| import Mathlib |
| import OAI.Geometry.CubicTorus.Estimates.SliceJensenWeight |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem isMinimizerAt_one_vertex_position {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hm : IsMinimizerAt V E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hp : ∀ m x,N (x+latticeVector m)=N x) |
| (ha : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) |
| (hs : ∀ i : Fin 3,coordinateReflection i ⁻¹' E =ᵐ[torusVolume] E) |
| {s : Fin 3 → Bool} |
| (hsg : ∀ (x : Cover) (i : Fin 3), |
| quotientMap x∈frontier (openRepresentative E)∩coordinateHalf i (if s i then 0 else 1/2) → |
| 0<N x i) |
| (hv : ∀ i : Fin 3,quotientMap (chamberVertex s (fun j => decide (j=i)))∉closure (openRepresentative E)) : |
| ∃ X : Cover → Cover,ContDiff ℝ (⊤ : ℕ∞) X ∧ |
| (∀ m x,X (x+latticeVector m)=X x) ∧ |
| ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → |
| fderiv ℝ X x=ContinuousLinearMap.id ℝ Cover ∧ 0<⟪N x,X x⟫ := by |
| have hsgc : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E)∩coordinateChamber s → |
| ∀ i : Fin 3,0≤N x i := by |
| intro x hx i |
| exact (hsg x i ⟨hx.1,mem_iInter.mp hx.2 i⟩).le |
| have hgap := fun i => isMinimizerAt_chamber_face_gap hV hE hm hμ ha hsgc hs i (hv i) |
| choose d hd hgap using hgap |
| let δ := min (1/4 : ℝ) (min (d 0) (min (d 1) (d 2))) |
| have hδ : 0<δ := lt_min (by norm_num) (lt_min (hd 0) (lt_min (hd 1) (hd 2))) |
| have hδq : δ≤1/4 := min_le_left _ _ |
| have hδd : ∀ i : Fin 3,δ≤d i := by |
| intro i |
| fin_cases i |
| · exact (min_le_right _ _).trans (min_le_left _ _) |
| · exact (min_le_right _ _).trans ((min_le_right _ _).trans (min_le_left _ _)) |
| · exact (min_le_right _ _).trans ((min_le_right _ _).trans (min_le_right _ _)) |
| let r := 1/2-δ/2 |
| have hr : 0<r := by dsimp [r]; linarith |
| have hr' : r<1/2 := by dsimp [r]; linarith |
| obtain ⟨X,hX,hXp,hXe⟩ := exists_periodic_position_box (chamberCorner s) hr hr' |
| have hsF := fun i => openRepresentative_reflection_invariant hE i (hs i) |
| have hcorner := (isMinimizerAt_chamber_vertex_order hV hE hm hμ ha hsgc hs).1 |
| rw [chamberVertex_bottom] at hcorner |
| refine ⟨X,hX,hXp,?_⟩ |
| intro x hx |
| let a := quotientMap x |
| let y := shortestLift (a-quotientMap (chamberCorner s))+chamberCorner s |
| have hyq : quotientMap y=quotientMap x := by |
| change quotientMap (shortestLift (a-quotientMap (chamberCorner s))+chamberCorner s)=a |
| rw [quotientMap_add,quotientMap_shortestLift,sub_add_cancel] |
| have hyb : ∀ i : Fin 3,|y i-chamberCorner s i|≤r := by |
| intro i |
| have hclosed : quotientMap (chamberFoldLift s a)∈closure (openRepresentative E) := |
| (chamberFoldLift_mem_iff (reflection_invariant_closure hsF) s a).mpr (frontier_subset_closure hx) |
| have hgi := hgap i (chamberFoldLift s a) (chamberFoldLift_mem s a) hclosed |
| rw [chamberFoldLift_apply] at hgi |
| change |shortestLift (a-quotientMap (chamberCorner s)) i+chamberCorner s i-chamberCorner s i|≤r |
| simp only [add_sub_cancel_right] |
| dsimp [r] |
| linarith [hδd i] |
| have hyS : quotientMap y∈frontier (openRepresentative E) := by rwa [hyq] |
| have hyne : y≠chamberCorner s := by |
| intro he |
| rw [he,frontier,(isOpen_openRepresentative E).interior_eq] at hyS |
| exact hyS.2 hcorner |
| have hnxy : N x=N y := by |
| rw [← periodic_representative hp x,← periodic_representative hp y,hyq] |
| have hXxy : X x=X y := by |
| rw [← periodic_representative hXp x,← periodic_representative hXp y,hyq] |
| have hDxy : fderiv ℝ X x=fderiv ℝ X y := by |
| rw [← periodic_representative (fderiv_periodic hXp) x, |
| ← periodic_representative (fderiv_periodic hXp) y,hyq] |
| have he := hXe y hyb |
| constructor |
| · rw [hDxy,he.fderiv_eq] |
| exact ((hasFDerivAt_id y).sub_const (chamberCorner s)).fderiv |
| · rw [hnxy,hXxy,he.eq_of_nhds] |
| exact isMinimizerAt_chamber_support_positive hV hE hm hμ hN hp ha hs hsg hyS |
| (fun i => (hyb i).trans_lt hr') hyne |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
| namespace Periodic |
|
|
| theorem isMinimizerAt_full_jacobi_system {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hm : IsMinimizerAt V E) : |
| ∃ (μ : FiniteMeasure NormalBundle) (H : ℝ) (N : Cover → Cover) |
| (hN : ContDiff ℝ (⊤ : ℕ∞) N) (hp : ∀ m x,N (x+latticeVector m)=N x), |
| IsNormalMeasure E μ ∧ |
| (∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) ∧ |
| (∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → |
| (∀ᶠ y in 𝓝 x,‖N y‖=1) ∧ divergence N x=H) ∧ |
| (∀ F : Cover → Cover,ContDiff ℝ (⊤ : ℕ∞) F → |
| (∀ m x,F (x+latticeVector m)=F x) → |
| (∫ z,tangentialDivergence F z.1 z.2 ∂(μ : Measure NormalBundle))= |
| H*(∫ z,⟪(z.2 : Cover),F z.1⟫ ∂(μ : Measure NormalBundle))) ∧ |
| (∀ u∈surfaceH1 μ,surfaceMean μ u=0 → |
| 0 ≤ surfaceJacobiB μ (surfaceCurvatureLp μ N hN) u u) ∧ |
| (∀ i : Fin 3,∀ w∈surfaceH1 μ,surfaceJacobiB μ (surfaceCurvatureLp μ N hN) |
| (surfaceSmoothJet μ (normalComponentScalar N hN hp i)) w=0) := by |
| obtain ⟨μ,H,hμ,hCMC,hs⟩ := isMinimizerAt_boundary_stability hV hE hm |
| obtain ⟨N,hN,hp,hn⟩ := isMinimizerAt_global_normal hV hE hm hμ |
| have hdiv := weakCMC_normal_extension hCMC hN hp hn |
| let R := regularizedNormal N |
| have hR : ContDiff ℝ (⊤ : ℕ∞) R := regularizedNormal_contDiff hN |
| have hRp : ∀ m x,R (x+latticeVector m)=R x := regularizedNormal_periodic hp |
| have he : ∀ᵐ z ∂(μ : Measure NormalBundle),R=ᶠ[𝓝 (z.1 : Cover)]N := |
| hn.mono fun z hz => regularizedNormal_eventuallyEq hz.2 |
| have hRa : ∀ᵐ z ∂(μ : Measure NormalBundle),R z.1=(z.2 : Cover) := by |
| filter_upwards [hn,he] with z hz hez |
| exact hez.eq_of_nhds.trans hz.1 |
| have hRu : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → |
| ∀ᶠ y in 𝓝 x,‖R y‖=1 := by |
| intro x hx |
| have hnorm : ‖N x‖=1 := by |
| apply isMinimizerAt_periodic_ae_eq_boundary hV hE hm hμ hN.continuous.norm continuous_const |
| (fun m x => by rw [hp]) (fun _ _ => rfl) _ x hx |
| filter_upwards [hn] with z hz |
| rw [hz.1,z.2.property] |
| exact regularizedNormal_unit_near hN.continuous (by rw [hnorm]; norm_num) |
| have hRd : ∀ᵐ z ∂(μ : Measure NormalBundle),divergence R z.1=H := by |
| filter_upwards [he,hdiv] with z hz hdz |
| simpa only [divergence_eq_coverTrace,hz.fderiv_eq] using hdz |
| have hRd' : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → divergence R x=H := |
| isMinimizerAt_periodic_ae_eq_boundary hV hE hm hμ (continuous_divergence hR) continuous_const |
| (divergence_periodic hRp) (fun _ _ => rfl) hRd |
| have hμU := hμ.congr_ae (isMinimizerAt_openRepresentative_ae hV hE hm).symm |
| have hbd := normalMeasure_frontier_ae (isOpen_openRepresentative E).measurableSet hμU |
| have hstab : ∀ f : Cover → ℝ,ContDiff ℝ (⊤ : ℕ∞) f → |
| (∀ m x,f (x+latticeVector m)=f x) → |
| (∫ z,f z.1 ∂(μ : Measure NormalBundle))=0 → 0≤normalExtensionJacobi μ f R := by |
| intro f hf hfp hz |
| have hgeom : ∀ᵐ z ∂(μ : Measure NormalBundle),z.1∈tsupport f → |
| R z.1=(z.2 : Cover) ∧ (∀ᶠ y in 𝓝 (z.1 : Cover),‖R y‖=1) ∧ divergence R z.1=H := by |
| filter_upwards [hRa,hbd,hRd] with z ha hb hd |
| exact fun _ => ⟨ha,hRu z.1 hb,hd⟩ |
| rw [← boundaryQuadratic_eq_normalExtension hf hR hgeom] |
| apply hs (fun y => f y • R y) (hf.smul hR) |
| · intro m x |
| simp only [hfp,hRp] |
| · rw [← hz] |
| apply integral_congr_ae |
| filter_upwards [hRa] with z ha |
| rw [ha,real_inner_smul_right,real_inner_self_eq_norm_sq,z.2.property,one_pow,mul_one] |
| have hmass : (μ : Measure NormalBundle).real univ≠0 := by |
| rw [Measure.real,← FiniteMeasure.ennreal_mass] |
| change (μ.mass : ℝ)≠0 |
| rw [hμ.1] |
| exact (ENNReal.toReal_pos (perimeter_pos_of_volume hV hm.1.1 hm.2.1).ne' hm.1.2.ne).ne' |
| have hH1 : ∀ u∈surfaceH1 μ,surfaceMean μ u=0 → |
| 0 ≤ surfaceJacobiB μ (surfaceCurvatureLp μ R hR) u u := |
| fun u hu hz => surfaceH1_stability_of_smooth μ hmass R hR hstab hu hz |
| exact ⟨μ,H,R,hR,hRp,hμ,hRa,fun x hx => ⟨hRu x hx,hRd' x hx⟩,hCMC,hH1, |
| fun i => normalComponentScalar_global_jacobi hV hE hm hμ hR hRp hRa hRu hH1 i⟩ |
|
|
| end Periodic |
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma tangentialDivergence_id_derivative {X : Cover → Cover} {x n : Cover} |
| (hd : fderiv ℝ X x=ContinuousLinearMap.id ℝ Cover) (hn : ‖n‖=1) : |
| tangentialDivergence X x n=2 := by |
| simp only [tangentialDivergence,divergence_eq_coverTrace,hd,coverTrace, |
| ContinuousLinearMap.id_apply,PiLp.single_apply,ite_true, |
| Finset.sum_const,Finset.card_fin,real_inner_self_eq_norm_sq,hn,one_pow] |
| norm_num |
|
|
| lemma tangentialDivergence_unit_normal {N : Cover → Cover} {x : Cover} |
| (hN : DifferentiableAt ℝ N x) (hu : ∀ᶠ y in 𝓝 x,‖N y‖=1) : |
| tangentialDivergence N x (N x)=divergence N x := by |
| have hz := smooth_unit_normal_derivative hN hu |
| rw [tangentialDivergence,← ContinuousLinearMap.adjoint_inner_left,hz,inner_zero_left,sub_zero] |
|
|
| theorem isMinimizerAt_positive_position_spherical {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hm : IsMinimizerAt V E) |
| (hc : IsPreconnected (frontier (openRepresentative E))) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) {H : ℝ} |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hpN : ∀ m x,N (x+latticeVector m)=N x) |
| (ha : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) |
| (hb : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → |
| (∀ᶠ y in 𝓝 x,‖N y‖=1) ∧ divergence N x=H) |
| (hCMC : ∀ F : Cover → Cover,ContDiff ℝ (⊤ : ℕ∞) F → |
| (∀ m x,F (x+latticeVector m)=F x) → |
| (∫ z,tangentialDivergence F z.1 z.2 ∂(μ : Measure NormalBundle))= |
| H*(∫ z,⟪(z.2 : Cover),F z.1⟫ ∂(μ : Measure NormalBundle))) |
| (hj : ∀ i : Fin 3,∀ w∈surfaceH1 μ,surfaceJacobiB μ (surfaceCurvatureLp μ N hN) |
| (surfaceSmoothJet μ (normalComponentScalar N hN hpN i)) w=0) |
| {X : Cover → Cover} (hX : ContDiff ℝ (⊤ : ℕ∞) X) |
| (hpX : ∀ m x,X (x+latticeVector m)=X x) |
| (hDX : ∀ᵐ z ∂(μ : Measure NormalBundle),fderiv ℝ X z.1=ContinuousLinearMap.id ℝ Cover) |
| (hpos : ∀ᵐ z ∂(μ : Measure NormalBundle),0<⟪N z.1,X z.1⟫) : |
| ∃ (c : Cover) (r : ℝ),0<r ∧ r<1/2 ∧ |
| frontier (openRepresentative E)=quotientMap '' Metric.sphere c r := by |
| let q : NormalBundle → ℝ := fun z => coverTrace ((fderiv ℝ N z.1).comp (fderiv ℝ N z.1)) |
| let u : NormalBundle → ℝ := fun z => ⟪N z.1,X z.1⟫ |
| let A := (μ : Measure NormalBundle).real univ |
| have hqc : Continuous q := (normal_curvature_continuous hN).comp normalEmbedding_continuous.fst |
| have huc : Continuous u := (hN.continuous.comp normalEmbedding_continuous.fst).inner (𝕜 := ℝ) |
| (hX.continuous.comp normalEmbedding_continuous.fst) |
| have hqi := hqc.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace q) (μ := (μ : Measure NormalBundle)) |
| have hui := huc.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace u) (μ := (μ : Measure NormalBundle)) |
| have hqui := (hqc.mul huc).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) (μ := (μ : Measure NormalBundle)) |
| have hμU := hμ.congr_ae (isMinimizerAt_openRepresentative_ae hV hE hm).symm |
| have hbd := normalMeasure_frontier_ae (isOpen_openRepresentative E).measurableSet hμU |
| have hdiv : ∀ᵐ z ∂(μ : Measure NormalBundle),divergence N z.1=H := |
| hbd.mono fun z hz => (hb z.1 hz).2 |
| have hunit : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover) ∧ |
| (∀ᶠ y in 𝓝 (z.1 : Cover),‖N y‖=1) := by |
| filter_upwards [ha,hbd] with z haz hbz |
| exact ⟨haz,(hb z.1 hbz).1⟩ |
| have htN : ∀ᵐ z ∂(μ : Measure NormalBundle),tangentialDivergence N z.1 z.2=H := by |
| filter_upwards [hunit,hdiv] with z hz hdz |
| rw [← hz.1,tangentialDivergence_unit_normal (hN.differentiable (by simp) _) hz.2,hdz] |
| have hQ : (∫ z,q z*u z ∂(μ : Measure NormalBundle))=H*A := by |
| rw [translation_jacobi_position_identity hN hX hpN hpX hj hDX,integral_congr_ae htN, |
| integral_const,smul_eq_mul,mul_comm] |
| have hM : H*(∫ z,u z ∂(μ : Measure NormalBundle))=2*A := by |
| have hh := hCMC X hX hpX |
| have hleft : (∫ z,tangentialDivergence X z.1 z.2 ∂(μ : Measure NormalBundle))=2*A := by |
| calc |
| _ = ∫ _ : NormalBundle,(2 : ℝ) ∂(μ : Measure NormalBundle) := integral_congr_ae |
| (hDX.mono fun z hz => tangentialDivergence_id_derivative hz z.2.property) |
| _ = _ := by rw [integral_const,smul_eq_mul,mul_comm] |
| rw [hleft] at hh |
| rw [hh] |
| congr 1 |
| apply integral_congr_ae |
| filter_upwards [ha] with z hz |
| exact congrArg (fun n : Cover => ⟪n,X z.1⟫) hz |
| have hA : 0<A := by |
| change 0<(μ : Measure NormalBundle).real univ |
| rw [Measure.real,← FiniteMeasure.ennreal_mass] |
| change 0<(μ.mass : ℝ) |
| rw [hμ.1] |
| exact ENNReal.toReal_pos (perimeter_pos_of_volume hV hm.1.1 hm.2.1).ne' hm.1.2.ne |
| have hH : H≠0 := by intro hz; rw [hz,zero_mul] at hM; linarith |
| have hbound : ∀ᵐ z ∂(μ : Measure NormalBundle),0≤2*q z-H^2 := by |
| filter_upwards [hbd,hunit,hdiv] with z hbz huz hdz |
| have hh := isMinimizerAt_mean_curvature_bound hV hE hm hμ hN hpN ha z.1 hbz huz.2 |
| rw [hdz] at hh |
| exact sub_nonneg.mpr hh |
| have hip : Integrable (fun z => (2*q z-H^2)*u z) (μ : Measure NormalBundle) := |
| (((continuous_const.mul hqc).sub continuous_const).mul huc).integrable_of_hasCompactSupport |
| (HasCompactSupport.of_compactSpace _) |
| have hzero : (∫ z,(2*q z-H^2)*u z ∂(μ : Measure NormalBundle))=0 := by |
| simp_rw [sub_mul,mul_assoc] |
| have hiq : Integrable (fun z => 2*(q z*u z)) (μ : Measure NormalBundle) := hqui.const_mul 2 |
| have hiu : Integrable (fun z => H^2*u z) (μ : Measure NormalBundle) := hui.const_mul _ |
| rw [integral_sub hiq hiu,integral_const_mul,integral_const_mul,hQ] |
| nlinarith [congrArg (fun t : ℝ => H*t) hM] |
| have hzp := (integral_eq_zero_iff_of_nonneg_ae (by |
| filter_upwards [hbound,hpos] with z hz hpz |
| exact mul_nonneg hz hpz.le) hip).mp hzero |
| have hqeq : ∀ᵐ z ∂(μ : Measure NormalBundle),q z=H^2/2 := by |
| filter_upwards [hzp,hpos] with z hz hpz |
| have he := (mul_eq_zero.mp hz).resolve_right hpz.ne' |
| linarith |
| have hint : (∫ z,q z ∂(μ : Measure NormalBundle))=H^2*isoperimetricProfile V := by |
| rw [integral_congr_ae hqeq,integral_const,smul_eq_mul] |
| have hmA : A=2*isoperimetricProfile V := by |
| change (μ : Measure NormalBundle).real univ=_ |
| rw [Measure.real,← FiniteMeasure.ennreal_mass] |
| change (μ.mass : ℝ)=_ |
| rw [hμ.1,isoperimetricProfile_eq_of_isMinimizerAt hm] |
| ring |
| change A*(H^2/2)=_ |
| rw [hmA]; ring |
| obtain ⟨hr,hr',c,hc'⟩ := isMinimizerAt_mean_square_equality_spherical hV hE hm hc hμ hN hpN |
| hunit hH hdiv hint |
| exact ⟨c,|2/H|,hr,hr',hc'⟩ |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem isMinimizerAt_one_vertex_spherical {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hm : IsMinimizerAt V E) |
| (hc : IsPreconnected (frontier (openRepresentative E))) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hp : ∀ m x,N (x+latticeVector m)=N x) |
| (ha : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) |
| (hs : ∀ i : Fin 3,coordinateReflection i ⁻¹' E =ᵐ[torusVolume] E) |
| {s : Fin 3 → Bool} |
| (hsg : ∀ (x : Cover) (i : Fin 3), |
| quotientMap x∈frontier (openRepresentative E)∩coordinateHalf i (if s i then 0 else 1/2) → |
| 0<N x i) |
| (hv : ∀ i : Fin 3,quotientMap (chamberVertex s (fun j => decide (j=i)))∉closure (openRepresentative E)) : |
| ∃ (c : Cover) (r : ℝ),0<r ∧ r<1/2 ∧ |
| frontier (openRepresentative E)=quotientMap '' Metric.sphere c r := by |
| obtain ⟨ν,H,M,hM,hpM,hν,haM,hb,hCMC,_,hj⟩ := |
| Periodic.isMinimizerAt_full_jacobi_system hV hE hm |
| have he : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → N x=M x := by |
| intro x hx |
| exact isMinimizerAt_normal_local_ae_eq hV hV hE hE hm hm isOpen_univ (Filter.EventuallyEq.rfl) |
| hμ hν hN hM hp hpM ha haM x ⟨hx,mem_univ _⟩ |
| have hsgM : ∀ (x : Cover) (i : Fin 3), |
| quotientMap x∈frontier (openRepresentative E)∩coordinateHalf i (if s i then 0 else 1/2) → |
| 0<M x i := by |
| intro x i hx |
| rw [← he x hx.1] |
| exact hsg x i hx |
| obtain ⟨X,hX,hpX,hXb⟩ := isMinimizerAt_one_vertex_position hV hE hm hν hM hpM haM hs hsgM hv |
| have hνU := hν.congr_ae (isMinimizerAt_openRepresentative_ae hV hE hm).symm |
| have hbd := normalMeasure_frontier_ae (isOpen_openRepresentative E).measurableSet hνU |
| exact isMinimizerAt_positive_position_spherical hV hE hm hc hν hM hpM haM hb hCMC hj hX hpX |
| (hbd.mono fun z hz => (hXb z.1 hz).1) (hbd.mono fun z hz => (hXb z.1 hz).2) |
|
|
| theorem isMinimizerAt_one_vertex_standard_ball {V : ℝ} (hV : V∈Ioc 0 (1/2)) |
| {E : Set Torus} (hE : MeasurableSet E) (hm : IsMinimizerAt V E) |
| (hc : IsPreconnected (frontier (openRepresentative E))) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hp : ∀ m x,N (x+latticeVector m)=N x) |
| (ha : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) |
| (hs : ∀ i : Fin 3,coordinateReflection i ⁻¹' E =ᵐ[torusVolume] E) |
| {s : Fin 3 → Bool} |
| (hsg : ∀ (x : Cover) (i : Fin 3), |
| quotientMap x∈frontier (openRepresentative E)∩coordinateHalf i (if s i then 0 else 1/2) → |
| 0<N x i) |
| (hv : ∀ i : Fin 3,quotientMap (chamberVertex s (fun j => decide (j=i)))∉closure (openRepresentative E)) : |
| V≤4*Real.pi/81 ∧ CongruentAE E (canonicalBall V) := by |
| obtain ⟨c,r,hr,hr',hbd⟩ := isMinimizerAt_one_vertex_spherical |
| ⟨hV.1,by linarith [hV.2]⟩ hE hm hc hμ hN hp ha hs hsg hv |
| exact isMinimizerAt_spherical_standard_ball hV hE hm hr hr' hbd |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem isMinimizerAt_coordinate_multiplier {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hm : IsMinimizerAt V E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) : |
| ∃ H : ℝ,∀ D : Periodic.Variation,Periodic.normalFirstVariation μ D= |
| H*directionalFlux E D.coordinate D.function := by |
| obtain ⟨i,f,hf,hp,hflux⟩ := exists_positive_directionalFlux |
| (perimeter_pos_of_volume hV hm.1.1 hm.2.1) |
| obtain ⟨D,hi,hfD⟩ := Periodic.exists_variation_of_function i hf hp |
| have hD : directionalFlux E D.coordinate D.function≠0 := by |
| rw [hi,hfD]; exact hflux.ne' |
| exact Periodic.isMinimizerAt_exists_coordinate_multiplier hm hE hμ D hD |
|
|
| theorem isMinimizerAt_nonstandard_section_system {V : ℝ} (hV : V∈Ioc 0 (1/2)) |
| {E : Set Torus} (hmE : IsMinimizerAt V E) (hnstd : ¬SmallStandardRegion V E) : |
| ∃ (b : Torus) (F : Set Torus) (μ : FiniteMeasure NormalBundle) (N : Cover → Cover) |
| (_ : ContDiff ℝ (⊤ : ℕ∞) N) (_ : ∀ m x,N (x+latticeVector m)=N x) |
| (s : Fin 3 → Bool) (H : ℝ), |
| MeasurableSet F ∧ F =ᵐ[torusVolume] ((fun z => z+b) ⁻¹' E) ∧ IsMinimizerAt V F ∧ |
| IsPreconnected (frontier (openRepresentative F)) ∧ IsNormalMeasure F μ ∧ |
| (∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) ∧ |
| (∀ i : Fin 3,coordinateReflection i ⁻¹' F =ᵐ[torusVolume] F) ∧ |
| (∀ (x : Cover) (i : Fin 3), |
| quotientMap x∈frontier (openRepresentative F)∩coordinateHalf i (if s i then 0 else 1/2) → 0<N x i) ∧ |
| (∀ D : Periodic.Variation,Periodic.normalFirstVariation μ D= |
| H*directionalFlux F D.coordinate D.function) ∧ |
| (0<(μ.mass : ℝ)) ∧ (∑ i : Fin 3,axialFluxWeight μ i)=2 ∧ |
| (∀ i : Fin 3,0≤axialFluxWeight μ i ∧ axialFluxWeight μ i<1) ∧ |
| (∃ i : Fin 3,quotientMap (chamberVertex s (fun j => decide (j=i)))∈closure (openRepresentative F)) ∧ |
| (∀ i : Fin 3,Continuous (axialSectionArea (openRepresentative F) i) ∧ |
| AntitoneOn (axialSectionArea (openRepresentative F) i) |
| (Icc (chamberCorner s i) (chamberCorner s i+1/2))) ∧ |
| (∀ i : Fin 3, |
| 2*(axialSectionArea (openRepresentative F) i (chamberCorner s i)- |
| axialSectionArea (openRepresentative F) i (chamberCorner s i+1/2))≤ |
| (μ.mass : ℝ)*Real.sqrt (1-axialFluxWeight μ i)) ∧ |
| (∀ (i : Fin 3) (a : Bool), |
| axialSlicePerimeter (openRepresentative F) i (if a then 1/2 else 0)= |
| ENNReal.ofReal ((μ.mass : ℝ)*axialFluxWeight μ i+ |
| H*(axialSectionArea (openRepresentative F) i (if a then 1/2 else 0)-V))) := by |
| classical |
| have hV' : V∈Ioo (0 : ℝ) 1 := ⟨hV.1,by linarith [hV.2]⟩ |
| obtain ⟨b,F,μ,N,hN,hp,hF,he,hmF,hc,hμ,ha,hs,_,hsg⟩ := |
| (isMinimizerAt_standard_or_strict_reflection hV hmE).resolve_left hnstd |
| obtain ⟨s,hsg'⟩ := isMinimizerAt_exists_positive_sides hV' hF hmF hμ hN hp ha hs hsg |
| obtain ⟨H,hH⟩ := isMinimizerAt_coordinate_multiplier hV' hF hmF hμ |
| have hmass : 0<(μ.mass : ℝ) := by |
| rw [hμ.1] |
| exact ENNReal.toReal_pos (perimeter_pos_of_volume hV' hmF.1.1 hmF.2.1).ne' hmF.1.2.ne |
| have hzero : ∀ i : Fin 3,∃ a∈frontier (openRepresentative F),N (representative a) i≠0 := by |
| intro i |
| by_contra hn |
| push Not at hn |
| obtain ⟨c,r,hr0,hr,hbd⟩ := isMinimizerAt_zero_coordinate_cylindrical hV' hF hmF hc hμ hN hp ha i hn |
| obtain ⟨hl,hu,hcong⟩ := isMinimizerAt_cylindrical_standard_tube hV hF hmF i hr0 hr hbd |
| exact hnstd (Or.inr (Or.inl ⟨hl,hu,congruentAE_of_translated he hcong⟩)) |
| have hv : ∃ i : Fin 3,quotientMap (chamberVertex s (fun j => decide (j=i)))∈closure (openRepresentative F) := by |
| by_contra hn |
| push Not at hn |
| obtain ⟨hsmall,hcong⟩ := isMinimizerAt_one_vertex_standard_ball hV hF hmF hc hμ hN hp ha hs hsg' hn |
| exact hnstd (Or.inl ⟨hsmall,congruentAE_of_translated he hcong⟩) |
| refine ⟨b,F,μ,N,hN,hp,s,H,hF,he,hmF,hc,hμ,ha,hs,hsg',hH,hmass, |
| sum_axialFluxWeight μ hmass.ne',?_,hv,?_,?_,?_⟩ |
| · intro i |
| exact ⟨axialFluxWeight_nonneg μ i, |
| (axialFluxWeight_lt_one_iff hV' hF hmF hμ hN.continuous hp ha hmass i).mpr (hzero i)⟩ |
| · intro i |
| exact ⟨isMinimizerAt_continuous_axial_sections hV' hF hmF hμ hN hp ha hs hsg' i, |
| isMinimizerAt_antitoneOn_axialSectionArea hV' hF hmF hμ hN hp ha hs hsg' i⟩ |
| · intro i |
| rw [← isMinimizerAt_axial_endFlux hV' hF hmF hμ hN hp ha hs hsg' hH i] |
| exact axialProjection_bound μ i |
| · exact isMinimizerAt_axial_endpoint_perimeter hV' hF hmF hμ hN hp ha hs hsg' hH |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem isMinimizerAt_axial_bv_coarea {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hm : IsMinimizerAt V E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hp : ∀ m x,N (x+latticeVector m)=N x) |
| (ha : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) |
| (hs : ∀ i : Fin 3,coordinateReflection i ⁻¹' E =ᵐ[torusVolume] E) |
| {s : Fin 3 → Bool} |
| (hsg : ∀ (x : Cover) (i : Fin 3), |
| quotientMap x∈frontier (openRepresentative E)∩coordinateHalf i (if s i then 0 else 1/2) → 0<N x i) |
| (i : Fin 3) : |
| Measurable (axialCirclePerimeter (openRepresentative E) i) ∧ |
| (∀ᵐ t,axialCirclePerimeter (openRepresentative E) i t≠∞) ∧ |
| Integrable (fun t => (axialCirclePerimeter (openRepresentative E) i t).toReal) ∧ |
| ∀ f : AddCircle (1 : ℝ) → ℝ,Measurable f → |
| (∫ t,(axialCirclePerimeter (openRepresentative E) i t).toReal*f t)= |
| ∫ z,axialTangentialSpeed i z*f (axialBundleHeight i z) ∂(μ : Measure NormalBundle) := by |
| obtain ⟨H,hH⟩ := isMinimizerAt_coordinate_multiplier hV hE hm hμ |
| exact axial_bv_coarea (isOpen_openRepresentative E).measurableSet |
| (hμ.congr_ae (isMinimizerAt_openRepresentative_ae hV hE hm).symm) hN hp ha i |
| (isMinimizerAt_axial_boundary_null hV hE hm hμ hN hp ha hs hsg i) |
| (isMinimizerAt_axialHeight_absolutelyContinuous hV hE hm hμ hN hp ha hs hsg hH i) |
|
|
| theorem isMinimizerAt_nonstandard_actual_coarea {V : ℝ} (hV : V∈Ioc 0 (1/2)) |
| {E : Set Torus} (hmE : IsMinimizerAt V E) (hnstd : ¬SmallStandardRegion V E) : |
| ∃ (b : Torus) (F : Set Torus) (μ : FiniteMeasure NormalBundle), |
| MeasurableSet F ∧ F =ᵐ[torusVolume] ((fun z => z+b) ⁻¹' E) ∧ IsMinimizerAt V F ∧ |
| IsPreconnected (frontier (openRepresentative F)) ∧ IsNormalMeasure F μ ∧ |
| ∀ i : Fin 3,Measurable (axialCirclePerimeter (openRepresentative F) i) ∧ |
| (∀ᵐ t,axialCirclePerimeter (openRepresentative F) i t≠∞) ∧ |
| Integrable (fun t => (axialCirclePerimeter (openRepresentative F) i t).toReal) ∧ |
| ∀ f : AddCircle (1 : ℝ) → ℝ,Measurable f → |
| (∫ t,(axialCirclePerimeter (openRepresentative F) i t).toReal*f t)= |
| ∫ z,axialTangentialSpeed i z*f (axialBundleHeight i z) ∂(μ : Measure NormalBundle) := by |
| have hV' : V∈Ioo (0 : ℝ) 1 := ⟨hV.1,by linarith [hV.2]⟩ |
| obtain ⟨b,F,μ,N,hN,hp,s,H,hF,he,hmF,hc,hμ,ha,hs,hsg,rest⟩ := |
| isMinimizerAt_nonstandard_section_system hV hmE hnstd |
| refine ⟨b,F,μ,hF,he,hmF,hc,hμ,?_⟩ |
| intro i |
| exact isMinimizerAt_axial_bv_coarea hV' hF hmF hμ hN hp ha hs hsg i |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem isMinimizerAt_integrated_jensen {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hm : IsMinimizerAt V E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hp : ∀ m x,N (x+latticeVector m)=N x) |
| (ha : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) |
| (hs : ∀ i : Fin 3,coordinateReflection i ⁻¹' E =ᵐ[torusVolume] E) |
| {s : Fin 3 → Bool} |
| (hsg : ∀ (x : Cover) (i : Fin 3), |
| quotientMap x∈frontier (openRepresentative E)∩coordinateHalf i (if s i then 0 else 1/2) → 0<N x i) |
| {H : ℝ} (hH : ∀ D : Periodic.Variation,Periodic.normalFirstVariation μ D= |
| H*directionalFlux E D.coordinate D.function) (i : Fin 3) : |
| let J : ℝ → ℝ := fun t => sliceJensenG (axialJensenFlux (openRepresentative E) μ H V i t/ |
| (axialCirclePerimeter (openRepresentative E) i t).toReal) |
| (∀ᵐ t ∂volume.restrict (Ioo (0 : ℝ) 1), |
| 0≤-deriv (axialSectionArea (openRepresentative E) i) t*chamberHeightSign (s i) t) ∧ |
| IntervalIntegrable (fun t => J t*(-deriv (axialSectionArea (openRepresentative E) i) t* |
| chamberHeightSign (s i) t)) volume 0 1 ∧ |
| (∫ t in (0 : ℝ)..1,J t*(-deriv (axialSectionArea (openRepresentative E) i) t* |
| chamberHeightSign (s i) t))≤ |
| (μ.mass : ℝ)-(49/50 : ℝ)*(μ.mass : ℝ)*axialFluxWeight μ i := by |
| dsimp only |
| have hae := isMinimizerAt_openRepresentative_ae hV hE hm |
| have hμo := hμ.congr_ae hae.symm |
| have hvol : torusVolume.real (openRepresentative E)=V := by |
| rw [Measure.real,measure_congr hae,hm.2.1,ENNReal.toReal_ofReal hV.1.le] |
| have hHo : ∀ D : Periodic.Variation,Periodic.normalFirstVariation μ D= |
| H*directionalFlux (openRepresentative E) D.coordinate D.function := by |
| intro D |
| rw [directionalFlux_congr_ae hae,hH] |
| have hc := isMinimizerAt_continuous_axial_sections hV hE hm hμ hN hp ha hs hsg i |
| have hac := isMinimizerAt_axialHeight_absolutelyContinuous hV hE hm hμ hN hp ha hs hsg hH i |
| obtain ⟨r,hr,hr0,hrm,hJi,hJ⟩ := axial_integrated_jensen |
| (isOpen_openRepresentative E).measurableSet hvol hμo hHo hN hp ha i hc |
| (isMinimizerAt_axial_boundary_null hV hE hm hμ hN hp ha hs hsg i) hac |
| obtain ⟨g,hg,hgm,hgd⟩ := axial_section_derivative_density |
| (isOpen_openRepresentative E).measurableSet hμo i hac hc |
| have hfront := normalMeasure_frontier_ae (isOpen_openRepresentative E).measurableSet hμo |
| have hsign : ∀ᵐ z ∂(μ : Measure NormalBundle), |
| ((z.2 : Cover) i)*chamberHeightSign (s i) (axialBundleHeight i z)=|((z.2 : Cover) i)| := by |
| filter_upwards [hfront,ha] with z hz hzn |
| have he := isMinimizerAt_normal_chamberHeightSign hV hE hm hμ hN hp ha hs hsg hz i |
| rw [hzn] at he |
| exact he |
| have hrg := circle_unsigned_normal_density i (s i) hr hg hrm hgm hsign |
| have hrgr := (AddCircle.measurePreserving_mk (1 : ℝ) 0).quasiMeasurePreserving.ae hrg |
| simp only [zero_add] at hrgr |
| rw [← restrict_Ioo_eq_restrict_Ioc] at hrgr |
| have heq : (fun t : ℝ => sliceJensenG (axialJensenFlux (openRepresentative E) μ H V i t/ |
| (axialCirclePerimeter (openRepresentative E) i t).toReal)*r t) =ᵐ[volume.restrict (Ioo (0 : ℝ) 1)] |
| (fun t => sliceJensenG (axialJensenFlux (openRepresentative E) μ H V i t/ |
| (axialCirclePerimeter (openRepresentative E) i t).toReal)* |
| (-deriv (axialSectionArea (openRepresentative E) i) t*chamberHeightSign (s i) t)) := by |
| filter_upwards [hrgr,hgd] with t ht htd |
| rw [ht,htd] |
| have hI : IntervalIntegrable (fun t : ℝ => sliceJensenG (axialJensenFlux (openRepresentative E) μ H V i t/ |
| (axialCirclePerimeter (openRepresentative E) i t).toReal)*r t) volume 0 1 := by |
| rw [intervalIntegrable_iff_integrableOn_Ioo_of_le (by norm_num : (0 : ℝ)≤1)] |
| exact (circle_integrable_preimage_Icc hJi 0).mono_set (by simpa only [zero_add] using Ioo_subset_Icc_self) |
| have heqi : _ = _ := intervalIntegral.integral_congr_ae_restrict |
| (f := fun t : ℝ => sliceJensenG (axialJensenFlux (openRepresentative E) μ H V i t/ |
| (axialCirclePerimeter (openRepresentative E) i t).toReal)*r t) |
| (g := fun t : ℝ => sliceJensenG (axialJensenFlux (openRepresentative E) μ H V i t/ |
| (axialCirclePerimeter (openRepresentative E) i t).toReal)* |
| (-deriv (axialSectionArea (openRepresentative E) i) t*chamberHeightSign (s i) t)) |
| (μ := volume) (a := 0) (b := 1) (by rwa [uIoc_of_le (by norm_num : (0 : ℝ)≤1),← restrict_Ioo_eq_restrict_Ioc]) |
| refine ⟨?_,?_,?_⟩ |
| · filter_upwards [hrgr,hgd] with t ht htd |
| rw [htd] |
| exact (hr0 (t : AddCircle (1 : ℝ))).trans_eq ht |
| · rw [intervalIntegrable_iff_integrableOn_Ioo_of_le (by norm_num : (0 : ℝ)≤1)] at hI ⊢ |
| exact hI.congr heq |
| · rw [← heqi] |
| rw [← AddCircle.intervalIntegral_preimage (1 : ℝ) 0] at hJ |
| simpa only [zero_add] using hJ |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology BoundedContinuousFunction |
| namespace CubicTorus |
|
|
| lemma absolutelyContinuous_comp_lipschitz {f : ℝ → ℝ} {a b : ℝ} |
| (hf : AbsolutelyContinuousOnInterval f a b) {G : ℝ → ℝ} {K : ℝ≥0} |
| (hG : LipschitzWith K G) : AbsolutelyContinuousOnInterval (G ∘ f) a b := by |
| rw [absolutelyContinuousOnInterval_iff] at hf ⊢ |
| intro ε hε |
| have hK : 0<(K : ℝ)+1 := by positivity |
| obtain ⟨δ,hδ,hd⟩ := hf (ε/((K : ℝ)+1)) (div_pos hε hK) |
| refine ⟨δ,hδ,?_⟩ |
| intro E hE he |
| have hsum := hd E hE he |
| calc |
| _≤∑ i∈Finset.range E.1,((K : ℝ)+1)*dist (f (E.2 i).1) (f (E.2 i).2) := by |
| apply Finset.sum_le_sum |
| intro i hi |
| exact (hG.dist_le_mul _ _).trans (by nlinarith [(dist_nonneg : 0≤dist (f (E.2 i).1) (f (E.2 i).2))]) |
| _=((K : ℝ)+1)*(∑ i∈Finset.range E.1,dist (f (E.2 i).1) (f (E.2 i).2)) := by |
| rw [Finset.mul_sum] |
| _<ε := by simpa only [mul_comm] using (lt_div_iff₀ hK).mp hsum |
|
|
| theorem ac_substitution_boundedContinuous {f : ℝ → ℝ} {a b : ℝ} |
| (hf : AbsolutelyContinuousOnInterval f a b) (γ : ℝ →ᵇ ℝ) : |
| (∫ t in a..b,γ (f t)*deriv f t)=∫ w in f a..f b,γ w := by |
| let G : ℝ → ℝ := fun x => ∫ w in f a..x,γ w |
| have hG (x : ℝ) : HasDerivAt G (γ x) x := |
| (γ.continuous.integral_hasStrictDerivAt (f a) x).hasDerivAt |
| have hl : LipschitzWith ‖γ‖₊ G := by |
| apply lipschitzWith_of_nnnorm_deriv_le (fun x => (hG x).differentiableAt) |
| intro x |
| rw [(hG x).deriv] |
| exact γ.nnnorm_coe_le_nnnorm x |
| have hcomp := absolutelyContinuous_comp_lipschitz hf hl |
| have hd : ∀ᵐ t : ℝ,t∈uIcc a b → deriv (G ∘ f) t=γ (f t)*deriv f t := by |
| filter_upwards [hf.ae_differentiableAt] with t ht htab |
| exact ((hG (f t)).comp t (ht htab).hasDerivAt).deriv |
| calc |
| _=∫ t in a..b,deriv (G ∘ f) t := by |
| apply intervalIntegral.integral_congr_ae |
| exact hd.mono fun t ht hh => (ht (uIoc_subset_uIcc hh)).symm |
| _=G (f b)-G (f a) := hcomp.integral_deriv_eq_sub |
| _=_ := by simp [G] |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology BoundedContinuousFunction |
| namespace CubicTorus |
|
|
| lemma chamber_sign_integral_substitution {f : ℝ → ℝ} |
| (hf : AbsolutelyContinuousOnInterval f 0 1) (hp : f 1=f 0) |
| (b : Bool) (γ : ℝ →ᵇ ℝ) : |
| (∫ t in (0 : ℝ)..1,γ (f t)*(-deriv f t*chamberHeightSign b t))= |
| 2*(∫ w in f ((if b then 1/2 else 0)+1/2)..f (if b then 1/2 else 0),γ w) := by |
| let a : ℝ := if b then 1/2 else 0 |
| let G : ℝ → ℝ := fun x => ∫ w in f (a+1/2)..x,γ w |
| have hG (x : ℝ) : HasDerivAt G (γ x) x := |
| (γ.continuous.integral_hasStrictDerivAt (f (a+1/2)) x).hasDerivAt |
| have hl : LipschitzWith ‖γ‖₊ G := by |
| apply lipschitzWith_of_nnnorm_deriv_le (fun x => (hG x).differentiableAt) |
| intro x |
| rw [(hG x).deriv] |
| exact γ.nnnorm_coe_le_nnnorm x |
| have hcomp := absolutelyContinuous_comp_lipschitz hf hl |
| have hd : ∀ᵐ t : ℝ,t∈uIcc (0 : ℝ) 1 → |
| deriv (G ∘ f) t=γ (f t)*deriv f t := by |
| filter_upwards [hf.ae_differentiableAt] with t ht htab |
| exact ((hG (f t)).comp t (ht htab).hasDerivAt).deriv |
| have hsub := chamberSign_integral_deriv hcomp (congrArg G hp) b |
| calc |
| _=-(∫ t in (0 : ℝ)..1,deriv (G ∘ f) t*chamberHeightSign b t) := by |
| rw [← intervalIntegral.integral_neg] |
| apply intervalIntegral.integral_congr_ae |
| filter_upwards [hd] with t ht htab |
| rw [ht (uIoc_subset_uIcc htab)] |
| ring |
| _=_ := by rw [hsub]; simp [G,a] |
|
|
| theorem chamber_ac_area_measure {f : ℝ → ℝ} (hc : Continuous f) |
| (hf : AbsolutelyContinuousOnInterval f 0 1) (hp : f 1=f 0) (b : Bool) |
| (hends : f ((if b then 1/2 else 0)+1/2)≤f (if b then 1/2 else 0)) |
| (hn : ∀ᵐ t ∂volume.restrict (Ioo (0 : ℝ) 1),0≤-deriv f t*chamberHeightSign b t) : |
| Measure.map f ((volume.restrict (Ioo (0 : ℝ) 1)).withDensity |
| (fun t => ENNReal.ofReal (-deriv f t*chamberHeightSign b t)))= |
| (2 : ℝ≥0∞) • volume.restrict |
| (Ioo (f ((if b then 1/2 else 0)+1/2)) (f (if b then 1/2 else 0))) := by |
| have hi : Integrable (fun t : ℝ => -deriv f t*chamberHeightSign b t) |
| (volume.restrict (Ioo (0 : ℝ) 1)) := |
| (hf.intervalIntegrable_deriv.1.mono_set Ioo_subset_Ioc_self).neg.mul_bdd |
| ((measurable_chamberHeightSign b).comp (AddCircle.continuous_mk' (1 : ℝ)).measurable).aestronglyMeasurable |
| (Eventually.of_forall fun t => (norm_chamberHeightSign b t).le) |
| let := isFiniteMeasure_withDensity_ofReal hi.hasFiniteIntegral |
| have hwd : Measurable (fun t : ℝ => ENNReal.ofReal (-deriv f t*chamberHeightSign b t)) := |
| ((measurable_deriv f).neg.mul ((measurable_chamberHeightSign b).comp |
| (AddCircle.continuous_mk' (1 : ℝ)).measurable)).ennreal_ofReal |
| let : IsFiniteMeasure ((2 : ℝ≥0∞) • volume.restrict |
| (Ioo (f ((if b then 1/2 else 0)+1/2)) (f (if b then 1/2 else 0)))) := |
| Measure.smul_finite _ (by norm_num) |
| apply ext_of_forall_integral_eq_of_IsFiniteMeasure |
| intro γ |
| rw [integral_map hc.measurable.aemeasurable γ.continuous.aestronglyMeasurable, |
| integral_withDensity_eq_integral_toReal_smul hwd (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)] |
| have he : (∫ t in Ioo (0 : ℝ) 1, |
| (ENNReal.ofReal (-deriv f t*chamberHeightSign b t)).toReal • γ (f t))= |
| ∫ t in (0 : ℝ)..1,γ (f t)*(-deriv f t*chamberHeightSign b t) := by |
| rw [intervalIntegral.integral_of_le (by norm_num : (0 : ℝ)≤1),← restrict_Ioo_eq_restrict_Ioc] |
| apply integral_congr_ae |
| filter_upwards [hn] with t ht |
| rw [ENNReal.toReal_ofReal ht,smul_eq_mul,mul_comm] |
| rw [he,chamber_sign_integral_substitution hf hp b γ, |
| intervalIntegral.integral_of_le hends,integral_smul_measure] |
| simp only [ENNReal.toReal_ofNat,smul_eq_mul] |
| rw [← restrict_Ioo_eq_restrict_Ioc] |
|
|
| lemma chamber_ac_area_substitution {f : ℝ → ℝ} (hc : Continuous f) |
| (hf : AbsolutelyContinuousOnInterval f 0 1) (hp : f 1=f 0) (b : Bool) |
| (hends : f ((if b then 1/2 else 0)+1/2)≤f (if b then 1/2 else 0)) |
| (hn : ∀ᵐ t ∂volume.restrict (Ioo (0 : ℝ) 1),0≤-deriv f t*chamberHeightSign b t) |
| {γ : ℝ → ℝ≥0∞} (hγ : Measurable γ) : |
| (∫⁻ t in Ioo (0 : ℝ) 1,γ (f t)*ENNReal.ofReal (-deriv f t*chamberHeightSign b t))= |
| 2*(∫⁻ w in Ioo (f ((if b then 1/2 else 0)+1/2)) (f (if b then 1/2 else 0)),γ w) := by |
| have hwd : Measurable (fun t : ℝ => ENNReal.ofReal (-deriv f t*chamberHeightSign b t)) := |
| ((measurable_deriv f).neg.mul ((measurable_chamberHeightSign b).comp |
| (AddCircle.continuous_mk' (1 : ℝ)).measurable)).ennreal_ofReal |
| have he := congrArg (fun ν : Measure ℝ => ∫⁻ w,γ w ∂ν) |
| (chamber_ac_area_measure hc hf hp b hends hn) |
| rw [lintegral_map hγ hc.measurable, |
| lintegral_withDensity_eq_lintegral_mul _ hwd (show Measurable (fun t => γ (f t)) from hγ.comp hc.measurable),lintegral_smul_measure] at he |
| simpa only [Pi.mul_apply,smul_eq_mul,mul_comm] using he |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology BoundedContinuousFunction |
| namespace CubicTorus |
|
|
| theorem chamber_area_lower_bound {f : ℝ → ℝ} (hc : Continuous f) |
| (hf : AbsolutelyContinuousOnInterval f 0 1) (hp : f 1=f 0) (b : Bool) |
| (hends : f ((if b then 1/2 else 0)+1/2)≤f (if b then 1/2 else 0)) |
| (hn : ∀ᵐ t ∂volume.restrict (Ioo (0 : ℝ) 1),0≤-deriv f t*chamberHeightSign b t) |
| {J : ℝ → ℝ} |
| (hJi : IntegrableOn (fun t => J t*(-deriv f t*chamberHeightSign b t)) (Ioo (0 : ℝ) 1)) |
| {γ : ℝ → ℝ} (hγ : Measurable γ) (hγ0 : ∀ v,0≤γ v) |
| (hγJ : ∀ᵐ t ∂volume.restrict (Ioo (0 : ℝ) 1),γ (f t)≤J t) : |
| IntegrableOn γ (Ioo (f ((if b then 1/2 else 0)+1/2)) (f (if b then 1/2 else 0))) ∧ |
| 2*(∫ v in Ioo (f ((if b then 1/2 else 0)+1/2)) (f (if b then 1/2 else 0)),γ v)≤ |
| ∫ t in Ioo (0 : ℝ) 1,J t*(-deriv f t*chamberHeightSign b t) := by |
| let w : ℝ → ℝ := fun t => -deriv f t*chamberHeightSign b t |
| have hwm : Measurable w := (measurable_deriv f).neg.mul |
| ((measurable_chamberHeightSign b).comp (AddCircle.continuous_mk' (1 : ℝ)).measurable) |
| have hle : ∀ᵐ t ∂volume.restrict (Ioo (0 : ℝ) 1),γ (f t)*w t≤J t*w t := by |
| filter_upwards [hn,hγJ] with t ht hj |
| exact mul_le_mul_of_nonneg_right hj ht |
| have hnonneg : ∀ᵐ t ∂volume.restrict (Ioo (0 : ℝ) 1),0≤γ (f t)*w t := |
| hn.mono fun t ht => mul_nonneg (hγ0 _) ht |
| have hgi : Integrable (fun t => γ (f t)*w t) (volume.restrict (Ioo (0 : ℝ) 1)) := |
| hJi.mono_nonneg ((hγ.comp hc.measurable).mul hwm).aestronglyMeasurable hnonneg hle |
| have hgid : Integrable (γ ∘ f) ((volume.restrict (Ioo (0 : ℝ) 1)).withDensity |
| (fun t => ENNReal.ofReal (w t))) := by |
| apply (integrable_withDensity_iff_integrable_smul' hwm.ennreal_ofReal |
| (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)).mpr |
| apply hgi.congr |
| filter_upwards [hn] with t ht |
| rw [ENNReal.toReal_ofReal ht,smul_eq_mul] |
| exact mul_comm _ _ |
| have hmeas := chamber_ac_area_measure hc hf hp b hends hn |
| have hmapInt : Integrable γ (Measure.map f ((volume.restrict (Ioo (0 : ℝ) 1)).withDensity |
| (fun t => ENNReal.ofReal (w t)))) := |
| (integrable_map_measure hγ.aestronglyMeasurable hc.measurable.aemeasurable).mpr hgid |
| change Integrable γ (Measure.map f ((volume.restrict (Ioo (0 : ℝ) 1)).withDensity |
| (fun t => ENNReal.ofReal (-deriv f t*chamberHeightSign b t)))) at hmapInt |
| rw [hmeas] at hmapInt |
| have hga := (integrable_smul_measure (by norm_num : (2 : ℝ≥0∞)≠0) |
| (by norm_num : (2 : ℝ≥0∞)≠∞)).mp hmapInt |
| refine ⟨hga,?_⟩ |
| have he := congrArg (fun ν : Measure ℝ => ∫ v,γ v ∂ν) hmeas |
| rw [integral_map hc.measurable.aemeasurable hγ.aestronglyMeasurable, |
| integral_withDensity_eq_integral_toReal_smul hwm.ennreal_ofReal |
| (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top),integral_smul_measure] at he |
| simp only [ENNReal.toReal_ofNat,smul_eq_mul] at he |
| have hex : (∫ t in Ioo (0 : ℝ) 1,(ENNReal.ofReal (w t)).toReal*γ (f t))= |
| ∫ t in Ioo (0 : ℝ) 1,γ (f t)*w t := by |
| apply integral_congr_ae |
| filter_upwards [hn] with t ht |
| rw [ENNReal.toReal_ofReal ht,mul_comm] |
| change (∫ t in Ioo (0 : ℝ) 1,(ENNReal.ofReal (w t)).toReal*γ (f t))=_ at he |
| rw [hex] at he |
| rw [← he] |
| exact integral_mono_ae hgi hJi hle |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| end OAI |
|
|
