| import Mathlib |
| import OAI.Geometry.CubicTorus.Estimates.BVGraphRescaling |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter Metric |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma columnGraphMismatch_disagreement_volume_le {U : Set Torus} (hU : IsOpen U) |
| (σ : FiniteMeasure Torus) {R c₀ : ℝ} (hR : 0 < R) (hc₀ : 0 < c₀) |
| (hl : ∀ x ∈ frontier U, ∀ ρ : ℝ, 0 < ρ → ρ ≤ R → |
| c₀*ρ^2 ≤ (σ : Measure Torus).real (Metric.ball x ρ)) |
| (a : Torus) {n : Cover} (hn : ‖n‖ = 1) {r α η : ℝ} |
| (hr : 0 < r) (hrs : r < 1/2) (hα : α < 1/4) (hη : η ≤ 1/2) |
| (htrap : normalCap a n r (α*r) ⊆ (closure U)ᶜ ∧ normalCap a (-n) r (α*r) ⊆ U) |
| (S : Set Torus) (g : Plane → ℝ) |
| (hS : ∀ x ∈ S, g (normalPlaneCoordinates n (shortestLift (x-a))) = ⟪n, shortestLift (x-a)⟫) |
| (hcone : ∀ c ∈ S, ∀ x ∈ frontier U, dist x c ≤ r → |
| |⟪n, shortestLift (x-c)⟫| ≤ η*dist x c) : |
| (volume : Measure Plane) {y ∈ Metric.ball (0 : Plane) (r/4) | columnGraphMismatch U a n r g y ≠ 0} ≤ |
| ((planeAreaFactor : ℝ≥0∞)*ENNReal.ofReal (64/c₀)) * |
| (σ : Measure Torus) ((frontier U ∩ Metric.ball a (r/2)) \ S) := by |
| let Q : Torus → Plane := fun x ↦ normalPlaneCoordinates n (shortestLift (x-a)) |
| have hs : {y ∈ Metric.ball (0 : Plane) (r/4) | columnGraphMismatch U a n r g y ≠ 0} ⊆ |
| Q '' ((frontier U ∩ Metric.ball a (r/2)) \ S) := by |
| intro y hy |
| obtain ⟨x,hx,hxy⟩ := trapped_boundary_projects_disk hU a hn hr hrs hα htrap hy.1 |
| have hxS : x ∉ S := by |
| intro hm |
| apply hy.2 |
| rw [← hxy] |
| exact columnGraphMismatch_eq_zero_good hU hn hr hrs hα hη htrap hx (hcone x hm) |
| (by simpa only [hxy] using hy.1) g (hS x hm) |
| exact ⟨x, ⟨hx,hxS⟩, hxy⟩ |
| apply (measure_mono hs).trans |
| apply plane_projection_volume_le_perimeter σ (frontier U) hR hc₀ hl n a _ (fun x hx ↦ hx.1.1) |
| intro x hx |
| exact (Metric.ball_subset_ball (by linarith : r/2 ≤ 1/4)) hx.1.2 |
|
|
| theorem columnGraphMismatch_L1_le {U : Set Torus} (hU : IsOpen U) |
| (σ : FiniteMeasure Torus) {R c₀ : ℝ} (hR : 0 < R) (hc₀ : 0 < c₀) |
| (hl : ∀ x ∈ frontier U, ∀ ρ : ℝ, 0 < ρ → ρ ≤ R → |
| c₀*ρ^2 ≤ (σ : Measure Torus).real (Metric.ball x ρ)) |
| (a : Torus) {n : Cover} (hn : ‖n‖ = 1) {r α η : ℝ} |
| (hr : 0 < r) (hrs : r < 1/2) (hα0 : 0 ≤ α) (hα : α < 1/4) (hη : η ≤ 1/2) |
| (htrap : normalCap a n r (α*r) ⊆ (closure U)ᶜ ∧ normalCap a (-n) r (α*r) ⊆ U) |
| (S : Set Torus) (g : Plane → ℝ) (hg : Measurable g) |
| (hheight : ∀ y ∈ Metric.ball (0 : Plane) (r/4), |g y| ≤ α*r) |
| (hS : ∀ x ∈ S, g (normalPlaneCoordinates n (shortestLift (x-a))) = ⟪n, shortestLift (x-a)⟫) |
| (hcone : ∀ c ∈ S, ∀ x ∈ frontier U, dist x c ≤ r → |
| |⟪n, shortestLift (x-c)⟫| ≤ η*dist x c) : |
| (∫⁻ y in Metric.ball (0 : Plane) (r/4), ENNReal.ofReal (columnGraphMismatch U a n r g y)) ≤ |
| ENNReal.ofReal (2*α*r) * (((planeAreaFactor : ℝ≥0∞)*ENNReal.ofReal (64/c₀)) * |
| (σ : Measure Torus) ((frontier U ∩ Metric.ball a (r/2)) \ S)) := by |
| let D : Set Plane := {y ∈ Metric.ball (0 : Plane) (r/4) | columnGraphMismatch U a n r g y ≠ 0} |
| have hD : MeasurableSet D := measurableSet_ball.inter |
| (measurableSet_eq_fun (measurable_columnGraphMismatch hU.measurableSet a n r hg) measurable_const).compl |
| have hb : ∀ y ∈ Metric.ball (0 : Plane) (r/4), ENNReal.ofReal (columnGraphMismatch U a n r g y) ≤ |
| D.indicator (fun _ ↦ ENNReal.ofReal (2*α*r)) y := by |
| intro y hy |
| by_cases he : columnGraphMismatch U a n r g y = 0 |
| · simp [he] |
| · rw [indicator_of_mem (show y ∈ D from ⟨hy,he⟩)] |
| exact ENNReal.ofReal_le_ofReal (columnGraphMismatch_trapped_le hU.measurableSet hn hr hrs hα0 hα htrap g hy (hheight y hy)) |
| calc |
| _ ≤ ∫⁻ y in Metric.ball (0 : Plane) (r/4), D.indicator (fun _ ↦ ENNReal.ofReal (2*α*r)) y := |
| setLIntegral_mono' measurableSet_ball hb |
| _ = ENNReal.ofReal (2*α*r) * (volume : Measure Plane) D := by |
| rw [lintegral_indicator hD, lintegral_const, Measure.restrict_restrict hD] |
| simp only [Measure.restrict_apply_univ, inter_eq_left.mpr (show D ⊆ Metric.ball (0 : Plane) (r/4) from fun _ h ↦ h.1)] |
| _ ≤ _ := mul_le_mul_right (columnGraphMismatch_disagreement_volume_le hU σ hR hc₀ hl a hn hr hrs hα hη htrap S g hS hcone) _ |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter Metric |
| open scoped ENNReal NNReal Topology RealInnerProductSpace symmDiff |
| namespace CubicTorus |
|
|
| lemma characteristic_symmDiff_eq_abs (U F : Set Torus) (x : Torus) : |
| characteristic (U ∆ F) x = |characteristic U x-characteristic F x| := by |
| by_cases hU : x ∈ U <;> by_cases hF : x ∈ F <;> |
| simp [characteristic, Set.mem_symmDiff, hU, hF] |
|
|
| lemma graphSublevelIndicator_column (a : Cover) {n : Cover} (hn : ‖n‖ = 1) |
| (g : Plane → ℝ) (t : ℝ) (y : Plane) : |
| graphSublevelIndicator a n g (normalColumnChart a n (t,y)) = columnPhase (g y) t := by |
| simp only [graphSublevelIndicator, graphHeightExtension, Set.indicator, |
| Set.mem_ofPred_eq, normalColumnChart_height a hn, normalColumnChart_projection a hn, |
| columnPhase, mem_Iio] |
|
|
| lemma cubeCenter_add_mem {x : Cover} (hx : ‖x‖ < 1/2) : |
| cubeCenter+x ∈ fundamentalCube := by |
| intro i |
| have hi := (PiLp.norm_apply_le x i).trans_lt hx |
| rw [Real.norm_eq_abs, abs_lt] at hi |
| change 0 ≤ 1/2+x i ∧ 1/2+x i < 1 |
| constructor <;> linarith |
|
|
| lemma torusGraphRegion_symmDiff_ball_le {U : Set Torus} (hU : MeasurableSet U) |
| {n : Cover} (hn : ‖n‖ = 1) {r s : ℝ} (hrs : r < 1) (hs : s ≤ r/4) |
| {g : Plane → ℝ} (hg : Continuous g) : |
| torusVolume ((U ∆ torusGraphRegion cubeCenter n g) ∩ |
| Metric.ball (quotientMap cubeCenter) s) ≤ |
| ∫⁻ y in Metric.ball (0 : Plane) (r/4), |
| ENNReal.ofReal (columnGraphMismatch U (quotientMap cubeCenter) n r g y) := by |
| let D := U ∆ torusGraphRegion cubeCenter n g |
| have hD : MeasurableSet D := hU.symmDiff (measurableSet_torusGraphRegion _ _ hg) |
| let f : Cover → ℝ≥0∞ := (Metric.ball (0 : Cover) s).indicator |
| (fun x ↦ ENNReal.ofReal (characteristic D (quotientMap x+quotientMap cubeCenter))) |
| have hf : Measurable f := (((measurable_characteristic hD).comp |
| (measurable_quotientMap.add_const _)).ennreal_ofReal).indicator measurableSet_ball |
| have he : torusVolume (D ∩ Metric.ball (quotientMap cubeCenter) s) = ∫⁻ x, f x := by |
| have hm := lintegral_map (μ := volume.restrict (Metric.ball (0 : Cover) s)) |
| ((measurable_characteristic hD).ennreal_ofReal) (measurable_quotientMap.add_const (quotientMap cubeCenter)) |
| rw [torusVolume_chart_map_restrict _ (by linarith : s ≤ 1/2)] at hm |
| rw [lintegral_indicator measurableSet_ball] |
| rw [← hm] |
| have hi : (fun x ↦ ENNReal.ofReal (characteristic D x)) = D.indicator (fun _ ↦ (1 : ℝ≥0∞)) := by |
| funext x |
| by_cases hx : x ∈ D <;> simp [characteristic, hx] |
| rw [hi, lintegral_indicator hD, lintegral_one, Measure.restrict_apply_univ, Measure.restrict_apply hD] |
| rw [he, ← (normalColumnChart_measurePreserving 0 n).lintegral_comp hf] |
| have hb (t : ℝ) (y : Plane) : f (normalColumnChart 0 n (t,y)) ≤ |
| (Metric.ball (0 : Plane) (r/4)).indicator (fun y ↦ (Icc (-r/4) (r/4)).indicator |
| (fun t ↦ ENNReal.ofReal |characteristic U |
| (quotientMap cubeCenter+quotientMap (normalPlaneEmbedding n y+t • n))- |
| columnPhase (g y) t|) t) y := by |
| let x := normalColumnChart (0 : Cover) n (t,y) |
| by_cases hx : x ∈ Metric.ball (0 : Cover) s |
| · have hxnorm : ‖x‖ < s := by simpa only [Metric.mem_ball, dist_zero_right] using hx |
| have ht : |t| ≤ ‖x‖ := by |
| have hh := abs_real_inner_le_norm n x |
| rw [hn, one_mul] at hh |
| have ht0 : ⟪n,x⟫ = t := by simpa only [x, sub_zero] using normalColumnChart_height 0 hn t y |
| rwa [ht0] at hh |
| have hy : ‖y‖ ≤ ‖x‖ := by |
| have hh := (normalPlaneCoordinates_lipschitz n).dist_le_mul x 0 |
| have hy0 : normalPlaneCoordinates n x = y := by |
| simpa only [x, sub_zero] using normalColumnChart_projection 0 hn t y |
| simpa only [map_zero, hy0, dist_zero_right, one_mul, NNReal.coe_one] using hh |
| rw [indicator_of_mem (show y ∈ Metric.ball (0 : Plane) (r/4) by |
| simpa only [Metric.mem_ball, dist_zero_right] using hy.trans_lt (hxnorm.trans_le hs))] |
| rw [indicator_of_mem (show t ∈ Icc (-r/4) (r/4) by |
| have hh := abs_le.mp (ht.trans (hxnorm.le.trans hs)); constructor <;> linarith)] |
| dsimp [f] |
| rw [indicator_of_mem hx, characteristic_symmDiff_eq_abs] |
| have hxc : cubeCenter+x ∈ fundamentalCube := cubeCenter_add_mem (by linarith) |
| have heq : quotientMap x+quotientMap cubeCenter = quotientMap (cubeCenter+x) := by |
| rw [quotientMap_add, add_comm] |
| rw [heq, characteristic_torusGraphRegion hxc] |
| have hc : cubeCenter+x = normalColumnChart cubeCenter n (t,y) := by |
| simp only [x, normalColumnChart_apply _ hn, zero_add] |
| rw [hc, graphSublevelIndicator_column _ hn] |
| rw [normalColumnChart_apply _ hn, quotientMap_add] |
| · change f x ≤ _ |
| dsimp [f] |
| rw [indicator_of_notMem hx] |
| exact zero_le |
| calc |
| _ ≤ ∫⁻ y : Plane, ∫⁻ t : ℝ, (Metric.ball (0 : Plane) (r/4)).indicator |
| (fun y ↦ (Icc (-r/4) (r/4)).indicator |
| (fun t ↦ ENNReal.ofReal |characteristic U |
| (quotientMap cubeCenter+quotientMap (normalPlaneEmbedding n y+t • n))- |
| columnPhase (g y) t|) t) y := by |
| rw [Measure.volume_eq_prod] |
| rw [lintegral_prod_symm (fun p : ℝ × Plane ↦ f (normalColumnChart 0 n p)) |
| (hf.comp (normalColumnChart 0 n).measurable).aemeasurable] |
| exact lintegral_mono fun y ↦ lintegral_mono fun t ↦ hb t y |
| _ = _ := by |
| rw [← lintegral_indicator measurableSet_ball] |
| apply lintegral_congr |
| intro y |
| by_cases hy : y ∈ Metric.ball (0 : Plane) (r/4) |
| · simp only [indicator_of_mem hy] |
| rw [lintegral_indicator measurableSet_Icc] |
| exact (ofReal_integral_eq_lintegral_ofReal (columnGraphMismatch_integrable hU _ n r g y) |
| (Eventually.of_forall fun _ ↦ abs_nonneg _)).symm |
| · simp only [indicator_of_notMem hy, lintegral_zero] |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter Metric |
| open scoped ENNReal NNReal Topology RealInnerProductSpace symmDiff |
| namespace CubicTorus |
|
|
| lemma columnPhase_abs_sub (p q t : ℝ) : |
| |columnPhase p t-columnPhase q t| = |
| (Ico (min p q) (max p q)).indicator (fun _ ↦ (1 : ℝ)) t := by |
| by_cases hp : t < p <;> by_cases hq : t < q <;> |
| simp only [columnPhase, Set.indicator, mem_Iio, mem_Ico, min_le_iff, lt_max_iff, hp, hq] <;> |
| split_ifs <;> simp_all |
| first | linarith | (rename_i hbound; rcases hbound with hbound | hbound <;> linarith) |
|
|
| lemma integral_abs_columnPhase_sub (p q : ℝ) : |
| Integrable (fun t ↦ |columnPhase p t-columnPhase q t|) ∧ |
| (∫ t : ℝ, |columnPhase p t-columnPhase q t|) = |p-q| := by |
| simp_rw [columnPhase_abs_sub] |
| constructor |
| · exact (integrableOn_const (ne_of_lt (show volume (Ico (min p q) (max p q)) < ⊤ by rw [Real.volume_Ico]; exact ENNReal.ofReal_lt_top))).integrable_indicator measurableSet_Ico |
| · rw [integral_indicator measurableSet_Ico, integral_const, measureReal_restrict_apply_univ, |
| smul_eq_mul, mul_one] |
| simp only [Measure.real, Real.volume_Ico, ENNReal.toReal_ofReal (sub_nonneg.mpr min_le_max)] |
| rw [abs_sub_comm] |
| exact max_sub_min_eq_abs _ _ |
|
|
| lemma columnGraphMismatch_change_le {U : Set Torus} (hU : MeasurableSet U) |
| (a : Torus) (n : Cover) (r : ℝ) (f g : Plane → ℝ) (y : Plane) : |
| columnGraphMismatch U a n r f y ≤ columnGraphMismatch U a n r g y + |f y-g y| := by |
| have hp := integral_abs_columnPhase_sub (g y) (f y) |
| have hpoint (t : ℝ) : |characteristic U (a+quotientMap (normalPlaneEmbedding n y+t • n))-columnPhase (f y) t| ≤ |
| |characteristic U (a+quotientMap (normalPlaneEmbedding n y+t • n))-columnPhase (g y) t|+ |
| |columnPhase (g y) t-columnPhase (f y) t| := abs_sub_le _ _ _ |
| have hi := integral_mono (columnGraphMismatch_integrable hU a n r f y) |
| ((columnGraphMismatch_integrable hU a n r g y).add hp.1.integrableOn) hpoint |
| simp only [Pi.add_apply] at hi |
| rw [integral_add (columnGraphMismatch_integrable hU a n r g y) hp.1.integrableOn] at hi |
| have hb : (∫ t in Icc (-r/4) (r/4), |columnPhase (g y) t-columnPhase (f y) t|) ≤ |f y-g y| := by |
| rw [abs_sub_comm, ← hp.2] |
| exact integral_mono_measure Measure.restrict_le_self (Eventually.of_forall fun _ ↦ abs_nonneg _) hp.1 |
| exact hi.trans (add_le_add le_rfl hb) |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter Metric |
| open scoped ENNReal NNReal Topology RealInnerProductSpace symmDiff |
| namespace CubicTorus |
|
|
| lemma torusGraphRegion_symmDiff_change_le {U : Set Torus} (hU : MeasurableSet U) |
| {n : Cover} (hn : ‖n‖ = 1) {r s : ℝ} (hrs : r < 1) (hs : s ≤ r/4) |
| {f g : Plane → ℝ} (hf : Continuous f) (hg : Measurable g) : |
| torusVolume ((U ∆ torusGraphRegion cubeCenter n f) ∩ |
| Metric.ball (quotientMap cubeCenter) s) ≤ |
| (∫⁻ y in Metric.ball (0 : Plane) (r/4), |
| ENNReal.ofReal (columnGraphMismatch U (quotientMap cubeCenter) n r g y)) + |
| ∫⁻ y in Metric.ball (0 : Plane) (r/4), ENNReal.ofReal |f y-g y| := by |
| apply (torusGraphRegion_symmDiff_ball_le hU hn hrs hs hf).trans |
| calc |
| _ ≤ ∫⁻ y in Metric.ball (0 : Plane) (r/4), |
| ENNReal.ofReal (columnGraphMismatch U (quotientMap cubeCenter) n r g y) + ENNReal.ofReal |f y-g y| := by |
| apply lintegral_mono |
| intro y |
| dsimp only |
| rw [← ENNReal.ofReal_add (columnGraphMismatch_nonneg _ _ _ _ _ _) (abs_nonneg _)] |
| exact ENNReal.ofReal_le_ofReal (columnGraphMismatch_change_le hU _ n r f g y) |
| _ = _ := lintegral_add_left |
| (measurable_columnGraphMismatch hU _ n r hg).ennreal_ofReal _ |
|
|
| lemma torusGraphRegion_graphSmooth_difference_le {U : Set Torus} (hU : MeasurableSet U) |
| {n : Cover} (hn : ‖n‖ = 1) {r s : ℝ} (hrs : r < 1) (hs : s ≤ r/4) |
| (b : ContDiffBump (0 : Plane)) {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) : |
| torusVolume ((U ∆ torusGraphRegion cubeCenter n (graphSmooth b g)) ∩ |
| Metric.ball (quotientMap cubeCenter) s) ≤ |
| (∫⁻ y in Metric.ball (0 : Plane) (r/4), |
| ENNReal.ofReal (columnGraphMismatch U (quotientMap cubeCenter) n r g y)) + |
| ENNReal.ofReal ((K : ℝ)*b.rOut) * (volume : Measure Plane) (Metric.ball 0 (r/4)) := by |
| apply (torusGraphRegion_symmDiff_change_le hU hn hrs hs |
| (graphSmooth_contDiff b hg).continuous hg.continuous.measurable).trans |
| apply add_le_add le_rfl |
| calc |
| _ ≤ ∫⁻ y in Metric.ball (0 : Plane) (r/4), ENNReal.ofReal ((K : ℝ)*b.rOut) := |
| lintegral_mono fun y ↦ ENNReal.ofReal_le_ofReal (graphSmooth_error b hg y) |
| _ = _ := by rw [lintegral_const, Measure.restrict_apply_univ] |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma columnPhase_integral {r q : ℝ} (hq : |q| ≤ r/4) : |
| (∫ t in Icc (-r/4) (r/4), columnPhase q t) = q+r/4 := by |
| have hset : Icc (-r/4) (r/4) ∩ Iio q = Ico (-r/4) q := by |
| ext t |
| simp only [mem_inter_iff,mem_Icc,mem_Iio,mem_Ico] |
| constructor |
| · exact fun h ↦ ⟨h.1.1,h.2⟩ |
| · exact fun h ↦ ⟨⟨h.1,h.2.le.trans (abs_le.mp hq).2⟩,h.2⟩ |
| unfold columnPhase |
| rw [setIntegral_indicator measurableSet_Iio,hset,integral_const,smul_eq_mul,mul_one, |
| measureReal_def,Measure.restrict_apply_univ,Real.volume_Ico, |
| ENNReal.toReal_ofReal (by linarith [(abs_le.mp hq).1] : 0 ≤ q- -r/4)] |
| ring |
|
|
| lemma normalColumnHeight_error_le_mismatch {U : Set Torus} (hU : MeasurableSet U) |
| (a : Torus) (n : Cover) (r : ℝ) {g : Plane → ℝ} (y : Plane) (hg : |g y| ≤ r/4) : |
| |g y-normalColumnHeight U a n r y| ≤ columnGraphMismatch U a n r g y := by |
| have hi : IntegrableOn (columnPhase (g y)) (Icc (-r/4) (r/4)) := |
| (integrable_const (1:ℝ)).indicator measurableSet_Iio |
| have heq : columnGraphMismatch U a n r g y = ∫ t in Icc (-r/4) (r/4), |
| |characteristic U (a+quotientMap (normalPlaneEmbedding n y+t • n))-columnPhase (g y) t| := rfl |
| rw [abs_sub_comm,normalColumnHeight] |
| have he : (∫ t in Icc (-r/4) (r/4), characteristic U |
| (a+quotientMap (normalPlaneEmbedding n y+t • n))) -r/4-g y = |
| ∫ t in Icc (-r/4) (r/4), (characteristic U |
| (a+quotientMap (normalPlaneEmbedding n y+t • n))-columnPhase (g y) t) := by |
| rw [integral_sub (normalColumn_integrable hU a n r y) hi,columnPhase_integral hg] |
| ring |
| rw [he,heq] |
| exact abs_integral_le_integral_abs |
|
|
| lemma columnGraphMismatch_bound {U : Set Torus} (hU : MeasurableSet U) |
| (a : Torus) (n : Cover) {r : ℝ} (hr : 0 ≤ r) (g : Plane → ℝ) (y : Plane) : |
| columnGraphMismatch U a n r g y ≤ r/2 := by |
| have hh := integral_mono (columnGraphMismatch_integrable hU a n r g y) |
| (integrable_const (1:ℝ)) (fun t ↦ columnPhase_difference_le U _ (g y) t) |
| rw [integral_const,smul_eq_mul,mul_one,measureReal_def,Measure.restrict_apply_univ, |
| Real.volume_Icc,ENNReal.toReal_ofReal (by linarith : 0 ≤ r/4- -r/4)] at hh |
| exact hh.trans_eq (by ring) |
|
|
| lemma columnGraphMismatch_integrable_ball {U : Set Torus} (hU : MeasurableSet U) |
| (a : Torus) (n : Cover) {r : ℝ} (hr : 0 ≤ r) {g : Plane → ℝ} (hg : Measurable g) |
| (R : ℝ) : IntegrableOn (columnGraphMismatch U a n r g) (Metric.ball 0 R) := by |
| let : IsFiniteMeasure (volume.restrict (Metric.ball (0 : Plane) R)) := |
| isFiniteMeasure_restrict.mpr measure_ball_lt_top.ne |
| apply (integrable_const (r/2)).mono' |
| · exact (measurable_columnGraphMismatch hU a n r hg).aestronglyMeasurable |
| · exact Eventually.of_forall fun y ↦ by |
| rw [Real.norm_eq_abs,abs_of_nonneg (columnGraphMismatch_nonneg _ _ _ _ _ _)] |
| exact columnGraphMismatch_bound hU a n hr g y |
|
|
| lemma columnGraphMismatch_integral_change_le {U : Set Torus} (hU : MeasurableSet U) |
| (a : Torus) (n : Cover) {r : ℝ} (hr : 0 ≤ r) {f g : Plane → ℝ} |
| (hf : Measurable f) (hg : Measurable g) {δ : ℝ} (he : ∀ y, |f y-g y| ≤ δ) (R : ℝ) : |
| (∫ y in Metric.ball (0 : Plane) R, columnGraphMismatch U a n r f y) ≤ |
| (∫ y in Metric.ball (0 : Plane) R, columnGraphMismatch U a n r g y) + |
| δ*(volume : Measure Plane).real (Metric.ball 0 R) := by |
| have hfin : (volume : Measure Plane) (Metric.ball 0 R) ≠ ⊤ := measure_ball_lt_top.ne |
| have hi := integral_mono (columnGraphMismatch_integrable_ball hU a n hr hf R) |
| ((columnGraphMismatch_integrable_ball hU a n hr hg R).add (integrableOn_const hfin)) |
| (fun y ↦ (columnGraphMismatch_change_le hU a n r f g y).trans (add_le_add le_rfl (he y))) |
| simp only [Pi.add_apply] at hi |
| rw [integral_add (columnGraphMismatch_integrable_ball hU a n hr hg R) |
| (integrableOn_const hfin),setIntegral_const] at hi |
| simpa only [smul_eq_mul,mul_comm] using hi |
|
|
| lemma normalColumnHeight_integral_error_le {U : Set Torus} (hU : MeasurableSet U) |
| (a : Torus) (n : Cover) {r : ℝ} (hr : 0 ≤ r) {g : Plane → ℝ} |
| (hg : Measurable g) (hb : ∀ y, |g y| ≤ r/4) (R : ℝ) : |
| (∫ y in Metric.ball (0 : Plane) R, |g y-normalColumnHeight U a n r y|) ≤ |
| ∫ y in Metric.ball (0 : Plane) R, columnGraphMismatch U a n r g y := by |
| have hi := columnGraphMismatch_integrable_ball hU a n hr hg R |
| have hb' := fun y ↦ normalColumnHeight_error_le_mismatch hU a n r y (hb y) |
| have hi' : IntegrableOn (fun y ↦ |g y-normalColumnHeight U a n r y|) (Metric.ball 0 R) := by |
| apply hi.mono' |
| · exact (hg.sub (measurable_normalColumnHeight hU a n r)).abs.aestronglyMeasurable |
| · exact Eventually.of_forall fun y ↦ by simpa only [Real.norm_eq_abs,abs_abs] using hb' y |
| exact integral_mono hi' hi hb' |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace symmDiff |
| namespace CubicTorus |
|
|
| lemma plane_ball_real_volume_scale {r : ℝ} (hr : 0 < r) (b : ℝ) : |
| (volume : Measure Plane).real (Metric.ball 0 (r*b)) = |
| r^2*(volume : Measure Plane).real (Metric.ball 0 b) := by |
| have hh := plane_integral_rescale_ball (fun _ ↦ (1:ℝ)) hr b |
| simpa only [setIntegral_const,smul_eq_mul,mul_one] using hh |
|
|
| lemma torusGraphRegion_symmDiff_real_ball_le {U : Set Torus} (hU : MeasurableSet U) |
| {n : Cover} (hn : ‖n‖ = 1) {r s : ℝ} (hr : 0 ≤ r) (hrs : r < 1) (hs : s ≤ r/4) |
| {g : Plane → ℝ} (hg : Continuous g) : |
| torusVolume.real ((U ∆ torusGraphRegion cubeCenter n g) ∩ Metric.ball (quotientMap cubeCenter) s) ≤ |
| ∫ y in Metric.ball (0 : Plane) (r/4), columnGraphMismatch U (quotientMap cubeCenter) n r g y := by |
| have hh := torusGraphRegion_symmDiff_ball_le hU hn hrs hs hg |
| rw [← ofReal_integral_eq_lintegral_ofReal |
| (columnGraphMismatch_integrable_ball hU _ n hr hg.measurable (r/4)) |
| (Eventually.of_forall fun y ↦ columnGraphMismatch_nonneg _ _ _ _ _ _)] at hh |
| have ht := ENNReal.toReal_mono (ENNReal.ofReal_ne_top) hh |
| rw [ENNReal.toReal_ofReal (integral_nonneg fun y ↦ columnGraphMismatch_nonneg _ _ _ _ _ _)] at ht |
| exact ht |
|
|
| theorem exists_physical_smooth_graph {U : Set Torus} (hU : MeasurableSet U) |
| {n : Cover} (hn : ‖n‖ = 1) {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) |
| {r δ α : ℝ} (hr : 0 < r) (hrs : r < 1) (hδ : 0 < δ) (hδr : δ ≤ r/8) |
| (hα : α ≤ 1/4) (hheight : ∀ y, |g y| ≤ α*r) : |
| ∃ G : Plane → ℝ, ContDiff ℝ (⊤ : ℕ∞) G ∧ LipschitzWith K G ∧ |
| (∀ y, |G y| ≤ α*r) ∧ |
| (∫ y in Metric.ball (0 : Plane) (r/8), ‖fderiv ℝ G y‖^2) ≤ |
| (∫ y in Metric.ball (0 : Plane) (r/4), ‖fderiv ℝ g y‖^2) ∧ |
| (let B := (∫ y in Metric.ball (0 : Plane) (r/4), |
| columnGraphMismatch U (quotientMap cubeCenter) n r g y)+ |
| δ*r^2*(volume : Measure Plane).real (Metric.ball 0 (1/4)) |
| (∫ y in Metric.ball (0 : Plane) (r/4), |
| |G y-normalColumnHeight U (quotientMap cubeCenter) n r y|) ≤ B ∧ |
| ∀ s : ℝ, s ≤ r/4 → torusVolume.real ((U ∆ torusGraphRegion cubeCenter n G) ∩ |
| Metric.ball (quotientMap cubeCenter) s) ≤ B) := by |
| obtain ⟨G,hG,hGL,he,henergy,hH⟩ := exists_sharp_smooth_graph hg hδ |
| have hh (y : Plane) : |G y| ≤ α*r := by |
| simpa only [zero_add,one_mul] using hH 0 1 (α*r) (by simpa only [zero_add,one_mul] using hheight) y |
| have hb (y : Plane) : |G y| ≤ r/4 := (hh y).trans (by nlinarith) |
| refine ⟨G,hG,hGL,hh,?_,?_⟩ |
| · exact (henergy (r/8)).trans (setIntegral_mono_set |
| (lipschitz_gradient_sq_integrable_ball hg (r/4)) |
| (Eventually.of_forall fun y ↦ sq_nonneg _) (Metric.ball_subset_ball (by linarith)).eventuallyLE) |
| · have herr := columnGraphMismatch_integral_change_le hU (quotientMap cubeCenter) n hr.le |
| hG.continuous.measurable hg.continuous.measurable he (r/4) |
| have hv : (volume : Measure Plane).real (Metric.ball 0 (r/4)) = |
| r^2*(volume : Measure Plane).real (Metric.ball 0 (1/4)) := by |
| simpa only [mul_one_div] using plane_ball_real_volume_scale hr (1/4) |
| rw [hv] at herr |
| have herr' : (∫ y in Metric.ball (0 : Plane) (r/4), columnGraphMismatch U |
| (quotientMap cubeCenter) n r G y) ≤ |
| (∫ y in Metric.ball (0 : Plane) (r/4), columnGraphMismatch U |
| (quotientMap cubeCenter) n r g y)+δ*r^2*(volume : Measure Plane).real (Metric.ball 0 (1/4)) := by |
| simpa only [mul_assoc] using herr |
| exact ⟨(normalColumnHeight_integral_error_le hU _ n hr.le hG.continuous.measurable hb (r/4)).trans herr', |
| fun s hs ↦ (torusGraphRegion_symmDiff_real_ball_le hU hn hr.le hrs hs hG.continuous).trans herr'⟩ |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter Metric |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem normalMeasure_lipschitz_approximation_with_cones {E B : Set Torus} (hE : MeasurableSet E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) {N : Torus → Cover} (hN : Measurable N) |
| (hpolar : ∀ᵐ z ∂(μ : Measure NormalBundle), N (quotientMap z.1) = (z.2 : Cover)) |
| (hn : ∀ᵐ x ∂(spatialNormalMeasure μ : Measure Torus), ‖N x‖ = 1) |
| {D δ R A : ℝ} (hδ : 0 < δ) (hA : 0 < A) |
| (hd : ∀ a ∈ B, ∀ t : ℝ, 0 < t → t ≤ D → |
| δ * t ^ 3 ≤ torusVolume.real (E ∩ Metric.ball a t) ∧ |
| δ * t ^ 3 ≤ torusVolume.real (Eᶜ ∩ Metric.ball a t)) |
| (hu : ∀ a : Torus, ∀ r : ℝ, 0 < r → r ≤ R → |
| (spatialNormalMeasure μ : Measure Torus).real (Metric.ball a r) ≤ A * r ^ 2) |
| {ρ ε β : ℝ} (hρ : 0 < ρ) (hρsmall : ρ ≤ 1 / 32) (hρD : 2 * ρ ≤ D) (hρR : 80 * ρ ≤ R) |
| (hε : 0 < ε) (hεhalf : ε ≤ 1 / 2) (hβ : 0 < β) |
| (hβsmall : (131072 : ℝ) ^ 2 * (4 * A) * β ≤ (δ ^ 2 * (ε / 8) ^ 6) ^ 2) |
| (a : Torus) {n : Cover} (hnorm : ‖n‖ = 1) : |
| ∃ S : Set Torus, S ⊆ B ∩ Metric.ball a (ρ / 2) ∧ |
| (spatialNormalMeasure μ : Measure Torus) ((B ∩ Metric.ball a (ρ / 2)) \ S) ≤ |
| ENNReal.ofReal (25 * A / β) * |
| ENNReal.ofReal (∫ x in Metric.closedBall a (17 * ρ), ‖N x - n‖ ^ 2 |
| ∂(spatialNormalMeasure μ : Measure Torus)) ∧ |
| (∀ c ∈ S, ∀ x ∈ B, dist x c ≤ ρ → |
| |⟪n, shortestLift (x - c)⟫| ≤ ε * dist x c) ∧ |
| ∃ g : Cover → ℝ, LipschitzWith ⟨2 * ε, by positivity⟩ g ∧ |
| ∀ x ∈ S, g (normalPlaneProjection n (shortestLift (x - a))) = |
| ⟪n, shortestLift (x - a)⟫ := by |
| let σ := spatialNormalMeasure μ |
| let S := {c : Torus | c ∈ B ∩ Metric.ball a (ρ / 2) ∧ |
| ∀ s : ℝ, 0 < s → s ≤ 16 * ρ → normalExcess σ N c n s < β} |
| have hS : S ⊆ B ∩ Metric.ball a (ρ / 2) := fun _ hc ↦ hc.1 |
| have hgood (c : Torus) (hc : c ∈ S) : ∀ y ∈ B, dist y c ≤ ρ → |
| |⟪n, shortestLift (y - c)⟫| ≤ ε * dist y c := by |
| apply good_scales_boundary_cone hE hμ hN hpolar hn hδ hd hc.1.1 hnorm hρsmall hρD hε (by linarith) |
| intro s hs hs16 |
| have hmass := (measureReal_mono (Metric.closedBall_subset_ball (by linarith : s < 2 * s))).trans |
| (hu c (2 * s) (by positivity) (by linarith)) |
| have hmass' : (σ : Measure Torus).real (Metric.closedBall c s) ≤ (4 * A) * s ^ 2 := by |
| change (σ : Measure Torus).real (Metric.closedBall c s) ≤ A * (2 * s) ^ 2 at hmass |
| nlinarith |
| have ho := normalOscillation_sq_le_excess σ hN hn hnorm c hs hmass' |
| have he := hc.2 s hs hs16 |
| have hh := ho.trans_lt (mul_lt_mul_of_pos_left he (by positivity : 0 < 4 * A)) |
| have hh' := mul_lt_mul_of_pos_left hh (by norm_num : (0 : ℝ) < 131072 ^ 2) |
| have hT : 0 < δ ^ 2 * (ε / 8) ^ 6 := by positivity |
| have hnosc := normalOscillation_nonneg σ N c n s |
| nlinarith |
| have hbad : ∀ c ∈ (B ∩ Metric.ball a (ρ / 2)) \ S, ∃ r : ℝ, |
| 0 < r ∧ r ≤ 16 * ρ ∧ β ≤ normalExcess σ N c n r := by |
| intro c hc |
| have hh : ¬ ∀ s : ℝ, 0 < s → s ≤ 16 * ρ → normalExcess σ N c n s < β := by |
| intro hh |
| exact hc.2 ⟨hc.1, hh⟩ |
| push Not at hh |
| exact hh |
| have hbadmass := normalExcess_bad_set_measure hN hn n hA hβ hρ hρR hu a |
| ((B ∩ Metric.ball a (ρ / 2)) \ S) |
| (fun c hc ↦ (Metric.ball_subset_ball (by linarith : ρ / 2 ≤ ρ)) hc.1.2) hbad |
| have hSsmall : S ⊆ Metric.ball a (1 / 4) := |
| hS.trans (inter_subset_right.trans (Metric.ball_subset_ball (by linarith))) |
| have hcone : ∀ x ∈ S, ∀ y ∈ S, |⟪n, shortestLift (x - y)⟫| ≤ ε * dist x y := by |
| intro x hx y hy |
| apply hgood y hy x hx.1.1 |
| have hxa : dist x a < ρ / 2 := hx.1.2 |
| have hya : dist y a < ρ / 2 := hy.1.2 |
| linarith [dist_triangle_right x y a] |
| obtain ⟨g, hg, hgeq⟩ := exists_lipschitz_height_of_cone |
| ((fun z ↦ shortestLift (z - a)) '' S) hnorm hε.le hεhalf (lifted_cone_of_torus_cone hSsmall hcone) |
| exact ⟨S, hS, hbadmass, hgood, g, hg, fun x hx ↦ hgeq _ ⟨x, hx, rfl⟩⟩ |
|
|
| theorem isMinimizerAt_lipschitz_approximation_with_cones {V : ℝ} (hV : V ∈ Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| {ε : ℝ} (hε : 0 < ε) (hεhalf : ε ≤ 1 / 2) : |
| ∃ ρ₀ C : ℝ, 0 < ρ₀ ∧ 0 < C ∧ |
| ∀ μ : FiniteMeasure NormalBundle, IsNormalMeasure E μ → |
| ∀ N : Torus → Cover, Measurable N → |
| (∀ᵐ z ∂(μ : Measure NormalBundle), N (quotientMap z.1) = (z.2 : Cover)) → |
| (∀ᵐ x ∂(spatialNormalMeasure μ : Measure Torus), ‖N x‖ = 1) → |
| ∀ (a : Torus) (n : Cover), ‖n‖ = 1 → ∀ ρ : ℝ, 0 < ρ → ρ ≤ ρ₀ → |
| ∃ S : Set Torus, S ⊆ frontier (openRepresentative E) ∩ Metric.ball a (ρ / 2) ∧ |
| (spatialNormalMeasure μ : Measure Torus) |
| ((frontier (openRepresentative E) ∩ Metric.ball a (ρ / 2)) \ S) ≤ |
| ENNReal.ofReal C * ENNReal.ofReal |
| (∫ x in Metric.closedBall a (17 * ρ), ‖N x - n‖ ^ 2 |
| ∂(spatialNormalMeasure μ : Measure Torus)) ∧ |
| (∀ c ∈ S, ∀ x ∈ frontier (openRepresentative E), dist x c ≤ ρ → |
| |⟪n, shortestLift (x - c)⟫| ≤ ε * dist x c) ∧ |
| ∃ g : Cover → ℝ, LipschitzWith ⟨2 * ε, by positivity⟩ g ∧ |
| ∀ x ∈ S, g (normalPlaneProjection n (shortestLift (x - a))) = |
| ⟪n, shortestLift (x - a)⟫ := by |
| obtain ⟨D, δ, hD, hδ, hd⟩ := isMinimizerAt_two_sided_density hV hE hmin |
| obtain ⟨R, A, hR, hA, hu⟩ := isMinimizerAt_perimeter_upper_density hV hE hmin |
| let β := (δ ^ 2 * (ε / 8) ^ 6) ^ 2 / ((131072 : ℝ) ^ 2 * (4 * A)) |
| have hβ : 0 < β := by dsimp [β]; positivity |
| let ρ₀ := min (1 / 32 : ℝ) (min (D / 2) (R / 80)) |
| have hρ₀ : 0 < ρ₀ := by dsimp [ρ₀]; positivity |
| refine ⟨ρ₀, 25 * A / β, hρ₀, by positivity, ?_⟩ |
| intro μ hμ N hN hp hn a n hnorm ρ hρ hρle |
| have hd' : ∀ a ∈ frontier (openRepresentative E), ∀ t : ℝ, 0 < t → t ≤ D → |
| δ * t ^ 3 ≤ torusVolume.real (E ∩ Metric.ball a t) ∧ |
| δ * t ^ 3 ≤ torusVolume.real (Eᶜ ∩ Metric.ball a t) := by |
| rw [isMinimizerAt_frontier_openRepresentative hV hE hmin] |
| exact hd |
| apply normalMeasure_lipschitz_approximation_with_cones hE hμ hN hp hn hδ hA hd' (hu μ hμ) |
| hρ (hρle.trans (min_le_left _ _)) |
| (by have := hρle.trans ((min_le_right _ _).trans (min_le_left _ _)); linarith) |
| (by have := hρle.trans ((min_le_right _ _).trans (min_le_right _ _)); linarith) |
| hε hεhalf hβ |
| · dsimp [β] |
| have hden : (131072 : ℝ) ^ 2 * (4 * A) ≠ 0 := by positivity |
| rw [mul_div_cancel₀ _ hden] |
| · exact hnorm |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter Metric |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma clamp_height_bound (f : Plane → ℝ) {r : ℝ} (hr : 0 ≤ r) (x : Plane) : |
| |max (-r) (min r (f x))| ≤ r := by |
| rw [abs_le] |
| exact ⟨le_max_left _ _, max_le (by linarith) (min_le_left _ _)⟩ |
|
|
| theorem isMinimizerAt_trapped_graph_approximation {V : ℝ} (hV : V ∈ Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| {η : ℝ} (hη : 0 < η) (hηhalf : η ≤ 1 / 2) : |
| ∃ r₀ : ℝ, ∃ C : ℝ≥0∞, 0 < r₀ ∧ C < ⊤ ∧ |
| ∀ μ : FiniteMeasure NormalBundle, IsNormalMeasure E μ → |
| ∀ N : Torus → Cover, Measurable N → |
| (∀ᵐ z ∂(μ : Measure NormalBundle), N (quotientMap z.1) = (z.2 : Cover)) → |
| (∀ᵐ x ∂(spatialNormalMeasure μ : Measure Torus), ‖N x‖ = 1) → |
| ∀ (a : Torus) (n : Cover), ‖n‖ = 1 → ∀ r α : ℝ, 0 < r → r ≤ r₀ → α < 1 / 4 → |
| (normalCap a n r (α * r) ⊆ (closure (openRepresentative E))ᶜ ∧ |
| normalCap a (-n) r (α * r) ⊆ openRepresentative E) → |
| ∃ (S : Set Torus) (g : Plane → ℝ), |
| LipschitzWith ⟨2 * η, by positivity⟩ g ∧ |
| (∀ y, |g y| ≤ r / 4) ∧ |
| S ⊆ frontier (openRepresentative E) ∩ Metric.ball a (r / 2) ∧ |
| (∀ x ∈ S, g (normalPlaneCoordinates n (shortestLift (x - a))) = |
| ⟪n, shortestLift (x - a)⟫) ∧ |
| (∀ c ∈ S, ∀ x ∈ frontier (openRepresentative E), dist x c ≤ r → |
| |⟪n, shortestLift (x - c)⟫| ≤ η * dist x c) ∧ |
| let e := ENNReal.ofReal (∫ x in Metric.closedBall a (17 * r), ‖N x - n‖ ^ 2 |
| ∂(spatialNormalMeasure μ : Measure Torus)) |
| (spatialNormalMeasure μ : Measure Torus) |
| ((frontier (openRepresentative E) ∩ Metric.ball a (r / 2)) \ S) ≤ C * e ∧ |
| (∫⁻ y in Metric.ball (0 : Plane) (r / 4), ENNReal.ofReal (‖fderiv ℝ g y‖ ^ 2)) ≤ C * e ∧ |
| (∫⁻ y in Metric.ball (0 : Plane) (r / 4), |
| ENNReal.ofReal |normalColumnHeight (openRepresentative E) a n r y - g y|) ≤ |
| C * ENNReal.ofReal r * e := by |
| obtain ⟨r₁, C₁, hr₁, hC₁, happ⟩ := isMinimizerAt_lipschitz_approximation_with_cones hV hE hmin hη hηhalf |
| obtain ⟨C₂, hC₂, henergy⟩ := isMinimizerAt_normalGraph_full_energy hV hE hmin |
| obtain ⟨R, c, hR, hc, hlower⟩ := isMinimizerAt_perimeter_lower_density hV hE hmin |
| let A : ℝ≥0∞ := (planeAreaFactor : ℝ≥0∞) * ENNReal.ofReal (64 / c) |
| let C : ℝ≥0∞ := max (ENNReal.ofReal C₁) (max (C₂ * (1 + ENNReal.ofReal C₁)) (A * ENNReal.ofReal C₁)) |
| have hCt : C < ⊤ := by dsimp [C, A]; finiteness |
| have hCbad : ENNReal.ofReal C₁ ≤ C := le_max_left _ _ |
| have hCgrad : C₂ * (1 + ENNReal.ofReal C₁) ≤ C := (le_max_left _ _).trans (le_max_right _ _) |
| have hCheight : A * ENNReal.ofReal C₁ ≤ C := (le_max_right _ _).trans (le_max_right _ _) |
| refine ⟨min r₁ (1 / 4), C, by positivity, hCt, ?_⟩ |
| intro μ hμ N hN hp hn a n hnorm r α hr hrr hα htrap |
| have hrs : r < 1 / 2 := lt_of_le_of_lt (hrr.trans (min_le_right _ _)) (by norm_num) |
| obtain ⟨S, hS, hbad, hcone, g₀, hg₀, heq₀⟩ := happ μ hμ N hN hp hn a n hnorm r hr |
| (hrr.trans (min_le_left _ _)) |
| let K : NNReal := ⟨2 * η, by positivity⟩ |
| have hg₀x : LipschitzWith K g₀ := hg₀ |
| let g₁ : Plane → ℝ := fun y ↦ g₀ (normalPlaneEmbedding n y) |
| have hg₁ : LipschitzWith K g₁ := by |
| apply LipschitzWith.of_dist_le_mul |
| intro x y |
| have hdist : dist (normalPlaneEmbedding n x) (normalPlaneEmbedding n y) = dist x y := by |
| simp only [dist_eq_norm] |
| change ‖(normalPlaneLinearIsometry n) x - (normalPlaneLinearIsometry n) y‖ = ‖x - y‖ |
| rw [← map_sub, (normalPlaneLinearIsometry n).norm_map] |
| exact (hg₀x.dist_le_mul (normalPlaneEmbedding n x) (normalPlaneEmbedding n y)).trans_eq |
| (congrArg (fun t : ℝ ↦ (K : ℝ) * t) hdist) |
| let g : Plane → ℝ := fun y ↦ max (-(r / 4)) (min (r / 4) (g₁ y)) |
| have hg : LipschitzWith ⟨2 * η, by positivity⟩ g := (hg₁.const_min (r / 4)).const_max (-(r / 4)) |
| have hheight (y : Plane) : |g y| ≤ r / 4 := clamp_height_bound g₁ (by positivity) y |
| have heq (x : Torus) (hx : x ∈ S) : g (normalPlaneCoordinates n (shortestLift (x - a))) = |
| ⟪n, shortestLift (x - a)⟫ := by |
| have hh := trapped_frontier_torus_height (isOpen_openRepresentative E) htrap |
| ⟨(hS hx).1, (Metric.ball_subset_ball (by linarith : r / 2 ≤ r)) (hS hx).2⟩ |
| have hb : |⟪n, shortestLift (x - a)⟫| ≤ r / 4 := hh.trans (by nlinarith) |
| dsimp [g, g₁] |
| rw [normalPlaneEmbedding_coordinates hnorm, heq₀ x hx, |
| min_eq_right (abs_le.mp hb).2, max_eq_right (abs_le.mp hb).1] |
| let e := ENNReal.ofReal (∫ x in Metric.closedBall a (17 * r), ‖N x - n‖ ^ 2 |
| ∂(spatialNormalMeasure μ : Measure Torus)) |
| have hbad' : (spatialNormalMeasure μ : Measure Torus) |
| ((frontier (openRepresentative E) ∩ Metric.ball a (r / 2)) \ S) ≤ ENNReal.ofReal C₁ * e := hbad |
| have hlocal : (∫⁻ x in Metric.ball a (r / 2), ENNReal.ofReal (‖N x - n‖ ^ 2) |
| ∂(spatialNormalMeasure μ : Measure Torus)) ≤ e := by |
| rw [← ofReal_integral_eq_lintegral_ofReal |
| (integrable_sq_norm_sub_normal hN hn n).integrableOn (Eventually.of_forall fun _ ↦ sq_nonneg _)] |
| apply ENNReal.ofReal_le_ofReal |
| apply integral_mono_measure |
| (Measure.restrict_mono (Metric.ball_subset_closedBall.trans |
| (Metric.closedBall_subset_closedBall (by linarith : r / 2 ≤ 17 * r))) le_rfl) |
| (Eventually.of_forall fun _ ↦ sq_nonneg _) |
| (integrable_sq_norm_sub_normal hN hn n).integrableOn |
| have he := henergy μ hμ N hN hp hn n hnorm g ⟨2 * η, by positivity⟩ hg |
| (by exact_mod_cast (show 2 * η ≤ 1 by linarith)) a r α hr hrs hα htrap |
| (fun y _ ↦ hheight y) S heq |
| have hh := normalColumnHeight_graph_L1_le (isOpen_openRepresentative E) |
| (spatialNormalMeasure μ) hR hc |
| (by rw [isMinimizerAt_frontier_openRepresentative hV hE hmin]; exact hlower μ hμ) |
| a hnorm hr hrs hα hηhalf htrap S g hg.continuous.measurable (fun y _ ↦ hheight y) heq hcone |
| refine ⟨S, g, hg, hheight, hS, heq, hcone, hbad'.trans (mul_le_mul_left hCbad _), ?_, ?_⟩ |
| · calc |
| _ ≤ C₂ * (e + ENNReal.ofReal C₁ * e) := he.trans (mul_le_mul_right (add_le_add hlocal hbad') _) |
| _ = (C₂ * (1 + ENNReal.ofReal C₁)) * e := by simp only [mul_add, add_mul, mul_one, mul_assoc] |
| _ ≤ C * e := mul_le_mul_left hCgrad _ |
| · calc |
| _ ≤ ENNReal.ofReal (r / 2) * (A * (ENNReal.ofReal C₁ * e)) := |
| hh.trans (mul_le_mul_right (mul_le_mul_right hbad' A) _) |
| _ ≤ ENNReal.ofReal r * (A * (ENNReal.ofReal C₁ * e)) := |
| mul_le_mul_left (ENNReal.ofReal_le_ofReal (by linarith)) _ |
| _ = (A * ENNReal.ofReal C₁) * ENNReal.ofReal r * e := by ac_rfl |
| _ ≤ C * ENNReal.ofReal r * e := mul_le_mul_left (mul_le_mul_left hCheight _) _ |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter Metric |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem isMinimizerAt_tightly_trapped_graph_approximation {V : ℝ} (hV : V ∈ Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| {η : ℝ} (hη : 0 < η) (hηhalf : η ≤ 1 / 2) : |
| ∃ r₀ : ℝ, ∃ C : ℝ≥0∞, 0 < r₀ ∧ C < ⊤ ∧ |
| ∀ μ : FiniteMeasure NormalBundle, IsNormalMeasure E μ → |
| ∀ N : Torus → Cover, Measurable N → |
| (∀ᵐ z ∂(μ : Measure NormalBundle), N (quotientMap z.1) = (z.2 : Cover)) → |
| (∀ᵐ x ∂(spatialNormalMeasure μ : Measure Torus), ‖N x‖ = 1) → |
| ∀ (a : Torus) (n : Cover), ‖n‖ = 1 → ∀ r α : ℝ, 0 < r → r ≤ r₀ → 0 ≤ α → α < 1 / 4 → |
| (normalCap a n r (α * r) ⊆ (closure (openRepresentative E))ᶜ ∧ |
| normalCap a (-n) r (α * r) ⊆ openRepresentative E) → |
| ∃ (S : Set Torus) (g : Plane → ℝ), |
| LipschitzWith ⟨2 * η, by positivity⟩ g ∧ |
| (∀ y, |g y| ≤ α * r) ∧ |
| S ⊆ frontier (openRepresentative E) ∩ Metric.ball a (r / 2) ∧ |
| (∀ x ∈ S, g (normalPlaneCoordinates n (shortestLift (x - a))) = |
| ⟪n, shortestLift (x - a)⟫) ∧ |
| (∀ c ∈ S, ∀ x ∈ frontier (openRepresentative E), dist x c ≤ r → |
| |⟪n, shortestLift (x - c)⟫| ≤ η * dist x c) ∧ |
| let e := ENNReal.ofReal (∫ x in Metric.closedBall a (17 * r), ‖N x - n‖ ^ 2 |
| ∂(spatialNormalMeasure μ : Measure Torus)) |
| (spatialNormalMeasure μ : Measure Torus) |
| ((frontier (openRepresentative E) ∩ Metric.ball a (r / 2)) \ S) ≤ C * e ∧ |
| (∫⁻ y in Metric.ball (0 : Plane) (r / 4), ENNReal.ofReal (‖fderiv ℝ g y‖ ^ 2)) ≤ C * e ∧ |
| (∫⁻ y in Metric.ball (0 : Plane) (r / 4), |
| ENNReal.ofReal (columnGraphMismatch (openRepresentative E) a n r g y)) ≤ |
| C * ENNReal.ofReal α * ENNReal.ofReal r * e := by |
| obtain ⟨r₁, C₁, hr₁, hC₁, happ⟩ := isMinimizerAt_lipschitz_approximation_with_cones hV hE hmin hη hηhalf |
| obtain ⟨C₂, hC₂, henergy⟩ := isMinimizerAt_normalGraph_full_energy hV hE hmin |
| obtain ⟨R, c, hR, hc, hlower⟩ := isMinimizerAt_perimeter_lower_density hV hE hmin |
| let A : ℝ≥0∞ := (planeAreaFactor : ℝ≥0∞) * ENNReal.ofReal (64 / c) |
| let C : ℝ≥0∞ := max (ENNReal.ofReal C₁) (max (C₂ * (1 + ENNReal.ofReal C₁)) (2 * A * ENNReal.ofReal C₁)) |
| have hCt : C < ⊤ := by dsimp [C, A]; finiteness |
| have hCbad : ENNReal.ofReal C₁ ≤ C := le_max_left _ _ |
| have hCgrad : C₂ * (1 + ENNReal.ofReal C₁) ≤ C := (le_max_left _ _).trans (le_max_right _ _) |
| have hCheight : 2 * A * ENNReal.ofReal C₁ ≤ C := (le_max_right _ _).trans (le_max_right _ _) |
| refine ⟨min r₁ (1 / 4), C, by positivity, hCt, ?_⟩ |
| intro μ hμ N hN hp hn a n hnorm r α hr hrr hα0 hα htrap |
| have hrs : r < 1 / 2 := lt_of_le_of_lt (hrr.trans (min_le_right _ _)) (by norm_num) |
| obtain ⟨S, hS, hbad, hcone, g₀, hg₀, heq₀⟩ := happ μ hμ N hN hp hn a n hnorm r hr |
| (hrr.trans (min_le_left _ _)) |
| let K : NNReal := ⟨2 * η, by positivity⟩ |
| have hg₀x : LipschitzWith K g₀ := hg₀ |
| let g₁ : Plane → ℝ := fun y ↦ g₀ (normalPlaneEmbedding n y) |
| have hg₁ : LipschitzWith K g₁ := by |
| apply LipschitzWith.of_dist_le_mul |
| intro x y |
| have hdist : dist (normalPlaneEmbedding n x) (normalPlaneEmbedding n y) = dist x y := by |
| simp only [dist_eq_norm] |
| change ‖(normalPlaneLinearIsometry n) x - (normalPlaneLinearIsometry n) y‖ = ‖x - y‖ |
| rw [← map_sub, (normalPlaneLinearIsometry n).norm_map] |
| exact (hg₀x.dist_le_mul (normalPlaneEmbedding n x) (normalPlaneEmbedding n y)).trans_eq |
| (congrArg (fun t : ℝ ↦ (K : ℝ) * t) hdist) |
| let g : Plane → ℝ := fun y ↦ max (-(α*r)) (min (α*r) (g₁ y)) |
| have hg : LipschitzWith ⟨2 * η, by positivity⟩ g := (hg₁.const_min (α*r)).const_max (-(α*r)) |
| have hheight (y : Plane) : |g y| ≤ α*r := clamp_height_bound g₁ (by positivity) y |
| have heq (x : Torus) (hx : x ∈ S) : g (normalPlaneCoordinates n (shortestLift (x - a))) = |
| ⟪n, shortestLift (x - a)⟫ := by |
| have hh := trapped_frontier_torus_height (isOpen_openRepresentative E) htrap |
| ⟨(hS hx).1, (Metric.ball_subset_ball (by linarith : r / 2 ≤ r)) (hS hx).2⟩ |
| have hb : |⟪n, shortestLift (x - a)⟫| ≤ α*r := hh |
| dsimp [g, g₁] |
| rw [normalPlaneEmbedding_coordinates hnorm, heq₀ x hx, |
| min_eq_right (abs_le.mp hb).2, max_eq_right (abs_le.mp hb).1] |
| let e := ENNReal.ofReal (∫ x in Metric.closedBall a (17 * r), ‖N x - n‖ ^ 2 |
| ∂(spatialNormalMeasure μ : Measure Torus)) |
| have hbad' : (spatialNormalMeasure μ : Measure Torus) |
| ((frontier (openRepresentative E) ∩ Metric.ball a (r / 2)) \ S) ≤ ENNReal.ofReal C₁ * e := hbad |
| have hlocal : (∫⁻ x in Metric.ball a (r / 2), ENNReal.ofReal (‖N x - n‖ ^ 2) |
| ∂(spatialNormalMeasure μ : Measure Torus)) ≤ e := by |
| rw [← ofReal_integral_eq_lintegral_ofReal |
| (integrable_sq_norm_sub_normal hN hn n).integrableOn (Eventually.of_forall fun _ ↦ sq_nonneg _)] |
| apply ENNReal.ofReal_le_ofReal |
| apply integral_mono_measure |
| (Measure.restrict_mono (Metric.ball_subset_closedBall.trans |
| (Metric.closedBall_subset_closedBall (by linarith : r / 2 ≤ 17 * r))) le_rfl) |
| (Eventually.of_forall fun _ ↦ sq_nonneg _) |
| (integrable_sq_norm_sub_normal hN hn n).integrableOn |
| have he := henergy μ hμ N hN hp hn n hnorm g ⟨2 * η, by positivity⟩ hg |
| (by exact_mod_cast (show 2 * η ≤ 1 by linarith)) a r α hr hrs hα htrap |
| (fun y _ ↦ (hheight y).trans (by nlinarith)) S heq |
| have hh := columnGraphMismatch_L1_le (isOpen_openRepresentative E) |
| (spatialNormalMeasure μ) hR hc |
| (by rw [isMinimizerAt_frontier_openRepresentative hV hE hmin]; exact hlower μ hμ) |
| a hnorm hr hrs hα0 hα hηhalf htrap S g hg.continuous.measurable (fun y _ ↦ hheight y) heq hcone |
| refine ⟨S, g, hg, hheight, hS, heq, hcone, hbad'.trans (mul_le_mul_left hCbad _), ?_, ?_⟩ |
| · calc |
| _ ≤ C₂ * (e + ENNReal.ofReal C₁ * e) := he.trans (mul_le_mul_right (add_le_add hlocal hbad') _) |
| _ = (C₂ * (1 + ENNReal.ofReal C₁)) * e := by simp only [mul_add, add_mul, mul_one, mul_assoc] |
| _ ≤ C * e := mul_le_mul_left hCgrad _ |
| · calc |
| _ ≤ ENNReal.ofReal (2*α*r) * (A * (ENNReal.ofReal C₁ * e)) := |
| hh.trans (mul_le_mul_right (mul_le_mul_right hbad' A) _) |
| _ = (2 * A * ENNReal.ofReal C₁) * ENNReal.ofReal α * ENNReal.ofReal r * e := by |
| rw [ENNReal.ofReal_mul (by positivity : 0 ≤ 2*α), ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 2), ENNReal.ofReal_ofNat] |
| ac_rfl |
| _ ≤ C * ENNReal.ofReal α * ENNReal.ofReal r * e := |
| mul_le_mul_left (mul_le_mul_left (mul_le_mul_left hCheight _) _) _ |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace symmDiff |
| namespace CubicTorus |
|
|
| theorem isMinimizerAt_smooth_trapped_approximation {V : ℝ} (hV : V ∈ Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) : |
| ∃ r₀ D : ℝ, 0 < r₀ ∧ 0 ≤ D ∧ |
| ∀ μ : FiniteMeasure NormalBundle, IsNormalMeasure E μ → |
| ∀ N : Torus → Cover, Measurable N → |
| (∀ᵐ z ∂(μ : Measure NormalBundle), N (quotientMap z.1) = (z.2 : Cover)) → |
| (∀ᵐ x ∂(spatialNormalMeasure μ : Measure Torus), ‖N x‖ = 1) → |
| ∀ n : Cover, ‖n‖ = 1 → ∀ r α e : ℝ, 0 < r → r ≤ r₀ → 0 ≤ α → α < 1/4 → |
| 0 < e → e ≤ 1/2 → |
| (normalCap (quotientMap cubeCenter) n r (α*r) ⊆ (closure (openRepresentative E))ᶜ ∧ |
| normalCap (quotientMap cubeCenter) (-n) r (α*r) ⊆ openRepresentative E) → |
| (∫ x in Metric.closedBall (quotientMap cubeCenter) (17*r), ‖N x-n‖^2 |
| ∂(spatialNormalMeasure μ : Measure Torus)) ≤ r^2*e^2 → |
| ∃ G : Plane → ℝ, ContDiff ℝ (⊤ : ℕ∞) G ∧ LipschitzWith 1 G ∧ |
| (∀ y, |G y| ≤ α*r) ∧ |
| (∫ y in Metric.ball (0 : Plane) (r/8), ‖fderiv ℝ G y‖^2) ≤ r^2*e^2*D ∧ |
| (let B := r^3*e^2*(D*α+e*(volume : Measure Plane).real (Metric.ball 0 (1/4))) |
| (∫ y in Metric.ball (0 : Plane) (r/4), |
| |G y-normalColumnHeight (openRepresentative E) (quotientMap cubeCenter) n r y|) ≤ B ∧ |
| ∀ s : ℝ, s ≤ r/4 → torusVolume.real |
| (((openRepresentative E) ∆ torusGraphRegion cubeCenter n G) ∩ |
| Metric.ball (quotientMap cubeCenter) s) ≤ B) := by |
| obtain ⟨R,C,hR,hC,happ⟩ := isMinimizerAt_tightly_trapped_graph_approximation |
| hV hE hmin (by norm_num : (0:ℝ)<1/2) le_rfl |
| refine ⟨min R (1/4),C.toReal,by positivity,ENNReal.toReal_nonneg,?_⟩ |
| intro μ hμ N hN hp hunit n hn r α e hr hrr hα0 hα he hes htrap htilt |
| have hrs : r < 1 := lt_of_le_of_lt (hrr.trans (min_le_right _ _)) (by norm_num) |
| obtain ⟨S,g,hg,hh,hS,hgood,hcone,hbad,henergy,hmismatch⟩ := |
| happ μ hμ N hN hp hunit (quotientMap cubeCenter) n hn r α hr |
| (hrr.trans (min_le_left _ _)) hα0 hα htrap |
| have hg1 : LipschitzWith 1 g := hg.weaken (by |
| change (2:ℝ)*(1/2) ≤ 1 |
| norm_num) |
| let T := ∫ x in Metric.closedBall (quotientMap cubeCenter) (17*r), ‖N x-n‖^2 |
| ∂(spatialNormalMeasure μ : Measure Torus) |
| have hT : 0 ≤ T := integral_nonneg fun x ↦ sq_nonneg _ |
| have hgE : (∫ y in Metric.ball (0 : Plane) (r/4), ‖fderiv ℝ g y‖^2) ≤ C.toReal*T := by |
| rw [← ofReal_integral_eq_lintegral_ofReal (lipschitz_gradient_sq_integrable_ball hg1 (r/4)) |
| (Eventually.of_forall fun y ↦ sq_nonneg _)] at henergy |
| have ht := ENNReal.toReal_mono (by finiteness) henergy |
| simpa only [ENNReal.toReal_mul,ENNReal.toReal_ofReal (integral_nonneg fun y ↦ sq_nonneg _)] using ht |
| have hgM : (∫ y in Metric.ball (0 : Plane) (r/4), |
| columnGraphMismatch (openRepresentative E) (quotientMap cubeCenter) n r g y) ≤ |
| C.toReal*α*r*T := by |
| rw [← ofReal_integral_eq_lintegral_ofReal |
| (columnGraphMismatch_integrable_ball (isOpen_openRepresentative E).measurableSet |
| _ n hr.le hg1.continuous.measurable (r/4)) |
| (Eventually.of_forall fun y ↦ columnGraphMismatch_nonneg _ _ _ _ _ _)] at hmismatch |
| have ht := ENNReal.toReal_mono (by finiteness) hmismatch |
| simpa only [ENNReal.toReal_mul,ENNReal.toReal_ofReal (integral_nonneg fun x ↦ sq_nonneg _),ENNReal.toReal_ofReal hα0, |
| ENNReal.toReal_ofReal hr.le,ENNReal.toReal_ofReal |
| (integral_nonneg fun y ↦ columnGraphMismatch_nonneg _ _ _ _ _ _)] using ht |
| have hδ : 0 < r*e^3 := by positivity |
| have he3 : e^3 ≤ 1/8 := by nlinarith [mul_le_mul_of_nonneg_left hes he.le, sq_nonneg (e-1/2)] |
| have hδr : r*e^3 ≤ r/8 := by nlinarith |
| obtain ⟨G,hG,hGL,hheight,hGE,hGM⟩ := exists_physical_smooth_graph |
| (isOpen_openRepresentative E).measurableSet hn hg1 hr hrs hδ hδr hα.le hh |
| have hB : (∫ y in Metric.ball (0 : Plane) (r/4), columnGraphMismatch |
| (openRepresentative E) (quotientMap cubeCenter) n r g y)+ |
| (r*e^3)*r^2*(volume : Measure Plane).real (Metric.ball 0 (1/4)) ≤ |
| r^3*e^2*(C.toReal*α+e*(volume : Measure Plane).real (Metric.ball 0 (1/4))) := by |
| have hx := hgM.trans (mul_le_mul_of_nonneg_left htilt (by positivity : 0 ≤ C.toReal*α*r)) |
| calc |
| _ ≤ C.toReal*α*r*(r^2*e^2)+(r*e^3)*r^2*(volume : Measure Plane).real (Metric.ball 0 (1/4)) := add_le_add hx le_rfl |
| _ = _ := by ring |
| refine ⟨G,hG,hGL,hheight,?_,hGM.1.trans hB,fun s hs ↦ (hGM.2 s hs).trans hB⟩ |
| exact hGE.trans (hgE.trans ((mul_le_mul_of_nonneg_left htilt ENNReal.toReal_nonneg).trans_eq (by ring))) |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
|
|
