAt_const.sub ((normalPlaneLinearIsometry n).continuous.continuousAt.comp |
| ((InnerProductSpace.toDual ℝ Plane).symm.continuous.continuousAt.comp hc)) |
|
|
| lemma graph_weighted_continuous (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| {g : Plane → ℝ} (hg : Continuous g) {R : ℝ} |
| {φ : Cover → ℝ} (hφ : Continuous φ) (hs : tsupport φ ⊆ Metric.closedBall a R) |
| {ψ : Plane → ℝ} (hψ : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt ψ y) : |
| Continuous (fun y : Plane ↦ φ (a+normalGraph n g y)*ψ y) := by |
| rw [continuous_iff_continuousAt] |
| intro y |
| by_cases hy : y ∈ Metric.closedBall (0 : Plane) R |
| · exact (graph_trace_continuous a n hg hφ).continuousAt.mul (hψ y hy) |
| · have he : (fun z : Plane ↦ φ (a+normalGraph n g z)*ψ z) =ᶠ[𝓝 y] fun _ ↦ 0 := by |
| filter_upwards [Metric.isClosed_closedBall.isOpen_compl.mem_nhds hy] with z hz |
| rw [graph_trace_zero_outside hn hs _ hz,zero_mul] |
| exact continuousAt_const.congr_of_eventuallyEq he |
|
|
| lemma graph_weighted_integrable (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| {g : Plane → ℝ} (hg : Continuous g) {R : ℝ} |
| {φ : Cover → ℝ} (hφ : Continuous φ) (hsφ : HasCompactSupport φ) |
| (hs : tsupport φ ⊆ Metric.closedBall a R) |
| {ψ : Plane → ℝ} (hψ : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt ψ y) : |
| Integrable (fun y : Plane ↦ φ (a+normalGraph n g y)*ψ y) := |
| (graph_weighted_continuous a hn hg hφ hs hψ).integrable_of_hasCompactSupport |
| (graph_trace_compactSupport a hn hg hsφ).mul_right |
|
|
| lemma setIntegral_graphSublevel_divergence (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) |
| {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y) |
| {F : Cover → Cover} (hF : ContDiff ℝ (⊤ : ℕ∞) F) |
| (hs : tsupport F ⊆ Metric.closedBall a R) : |
| (∫ x in {x : Cover | ⟪n,x-a⟫ < graphHeightExtension a n g x}, divergence F x) = |
| ∫ y : Plane, ⟪graphAreaVector n g y,F (a+normalGraph n g y)⟫ := by |
| rw [← integral_C1graphSublevel_divergence a hn hg hc hF hs] |
| have hm : MeasurableSet {x : Cover | ⟪n,x-a⟫ < graphHeightExtension a n g x} := |
| measurableSet_lt (by fun_prop) (hg.continuous.measurable.comp |
| ((normalPlaneCoordinates n).measurable.comp (measurable_id.sub measurable_const))) |
| rw [← integral_indicator hm] |
| congr 1 |
| funext x |
| classical |
| by_cases hx : ⟪n,x-a⟫ < graphHeightExtension a n g x <;> |
| simp [graphSublevelIndicator,hx] |
|
|
| theorem blowup_graph_patch_weighted_flux {A O U : Set Cover} (hA : MeasurableSet A) |
| (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) |
| {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y) |
| (hgraph : ∀ᵐ x : Cover ∂volume, x ∈ U → (x ∈ A ↔ ⟪n,x-a⟫ < graphHeightExtension a n g x)) |
| {ν : FiniteMeasure BlowupNormalBundle} |
| (hflux : ∀ F : Cover → Cover, ContDiff ℝ (⊤ : ℕ∞) F → HasCompactSupport F → |
| tsupport F ⊆ O → (∀ x, ‖F x‖ ≤ 1) → |
| (∫ z, ⟪(z.2 : Cover),F z.1⟫ ∂(ν : Measure BlowupNormalBundle)) = |
| ∫ x in A, divergence F x) |
| {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ) |
| (hφU : tsupport φ ⊆ U) (hφO : tsupport φ ⊆ O) |
| (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφb : ∀ x, φ x ∈ Icc 0 1) |
| {F : Cover → Cover} (hF : ContDiff ℝ (⊤ : ℕ∞) F) (hnF : ∀ x, ‖F x‖ ≤ 1) : |
| (∫ z in {z : BlowupNormalBundle | z.1 ∈ O}, φ z.1*⟪(z.2 : Cover),F z.1⟫ |
| ∂(ν : Measure BlowupNormalBundle)) = |
| ∫ y : Plane, φ (a+normalGraph n g y)*⟪graphAreaVector n g y,F (a+normalGraph n g y)⟫ := by |
| let H : Cover → Cover := fun x ↦ φ x • F x |
| have hH : ContDiff ℝ (⊤ : ℕ∞) H := hφ.smul hF |
| have hsH : HasCompactSupport H := hsφ.smul_right |
| have hHO : tsupport H ⊆ O := (tsupport_smul_subset_left _ _).trans hφO |
| have hnH (x) : ‖H x‖ ≤ 1 := by |
| rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg (hφb x).1] |
| exact (mul_le_mul (hφb x).2 (hnF x) (norm_nonneg _) zero_le_one).trans_eq (one_mul 1) |
| have he : (∫ z in {z : BlowupNormalBundle | z.1 ∈ O}, ⟪(z.2 : Cover),H z.1⟫ |
| ∂(ν : Measure BlowupNormalBundle)) = ∫ z, ⟪(z.2 : Cover),H z.1⟫ ∂(ν : Measure BlowupNormalBundle) := by |
| apply setIntegral_eq_integral_of_forall_compl_eq_zero |
| intro z hz |
| have hh : H z.1 = 0 := image_eq_zero_of_notMem_tsupport (fun h ↦ hz (hHO h)) |
| simp only [hh,inner_zero_right] |
| have hm : MeasurableSet {x : Cover | ⟪n,x-a⟫ < graphHeightExtension a n g x} := |
| measurableSet_lt (by fun_prop) (hg.continuous.measurable.comp |
| ((normalPlaneCoordinates n).measurable.comp (measurable_id.sub measurable_const))) |
| have hh := hflux H hH hsH hHO hnH |
| rw [setIntegral_divergence_congr_local hA hm hgraph |
| ((tsupport_smul_subset_left _ _).trans hφU), |
| setIntegral_graphSublevel_divergence a hn hg hc hH |
| ((tsupport_smul_subset_left _ _).trans hφR)] at hh |
| rw [← he] at hh |
| simpa only [H,inner_smul_right] using hh |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma graphSlopeNormal_contDiff {n : Cover} (hn : ‖n‖ = 1) : |
| ContDiff ℝ (⊤ : ℕ∞) (graphSlopeNormal n) := by |
| have hv : ContDiff ℝ (⊤ : ℕ∞) (fun p : Plane ↦ n-normalPlaneEmbedding n p) := |
| contDiff_const.sub (normalPlaneLinearIsometry n).contDiff |
| have hz (p : Plane) : n-normalPlaneEmbedding n p ≠ 0 := by |
| intro h |
| have hh := graphSlopeVector_norm_ge hn p |
| rw [h,norm_zero] at hh |
| norm_num at hh |
| exact ((hv.norm ℝ hz).inv (fun p ↦ norm_ne_zero_iff.mpr (hz p))).smul hv |
|
|
| def graphUnitNormalExtension (a n : Cover) (g : Plane → ℝ) (x : Cover) : Cover := |
| graphSlopeNormal n (gradient g (normalPlaneCoordinates n (x-a))) |
|
|
| lemma graphUnitNormalExtension_unit (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| (g : Plane → ℝ) (x : Cover) : ‖graphUnitNormalExtension a n g x‖ = 1 := |
| graphSlopeNormal_unit hn _ |
|
|
| lemma graphUnitNormalExtension_contDiff (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| {g : Plane → ℝ} (hg : ContDiff ℝ (⊤ : ℕ∞) g) : |
| ContDiff ℝ (⊤ : ℕ∞) (graphUnitNormalExtension a n g) := by |
| have hd := hg.fderiv_right (m := (⊤ : ℕ∞)) (by simp) |
| have hgrad : ContDiff ℝ (⊤ : ℕ∞) (gradient g) := |
| (InnerProductSpace.toDual ℝ Plane).symm.contDiff.comp hd |
| exact (graphSlopeNormal_contDiff hn).comp (hgrad.comp |
| ((normalPlaneCoordinates n).contDiff.comp (contDiff_id.sub contDiff_const))) |
|
|
| lemma graphUnitNormalExtension_graph (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| (g f : Plane → ℝ) (y : Plane) : |
| graphUnitNormalExtension a n f (a+normalGraph n g y) = graphSlopeNormal n (gradient f y) := by |
| unfold graphUnitNormalExtension |
| rw [add_sub_cancel_left,normalGraph, map_add,map_smul, |
| normalPlaneCoordinates_embedding,normalPlaneCoordinates_normal hn,smul_zero,add_zero] |
|
|
| lemma graphUnitNormalExtension_tendsto (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) {x : Cover} |
| (hc : ContinuousAt (fderiv ℝ g) (normalPlaneCoordinates n (x-a))) : |
| Tendsto (fun i ↦ graphUnitNormalExtension a n (graphSmooth (graphMollifier i) g) x) |
| atTop (𝓝 (graphUnitNormalExtension a n g x)) := |
| ((graphSlopeNormal_contDiff hn).continuous.tendsto _).comp (graphSmooth_gradient_tendsto hg hc) |
|
|
| lemma graphAreaVector_inner_unit {n : Cover} (hn : ‖n‖ = 1) (g : Plane → ℝ) (y : Plane) : |
| ⟪graphAreaVector n g y,graphSlopeNormal n (gradient g y)⟫ = ‖graphAreaVector n g y‖ := by |
| have hh : 0 < ‖graphAreaVector n g y‖ := lt_of_lt_of_le (by norm_num) (graphSlopeVector_norm_ge hn _) |
| rw [graphSlopeNormal,inner_smul_right,show n-normalPlaneEmbedding n (gradient g y)=graphAreaVector n g y from rfl, |
| real_inner_self_eq_norm_sq] |
| field_simp |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma graph_area_trace_mollified_normal_tendsto (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) |
| {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y) |
| {φ : Cover → ℝ} (hφ : Continuous φ) (hsφ : HasCompactSupport φ) |
| (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφ0 : ∀ x, 0 ≤ φ x) : |
| Tendsto (fun i ↦ ∫ y : Plane, φ (a+normalGraph n g y)* |
| ⟪graphAreaVector n g y,graphUnitNormalExtension a n (graphSmooth (graphMollifier i) g) (a+normalGraph n g y)⟫) |
| atTop (𝓝 (∫ y : Plane, φ (a+normalGraph n g y)*‖graphAreaVector n g y‖)) := by |
| have hi := graph_weighted_integrable a hn hg.continuous hφ hsφ hφR |
| (fun y hy ↦ (graphAreaVector_continuousAt n (hc y hy)).norm) |
| apply tendsto_integral_of_dominated_convergence |
| (fun y ↦ φ (a+normalGraph n g y)*‖graphAreaVector n g y‖) ?_ hi |
| · intro i |
| filter_upwards [] with y |
| rw [norm_mul,Real.norm_eq_abs,abs_of_nonneg (hφ0 _)] |
| apply mul_le_mul_of_nonneg_left _ (hφ0 _) |
| exact (norm_inner_le_norm _ _).trans_eq (by rw [graphUnitNormalExtension_unit a hn,mul_one]) |
| · filter_upwards [] with y |
| by_cases hy : y ∈ Metric.closedBall (0 : Plane) R |
| · have ht : Tendsto (fun i ↦ graphUnitNormalExtension a n (graphSmooth (graphMollifier i) g) |
| (a+normalGraph n g y)) atTop (𝓝 (graphSlopeNormal n (gradient g y))) := by |
| simp only [graphUnitNormalExtension_graph a hn] |
| exact ((graphSlopeNormal_contDiff hn).continuous.tendsto _).comp |
| (graphSmooth_gradient_tendsto hg (hc y hy)) |
| simpa only [graphAreaVector_inner_unit hn] using ((show Tendsto (fun _ : ℕ ↦ graphAreaVector n g y) atTop |
| (𝓝 (graphAreaVector n g y)) from tendsto_const_nhds).inner ht).const_mul (φ (a+normalGraph n g y)) |
| · simp only [graph_trace_zero_outside hn hφR _ hy,zero_mul] |
| exact tendsto_const_nhds |
| · intro i |
| exact (graph_weighted_continuous a hn hg.continuous hφ hφR (fun y hy ↦ |
| (graphAreaVector_continuousAt n (hc y hy)).inner |
| (graph_trace_continuous a n hg.continuous |
| (graphUnitNormalExtension_contDiff a hn (graphSmooth_contDiff _ hg)).continuous).continuousAt)).aestronglyMeasurable |
|
|
| theorem blowup_graph_patch_weighted_mass {A O U : Set Cover} (hA : MeasurableSet A) |
| (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) |
| {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y) |
| (hgraph : ∀ᵐ x : Cover ∂volume, x ∈ U → (x ∈ A ↔ ⟪n,x-a⟫ < graphHeightExtension a n g x)) |
| {ν : FiniteMeasure BlowupNormalBundle} |
| (heq : coverLocalPerimeter A O = (ν : Measure BlowupNormalBundle) {z | z.1 ∈ O}) |
| (hflux : ∀ F : Cover → Cover, ContDiff ℝ (⊤ : ℕ∞) F → HasCompactSupport F → |
| tsupport F ⊆ O → (∀ x, ‖F x‖ ≤ 1) → |
| (∫ z, ⟪(z.2 : Cover),F z.1⟫ ∂(ν : Measure BlowupNormalBundle)) = |
| ∫ x in A, divergence F x) |
| {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ) |
| (hφU : tsupport φ ⊆ U) (hφO : tsupport φ ⊆ O) |
| (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφb : ∀ x, φ x ∈ Icc 0 1) : |
| (∫ z in {z : BlowupNormalBundle | z.1 ∈ O}, φ z.1 ∂(ν : Measure BlowupNormalBundle)) = |
| ∫ y : Plane, φ (a+normalGraph n g y)*‖graphAreaVector n g y‖ := by |
| let μ := (ν : Measure BlowupNormalBundle).restrict {z | z.1 ∈ O} |
| have hiφ : Integrable (fun z : BlowupNormalBundle ↦ φ z.1) μ := |
| Integrable.of_bound (hφ.continuous.comp continuous_fst).aestronglyMeasurable 1 |
| (ae_of_all μ fun z ↦ by rw [Real.norm_eq_abs,abs_of_nonneg (hφb z.1).1]; exact (hφb z.1).2) |
| have hiArea := graph_weighted_integrable a hn hg.continuous hφ.continuous hsφ hφR |
| (fun y hy ↦ (graphAreaVector_continuousAt n (hc y hy)).norm) |
| have hiF (F : Cover → Cover) (hF : ContDiff ℝ (⊤ : ℕ∞) F) (hnF : ∀ x, ‖F x‖ ≤ 1) : |
| Integrable (fun z : BlowupNormalBundle ↦ φ z.1*⟪(z.2 : Cover),F z.1⟫) μ := by |
| simpa only [mul_comm] using hiφ.bdd_mul |
| (show Continuous (fun z : BlowupNormalBundle ↦ ⟪(z.2 : Cover),F z.1⟫) from by fun_prop).aestronglyMeasurable |
| (ae_of_all μ fun z ↦ blowup_test_inner_bound hnF z) |
| have hfield (F : Cover → Cover) (hF : ContDiff ℝ (⊤ : ℕ∞) F) (hnF : ∀ x, ‖F x‖ ≤ 1) : |
| (∫ z, φ z.1*⟪(z.2 : Cover),F z.1⟫ ∂μ) ≤ |
| ∫ y : Plane, φ (a+normalGraph n g y)*‖graphAreaVector n g y‖ := by |
| rw [show (∫ z, φ z.1*⟪(z.2 : Cover),F z.1⟫ ∂μ) = _ from |
| blowup_graph_patch_weighted_flux hA a hn hg hc hgraph hflux hφ hsφ hφU hφO hφR hφb hF hnF] |
| apply integral_mono (graph_weighted_integrable a hn hg.continuous hφ.continuous hsφ hφR |
| (fun y hy ↦ (graphAreaVector_continuousAt n (hc y hy)).inner |
| (graph_trace_continuous a n hg.continuous hF.continuous).continuousAt)) hiArea |
| intro y |
| apply mul_le_mul_of_nonneg_left _ (hφb _).1 |
| exact (real_inner_le_norm _ _).trans (mul_le_of_le_one_right (norm_nonneg _) (hnF _)) |
| refine le_antisymm ?_ ?_ |
| · exact blowupNormal_weighted_saturation heq hflux hiφ (by norm_num : (0 : ℝ)<1) |
| (fun z ↦ (hφb z.1).2) (fun F hF _ _ hnF ↦ hfield F hF hnF) |
| · apply le_of_tendsto (graph_area_trace_mollified_normal_tendsto a hn hg hc |
| hφ.continuous hsφ hφR (fun x ↦ (hφb x).1)) |
| apply Eventually.of_forall |
| intro i |
| let F := graphUnitNormalExtension a n (graphSmooth (graphMollifier i) g) |
| have hF : ContDiff ℝ (⊤ : ℕ∞) F := graphUnitNormalExtension_contDiff a hn (graphSmooth_contDiff _ hg) |
| have hnF (x : Cover) : ‖F x‖ ≤ 1 := (graphUnitNormalExtension_unit a hn _ x).le |
| rw [← blowup_graph_patch_weighted_flux hA a hn hg hc hgraph hflux hφ hsφ hφU hφO hφR hφb hF hnF] |
| apply integral_mono (hiF F hF hnF) hiφ |
| intro z |
| apply mul_le_of_le_one_right (hφb _).1 |
| exact (real_inner_le_norm (z.2 : Cover) (F z.1)).trans (by rw [z.2.property,one_mul]; exact hnF _) |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma normalPlaneCoordinates_norm_le (n x : Cover) : ‖normalPlaneCoordinates n x‖ ≤ ‖x‖ := by |
| simpa only [map_zero,dist_zero_right,NNReal.coe_one,one_mul] using |
| (normalPlaneCoordinates_lipschitz n).dist_le_mul x 0 |
|
|
| lemma graph_weighted_normal_tendsto (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) |
| {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y) |
| {φ : Cover → ℝ} (hφR : tsupport φ ⊆ Metric.closedBall a R) (z : BlowupNormalBundle) : |
| Tendsto (fun i ↦ φ z.1*⟪(z.2 : Cover),graphUnitNormalExtension a n |
| (graphSmooth (graphMollifier i) g) z.1⟫) atTop |
| (𝓝 (φ z.1*⟪(z.2 : Cover),graphUnitNormalExtension a n g z.1⟫)) := by |
| by_cases hz : z.1 ∈ tsupport φ |
| · have hcoord : normalPlaneCoordinates n (z.1-a) ∈ Metric.closedBall (0 : Plane) R := by |
| simp only [Metric.mem_closedBall,dist_zero_right] |
| exact (normalPlaneCoordinates_norm_le n _).trans (by simpa [Metric.mem_closedBall,dist_eq_norm] using hφR hz) |
| exact ((show Tendsto (fun _ : ℕ ↦ (z.2 : Cover)) atTop (𝓝 (z.2 : Cover)) from |
| tendsto_const_nhds).inner (graphUnitNormalExtension_tendsto a hn hg (hc _ hcoord))).const_mul _ |
| · simp only [image_eq_zero_of_notMem_tsupport hz,zero_mul] |
| exact tendsto_const_nhds |
|
|
| theorem blowup_graph_patch_normal_ae {A O U : Set Cover} (hA : MeasurableSet A) |
| (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) |
| {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y) |
| (hgraph : ∀ᵐ x : Cover ∂volume, x ∈ U → (x ∈ A ↔ ⟪n,x-a⟫ < graphHeightExtension a n g x)) |
| {ν : FiniteMeasure BlowupNormalBundle} |
| (heq : coverLocalPerimeter A O = (ν : Measure BlowupNormalBundle) {z | z.1 ∈ O}) |
| (hflux : ∀ F : Cover → Cover, ContDiff ℝ (⊤ : ℕ∞) F → HasCompactSupport F → |
| tsupport F ⊆ O → (∀ x, ‖F x‖ ≤ 1) → |
| (∫ z, ⟪(z.2 : Cover),F z.1⟫ ∂(ν : Measure BlowupNormalBundle)) = ∫ x in A, divergence F x) |
| {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ) |
| (hφU : tsupport φ ⊆ U) (hφO : tsupport φ ⊆ O) |
| (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφb : ∀ x, φ x ∈ Icc 0 1) : |
| ∀ᵐ z ∂(ν : Measure BlowupNormalBundle).restrict {z | z.1 ∈ O}, |
| 0 < φ z.1 → (z.2 : Cover)=graphUnitNormalExtension a n g z.1 := by |
| let μ := (ν : Measure BlowupNormalBundle).restrict {z | z.1 ∈ O} |
| let F : ℕ → Cover → Cover := fun i ↦ graphUnitNormalExtension a n (graphSmooth (graphMollifier i) g) |
| let f : ℕ → BlowupNormalBundle → ℝ := fun i z ↦ φ z.1*⟪(z.2 : Cover),F i z.1⟫ |
| let f₀ : BlowupNormalBundle → ℝ := fun z ↦ φ z.1*⟪(z.2 : Cover),graphUnitNormalExtension a n g z.1⟫ |
| have hF (i) : ContDiff ℝ (⊤ : ℕ∞) (F i) := graphUnitNormalExtension_contDiff a hn (graphSmooth_contDiff _ hg) |
| have hFn (i) (x) : ‖F i x‖ ≤ 1 := (graphUnitNormalExtension_unit a hn _ x).le |
| have hf (i) : AEStronglyMeasurable (f i) μ := |
| (show Continuous (f i) from (hφ.continuous.comp continuous_fst).mul |
| (continuous_subtype_val.comp continuous_snd |>.inner ((hF i).continuous.comp continuous_fst))).aestronglyMeasurable |
| have ht : ∀ᵐ z ∂μ, Tendsto (fun i ↦ f i z) atTop (𝓝 (f₀ z)) := |
| ae_of_all μ (graph_weighted_normal_tendsto a hn hg hc hφR) |
| have hb (i) (z : BlowupNormalBundle) : ‖f i z‖ ≤ 1 := by |
| dsimp only [f] |
| rw [norm_mul,Real.norm_eq_abs,abs_of_nonneg (hφb _).1] |
| exact (mul_le_mul (hφb _).2 (blowup_test_inner_bound (hFn i) z) (abs_nonneg _) zero_le_one).trans_eq (one_mul 1) |
| have hb₀ (z : BlowupNormalBundle) : ‖f₀ z‖ ≤ 1 := by |
| dsimp only [f₀] |
| rw [norm_mul,Real.norm_eq_abs,abs_of_nonneg (hφb _).1] |
| exact (mul_le_mul (hφb _).2 (blowup_test_inner_bound |
| (fun x ↦ (graphUnitNormalExtension_unit a hn _ x).le) z) (abs_nonneg _) zero_le_one).trans_eq (one_mul 1) |
| have hi₀ : Integrable f₀ μ := Integrable.of_bound |
| (aestronglyMeasurable_of_tendsto_ae atTop hf ht) 1 (ae_of_all μ hb₀) |
| have hilim : Tendsto (fun i ↦ ∫ z, f i z ∂μ) atTop (𝓝 (∫ z, f₀ z ∂μ)) := |
| tendsto_integral_of_dominated_convergence (fun _ ↦ (1:ℝ)) hf |
| (integrable_const 1) (fun i ↦ ae_of_all μ (hb i)) ht |
| have heint : (∫ z, f₀ z ∂μ) = ∫ z, φ z.1 ∂μ := by |
| apply tendsto_nhds_unique hilim |
| rw [show (∫ z, φ z.1 ∂μ) = _ from |
| blowup_graph_patch_weighted_mass hA a hn hg hc hgraph heq hflux hφ hsφ hφU hφO hφR hφb] |
| have hh := graph_area_trace_mollified_normal_tendsto a hn hg hc hφ.continuous hsφ hφR |
| (fun x ↦ (hφb x).1) |
| convert hh using 1 |
| funext i |
| exact blowup_graph_patch_weighted_flux hA a hn hg hc hgraph hflux hφ hsφ hφU hφO hφR hφb (hF i) (hFn i) |
| have hiφ : Integrable (fun z : BlowupNormalBundle ↦ φ z.1) μ := |
| Integrable.of_bound (hφ.continuous.comp continuous_fst).aestronglyMeasurable 1 |
| (ae_of_all μ fun z ↦ by rw [Real.norm_eq_abs,abs_of_nonneg (hφb _).1]; exact (hφb _).2) |
| have hinon (z : BlowupNormalBundle) : 0 ≤ φ z.1-f₀ z := by |
| have hh : ⟪(z.2 : Cover),graphUnitNormalExtension a n g z.1⟫ ≤ 1 := by |
| exact (real_inner_le_norm _ _).trans_eq (by rw [z.2.property,graphUnitNormalExtension_unit a hn,one_mul]) |
| exact sub_nonneg.mpr (mul_le_of_le_one_right (hφb _).1 hh) |
| have hezero : (∫ z, (φ z.1-f₀ z) ∂μ)=0 := by rw [integral_sub hiφ hi₀,heint,sub_self] |
| have hae := (integral_eq_zero_iff_of_nonneg_ae (ae_of_all μ hinon) (hiφ.sub hi₀)).mp hezero |
| filter_upwards [hae] with z hz |
| intro hzφ |
| have hi : ⟪(z.2 : Cover),graphUnitNormalExtension a n g z.1⟫ = 1 := by |
| have hh : φ z.1*(1-⟪(z.2 : Cover),graphUnitNormalExtension a n g z.1⟫)=0 := by |
| simpa only [mul_sub,mul_one,f₀,Pi.zero_apply] using hz |
| exact (sub_eq_zero.mp ((mul_eq_zero.mp hh).resolve_left hzφ.ne')).symm |
| exact (inner_eq_one_iff_of_norm_eq_one z.2.property (graphUnitNormalExtension_unit a hn g z.1)).mp hi |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem blowup_graph_patch_signed_mass {A O U : Set Cover} (hA : MeasurableSet A) |
| (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) |
| {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y) |
| (hgraph : ∀ᵐ x : Cover ∂volume, x ∈ U → (x ∈ A ↔ ⟪n,x-a⟫ < graphHeightExtension a n g x)) |
| {ν : FiniteMeasure BlowupNormalBundle} |
| (heq : coverLocalPerimeter A O = (ν : Measure BlowupNormalBundle) {z | z.1 ∈ O}) |
| (hflux : ∀ F : Cover → Cover, ContDiff ℝ (⊤ : ℕ∞) F → HasCompactSupport F → |
| tsupport F ⊆ O → (∀ x, ‖F x‖ ≤ 1) → |
| (∫ z, ⟪(z.2 : Cover),F z.1⟫ ∂(ν : Measure BlowupNormalBundle)) = ∫ x in A, divergence F x) |
| {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ) |
| (hφU : tsupport φ ⊆ U) (hφO : tsupport φ ⊆ O) |
| (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφb : ∀ x, φ x ∈ Icc 0 1) |
| {f : Cover → ℝ} (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hfb : ∀ x, |f x| ≤ 1) : |
| (∫ z in {z : BlowupNormalBundle | z.1 ∈ O}, φ z.1*f z.1 ∂(ν : Measure BlowupNormalBundle)) = |
| ∫ y : Plane, φ (a+normalGraph n g y)*f (a+normalGraph n g y)*‖graphAreaVector n g y‖ := by |
| let μ := (ν : Measure BlowupNormalBundle).restrict {z | z.1 ∈ O} |
| let ψ : Cover → ℝ := fun x ↦ φ x*((1+f x)/2) |
| have hψ : ContDiff ℝ (⊤ : ℕ∞) ψ := hφ.mul ((contDiff_const.add hf).div_const 2) |
| have hsψ : tsupport ψ ⊆ tsupport φ := tsupport_mul_subset_left |
| have hψb (x : Cover) : ψ x ∈ Icc 0 1 := by |
| have hx := abs_le.mp (hfb x) |
| have hhalf : (1+f x)/2 ∈ Icc 0 1 := ⟨by linarith,by linarith⟩ |
| exact ⟨mul_nonneg (hφb x).1 hhalf.1, |
| (mul_le_mul (hφb x).2 hhalf.2 hhalf.1 zero_le_one).trans_eq (one_mul 1)⟩ |
| have hsψ' : HasCompactSupport ψ := hsφ.of_isClosed_subset (isClosed_tsupport _) hsψ |
| have hmψ := blowup_graph_patch_weighted_mass hA a hn hg hc hgraph heq hflux hψ hsψ' |
| (hsψ.trans hφU) (hsψ.trans hφO) (hsψ.trans hφR) hψb |
| have hmφ := blowup_graph_patch_weighted_mass hA a hn hg hc hgraph heq hflux hφ hsφ hφU hφO hφR hφb |
| have hI (w : Cover → ℝ) (hw : Continuous w) (hb : ∀ x, w x ∈ Icc 0 1) : |
| Integrable (fun z : BlowupNormalBundle ↦ w z.1) μ := |
| Integrable.of_bound (hw.comp continuous_fst).aestronglyMeasurable 1 |
| (ae_of_all μ fun z ↦ by rw [Real.norm_eq_abs,abs_of_nonneg (hb _).1]; exact (hb _).2) |
| have hGφ := graph_weighted_integrable a hn hg.continuous hφ.continuous hsφ hφR |
| (fun y hy ↦ (graphAreaVector_continuousAt n (hc y hy)).norm) |
| have hGψ := graph_weighted_integrable a hn hg.continuous hψ.continuous hsψ' (hsψ.trans hφR) |
| (fun y hy ↦ (graphAreaVector_continuousAt n (hc y hy)).norm) |
| calc |
| (∫ z, φ z.1*f z.1 ∂μ) = ∫ z, (2*ψ z.1-φ z.1) ∂μ := by |
| congr 1; funext z; dsimp only [ψ]; ring |
| _ = 2*(∫ z, ψ z.1 ∂μ)-(∫ z, φ z.1 ∂μ) := by |
| rw [integral_sub ((hI ψ hψ.continuous hψb).const_mul 2) (hI φ hφ.continuous hφb),integral_const_mul] |
| _ = 2*(∫ y : Plane, ψ (a+normalGraph n g y)*‖graphAreaVector n g y‖)- |
| (∫ y : Plane, φ (a+normalGraph n g y)*‖graphAreaVector n g y‖) := by rw [hmψ,hmφ] |
| _ = ∫ y : Plane, (2*(ψ (a+normalGraph n g y)*‖graphAreaVector n g y‖)- |
| φ (a+normalGraph n g y)*‖graphAreaVector n g y‖) := by |
| rw [integral_sub (hGψ.const_mul 2) hGφ,integral_const_mul] |
| _ = _ := by congr 1; funext y; dsimp only [ψ]; ring |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma weighted_tendsto_on_tsupport {X : Type*} [TopologicalSpace X] |
| {φ : X → ℝ} {f : ℕ → X → ℝ} {g : X → ℝ} |
| (h : ∀ x ∈ tsupport φ, Tendsto (fun i ↦ f i x) atTop (𝓝 (g x))) (x : X) : |
| Tendsto (fun i ↦ φ x*f i x) atTop (𝓝 (φ x*g x)) := by |
| by_cases hx : x ∈ tsupport φ |
| · exact (h x hx).const_mul _ |
| · simp only [image_eq_zero_of_notMem_tsupport hx,zero_mul] |
| exact tendsto_const_nhds |
|
|
| theorem blowup_graph_patch_limit_mass {A O U : Set Cover} (hA : MeasurableSet A) |
| (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) |
| {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y) |
| (hgraph : ∀ᵐ x : Cover ∂volume, x ∈ U → (x ∈ A ↔ ⟪n,x-a⟫ < graphHeightExtension a n g x)) |
| {ν : FiniteMeasure BlowupNormalBundle} |
| (heq : coverLocalPerimeter A O = (ν : Measure BlowupNormalBundle) {z | z.1 ∈ O}) |
| (hflux : ∀ F : Cover → Cover, ContDiff ℝ (⊤ : ℕ∞) F → HasCompactSupport F → |
| tsupport F ⊆ O → (∀ x, ‖F x‖ ≤ 1) → |
| (∫ z, ⟪(z.2 : Cover),F z.1⟫ ∂(ν : Measure BlowupNormalBundle)) = ∫ x in A, divergence F x) |
| {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ) |
| (hφU : tsupport φ ⊆ U) (hφO : tsupport φ ⊆ O) |
| (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφb : ∀ x, φ x ∈ Icc 0 1) |
| {f : ℕ → Cover → ℝ} {f₀ : Cover → ℝ} |
| (hf : ∀ i, ContDiff ℝ (⊤ : ℕ∞) (f i)) (hfb : ∀ i x, |f i x| ≤ 1) |
| (hlim : ∀ x ∈ tsupport φ, Tendsto (fun i ↦ f i x) atTop (𝓝 (f₀ x))) : |
| (∫ z in {z : BlowupNormalBundle | z.1 ∈ O}, φ z.1*f₀ z.1 ∂(ν : Measure BlowupNormalBundle)) = |
| ∫ y : Plane, φ (a+normalGraph n g y)*f₀ (a+normalGraph n g y)*‖graphAreaVector n g y‖ := by |
| let μ := (ν : Measure BlowupNormalBundle).restrict {z | z.1 ∈ O} |
| have hleft : Tendsto (fun i ↦ ∫ z, φ z.1*f i z.1 ∂μ) atTop (𝓝 (∫ z, φ z.1*f₀ z.1 ∂μ)) := by |
| apply tendsto_integral_of_dominated_convergence (fun _ ↦ (1:ℝ)) |
| · intro i |
| exact ((hφ.continuous.mul (hf i).continuous).comp continuous_fst).aestronglyMeasurable |
| · exact integrable_const 1 |
| · intro i |
| filter_upwards [] with z |
| rw [norm_mul,Real.norm_eq_abs,abs_of_nonneg (hφb _).1,Real.norm_eq_abs] |
| exact (mul_le_mul (hφb _).2 (hfb i _) (abs_nonneg _) zero_le_one).trans_eq (one_mul 1) |
| · exact ae_of_all μ fun z ↦ weighted_tendsto_on_tsupport hlim z.1 |
| have hright : Tendsto (fun i ↦ ∫ y : Plane, |
| φ (a+normalGraph n g y)*f i (a+normalGraph n g y)*‖graphAreaVector n g y‖) atTop |
| (𝓝 (∫ y : Plane, φ (a+normalGraph n g y)*f₀ (a+normalGraph n g y)*‖graphAreaVector n g y‖)) := by |
| apply tendsto_integral_of_dominated_convergence (fun y ↦ φ (a+normalGraph n g y)*‖graphAreaVector n g y‖) |
| · intro i |
| have hh := graph_weighted_continuous a hn hg.continuous hφ.continuous hφR |
| (fun y hy ↦ (graphAreaVector_continuousAt n (hc y hy)).norm) |
| have hmeas : AEStronglyMeasurable (fun y : Plane ↦ |
| (φ (a+normalGraph n g y)*‖graphAreaVector n g y‖)*f i (a+normalGraph n g y)) |
| (volume : Measure Plane) := |
| (hh.mul (graph_trace_continuous a n hg.continuous (hf i).continuous)).aestronglyMeasurable |
| simpa only [mul_assoc,mul_comm,mul_left_comm] using hmeas |
| · exact graph_weighted_integrable a hn hg.continuous hφ.continuous hsφ hφR |
| (fun y hy ↦ (graphAreaVector_continuousAt n (hc y hy)).norm) |
| · intro i |
| filter_upwards [] with y |
| rw [norm_mul,norm_mul,Real.norm_eq_abs,abs_of_nonneg (hφb _).1,Real.norm_eq_abs, |
| norm_norm] |
| exact mul_le_mul_of_nonneg_right (mul_le_of_le_one_right (hφb _).1 (hfb i _)) (norm_nonneg _) |
| · filter_upwards [] with y |
| exact (weighted_tendsto_on_tsupport hlim _).mul_const _ |
| apply tendsto_nhds_unique hleft |
| convert hright using 1 |
| funext i |
| exact blowup_graph_patch_signed_mass hA a hn hg hc hgraph heq hflux hφ hsφ hφU hφO hφR hφb (hf i) (hfb i) |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma contDiff_divergence {F : Cover → Cover} (hF : ContDiff ℝ (⊤ : ℕ∞) F) : |
| ContDiff ℝ (⊤ : ℕ∞) (divergence F) := by |
| have hd := hF.fderiv_right (m := (⊤ : ℕ∞)) (by simp) |
| unfold divergence |
| apply ContDiff.sum |
| intro i _ |
| exact (EuclideanSpace.proj i).contDiff.comp (hd.clm_apply contDiff_const) |
|
|
| lemma contDiff_tangentialDivergence_normal {F N : Cover → Cover} |
| (hF : ContDiff ℝ (⊤ : ℕ∞) F) (hN : ContDiff ℝ (⊤ : ℕ∞) N) : |
| ContDiff ℝ (⊤ : ℕ∞) (fun x ↦ tangentialDivergence F x (N x)) := |
| (contDiff_divergence hF).sub (hN.inner ℝ ((hF.fderiv_right (m := (⊤ : ℕ∞)) (by simp)).clm_apply hN)) |
|
|
| lemma continuous_tangentialDivergence_normal (F : Cover → Cover) (x : Cover) : |
| Continuous (fun v : Cover ↦ tangentialDivergence F x v) := |
| continuous_const.sub (continuous_id.inner ((fderiv ℝ F x).continuous.comp continuous_id)) |
|
|
| theorem blowup_C1graph_stationary {A O U : Set Cover} (hA : MeasurableSet A) |
| (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) |
| {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y) |
| (hgraph : ∀ᵐ x : Cover ∂volume, x ∈ U → (x ∈ A ↔ ⟪n,x-a⟫ < graphHeightExtension a n g x)) |
| {ν : FiniteMeasure BlowupNormalBundle} |
| (heq : coverLocalPerimeter A O = (ν : Measure BlowupNormalBundle) {z | z.1 ∈ O}) |
| (hflux : ∀ F : Cover → Cover, ContDiff ℝ (⊤ : ℕ∞) F → HasCompactSupport F → |
| tsupport F ⊆ O → (∀ x, ‖F x‖ ≤ 1) → |
| (∫ z, ⟪(z.2 : Cover),F z.1⟫ ∂(ν : Measure BlowupNormalBundle)) = ∫ x in A, divergence F x) |
| {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ) |
| (hφU : tsupport φ ⊆ U) (hφO : tsupport φ ⊆ O) |
| (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφb : ∀ x, φ x ∈ Icc 0 1) |
| {F : Cover → Cover} (hF : ContDiff ℝ (⊤ : ℕ∞) F) (hsF : HasCompactSupport F) |
| (hφF : ∀ x ∈ tsupport F, φ x = 1) |
| (hstat : (∫ z, tangentialDivergence F z.1 z.2 ∂(ν : Measure BlowupNormalBundle))=0) : |
| (∫ y : Plane, tangentialDivergence F (a+normalGraph n g y) |
| (graphSlopeNormal n (gradient g y))*‖graphAreaVector n g y‖)=0 := by |
| let μ := (ν : Measure BlowupNormalBundle).restrict {z | z.1 ∈ O} |
| let M : ℝ := ‖blowupTangentTest hF hsF‖+1 |
| have hM : 0 < M := by dsimp only [M]; positivity |
| let N : ℕ → Cover → Cover := fun i ↦ graphUnitNormalExtension a n (graphSmooth (graphMollifier i) g) |
| let N₀ := graphUnitNormalExtension a n g |
| let f : ℕ → Cover → ℝ := fun i x ↦ M⁻¹*tangentialDivergence F x (N i x) |
| let f₀ : Cover → ℝ := fun x ↦ M⁻¹*tangentialDivergence F x (N₀ x) |
| have hN (i) : ContDiff ℝ (⊤ : ℕ∞) (N i) := graphUnitNormalExtension_contDiff a hn (graphSmooth_contDiff _ hg) |
| have hf (i) : ContDiff ℝ (⊤ : ℕ∞) (f i) := |
| contDiff_const.mul (contDiff_tangentialDivergence_normal hF (hN i)) |
| have hfb (i) (x) : |f i x| ≤ 1 := by |
| have hh := (blowupTangentTest hF hsF).norm_coe_le_norm |
| (x,⟨N i x,graphUnitNormalExtension_unit a hn _ x⟩) |
| change ‖tangentialDivergence F x (N i x)‖ ≤ ‖blowupTangentTest hF hsF‖ at hh |
| dsimp only [f] |
| rw [abs_mul,abs_of_pos (inv_pos.mpr hM)] |
| apply (mul_le_mul_of_nonneg_left (hh.trans (le_add_of_nonneg_right zero_le_one)) (inv_nonneg.mpr hM.le)).trans_eq |
| exact inv_mul_cancel₀ hM.ne' |
| have hlim (x : Cover) (hx : x ∈ tsupport φ) : Tendsto (fun i ↦ f i x) atTop (𝓝 (f₀ x)) := by |
| have hcoord : normalPlaneCoordinates n (x-a) ∈ Metric.closedBall (0 : Plane) R := by |
| simp only [Metric.mem_closedBall,dist_zero_right] |
| exact (normalPlaneCoordinates_norm_le n _).trans (by simpa [Metric.mem_closedBall,dist_eq_norm] using hφR hx) |
| exact (((continuous_tangentialDivergence_normal F x).tendsto _).comp |
| (graphUnitNormalExtension_tendsto a hn hg (hc _ hcoord))).const_mul M⁻¹ |
| have he := blowup_graph_patch_limit_mass hA a hn hg hc hgraph heq hflux hφ hsφ hφU hφO hφR hφb hf hfb hlim |
| have hae := blowup_graph_patch_normal_ae hA a hn hg hc hgraph heq hflux hφ hsφ hφU hφO hφR hφb |
| have hw (x v) : φ x*tangentialDivergence F x v=tangentialDivergence F x v := by |
| by_cases hx : x ∈ tsupport F |
| · rw [hφF x hx,one_mul] |
| · rw [tangentialDivergence_zero_of_notMem_tsupport hx,mul_zero] |
| have hFO : tsupport F ⊆ O := by |
| intro x hx |
| apply hφO |
| exact subset_tsupport _ (by rw [Function.mem_support,hφF x hx]; exact one_ne_zero) |
| have hleft : (∫ z, φ z.1*f₀ z.1 ∂μ)=0 := by |
| calc |
| (∫ z, φ z.1*f₀ z.1 ∂μ) = ∫ z, M⁻¹*tangentialDivergence F z.1 z.2 ∂μ := by |
| apply integral_congr_ae |
| filter_upwards [hae] with z hz |
| by_cases hzφ : 0 < φ z.1 |
| · dsimp only [f₀,N₀] |
| rw [← hz hzφ,← mul_assoc,mul_comm (φ z.1) M⁻¹,mul_assoc,hw] |
| · have hφz : φ z.1=0 := le_antisymm (le_of_not_gt hzφ) (hφb _).1 |
| dsimp only [f₀] |
| rw [hφz,zero_mul,← hw z.1 (z.2 : Cover),hφz,zero_mul,mul_zero] |
| _ = M⁻¹*(∫ z, tangentialDivergence F z.1 z.2 ∂μ) := integral_const_mul _ _ |
| _ = M⁻¹*(∫ z, tangentialDivergence F z.1 z.2 ∂(ν : Measure BlowupNormalBundle)) := by |
| congr 1 |
| apply setIntegral_eq_integral_of_forall_compl_eq_zero |
| intro z hz |
| exact tangentialDivergence_zero_of_notMem_tsupport (fun h ↦ hz (hFO h)) _ |
| _ = 0 := by rw [hstat,mul_zero] |
| have hright : (∫ y : Plane, φ (a+normalGraph n g y)*f₀ (a+normalGraph n g y)*‖graphAreaVector n g y‖) = |
| M⁻¹*(∫ y : Plane, tangentialDivergence F (a+normalGraph n g y) |
| (graphSlopeNormal n (gradient g y))*‖graphAreaVector n g y‖) := by |
| rw [← integral_const_mul] |
| congr 1 |
| funext y |
| dsimp only [f₀,N₀] |
| rw [graphUnitNormalExtension_graph a hn,← mul_assoc (φ _) M⁻¹,mul_comm (φ _) M⁻¹,mul_assoc M⁻¹,hw] |
| ring |
| rw [hright,hleft] at he |
| exact (mul_eq_zero.mp he.symm).resolve_left (inv_ne_zero hM.ne') |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| end CubicTorus |
| end |
| end |
| end |
|
|
| end OAI |
|
|
