| import Mathlib |
| import OAI.Geometry.BallPacking.Moser.AnnularFinalPacking |
| import OAI.Geometry.BallPacking.SurfaceArea.SurfaceInteriorBoxes |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace PackingSufficiencySupport.Hamiltonian |
|
|
| section |
|
|
| open scoped ContDiff Manifold Topology |
| open Set Function Manifold MeasureTheory |
| variable {P : Type} [NormedAddCommGroup P] [NormedSpace ℝ P] [FiniteDimensional ℝ P] |
| {M : Type*} [TopologicalSpace M] [ChartedSpace Plane M] |
| [IsManifold 𝓘(ℝ,Plane) ∞ M] [T2Space M] [SigmaCompactSpace M] |
| {I : Type*} [Fintype I] {K L T : Set M} |
|
|
| def redistributedDifference (B : I → SurfaceCoordinateBox M) |
| (ρ : SmoothPartitionOfUnity I 𝓘(ℝ,Plane) M K) |
| (D : P → ManifoldTwoForm Plane M) (χ : ManifoldTwoForm Plane M) : |
| P → ManifoldTwoForm Plane M := fun p => partitionFormMass B ρ (D p) • χ-D p |
|
|
| theorem surface_interior_redistribution |
| (hK : IsCompact K) (hc : IsPreconnected (interior K)) (hne : (interior K).Nonempty) |
| (hpos : PositivePlaneTransitions M) |
| (B : I → SurfaceCoordinateBox M) (ρ : SmoothPartitionOfUnity I 𝓘(ℝ,Plane) M K) |
| (hρ : ρ.IsSubordinate (fun i => (B i).carrier)) |
| {D : P → ManifoldTwoForm Plane M} (hD : SmoothTwoFormFamily D) |
| (hsD : ∀ p x u v,D p x u v= -D p x v u) |
| (hL : IsCompact L) (hLK : L⊆interior K) (hzD : ∀ p x,x∉L→D p x=0) |
| {χ : ManifoldTwoForm Plane M} (hχ : SmoothTwoForm χ) |
| (hsχ : ∀ x u v,χ x u v= -χ x v u) |
| (hT : IsCompact T) (hTK : T⊆interior K) (hzχ : ∀ x,x∉T→χ x=0) |
| (hmχ : partitionFormMass B ρ χ=1) : |
| ContDiff ℝ ∞ (fun p => partitionFormMass B ρ (D p)) ∧ |
| SmoothTwoFormFamily (redistributedDifference B ρ D χ) ∧ |
| (∀ p,partitionFormMass B ρ (redistributedDifference B ρ D χ p)=0) ∧ |
| HasInteriorPrimitiveFamily (interior K) (redistributedDifference B ρ D χ) := by |
| have hzDK : ∀ p x,x∉K→D p x=0 := |
| fun p x hx => hzD p x (fun hl => hx (interior_subset (hLK hl))) |
| have hzχK : ∀ x,x∉K→χ x=0 := |
| fun x hx => hzχ x (fun ht => hx (interior_subset (hTK ht))) |
| let H : P → ℝ := fun p => partitionFormMass B ρ (D p) |
| have hH : ContDiff ℝ ∞ H := partitionFormMass_smooth hK B ρ hρ hD hzDK |
| have hHχ : SmoothTwoFormFamily (fun p => H p • χ) := |
| (SmoothTwoFormFamily.const hχ).smul hH |
| have hΞ : SmoothTwoFormFamily (redistributedDifference B ρ D χ) := hHχ.sub hD |
| have hs : ∀ p x u v,redistributedDifference B ρ D χ p x u v= |
| -redistributedDifference B ρ D χ p x v u := by |
| intro p x u v |
| change H p*χ x u v-D p x u v= -(H p*χ x v u-D p x v u) |
| rw [hsχ x u v,hsD p x u v]; ring |
| have hz : ∀ p x,x∉L∪T→ redistributedDifference B ρ D χ p x=0 := by |
| intro p x hx |
| change H p • χ x-D p x=0 |
| rw [hzχ x (fun ht => hx (Or.inr ht)),hzD p x (fun hl => hx (Or.inl hl))] |
| apply ContinuousLinearMap.ext |
| intro u |
| apply ContinuousLinearMap.ext |
| intro v |
| change H p*0-0=0 |
| ring |
| have hm (p : P) : partitionFormMass B ρ (redistributedDifference B ρ D χ p)=0 := by |
| change partitionFormMass B ρ (H p • χ-D p)=0 |
| have hzHχ : ∀ x,x∉K→(H p • χ) x=0 := by |
| intro x hx |
| change H p • χ x=0 |
| rw [hzχK x hx] |
| apply ContinuousLinearMap.ext |
| intro u |
| apply ContinuousLinearMap.ext |
| intro v |
| exact mul_zero (H p) |
| rw [partitionFormMass_sub hK B ρ hρ (hHχ.eval p) (hD.eval p) hzHχ (hzDK p), |
| partitionFormMass_smul,hmχ,mul_one] |
| exact sub_self _ |
| refine ⟨hH,hΞ,hm,?_⟩ |
| exact interior_surface_primitive isOpen_interior hc hne hpos (hL.union hT) |
| (union_subset hLK hTK) ((union_subset hLK hTK).trans interior_subset) |
| B ρ hρ hΞ hs hz hm |
|
|
|
|
| end |
|
|
| section |
| open scoped ContDiff Manifold Topology ENNReal |
| open Set Function Manifold MeasureTheory |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| {M : Type*} [TopologicalSpace M] [ChartedSpace E M] |
| [IsManifold 𝓘(ℝ,E) ∞ M] [T2Space M] [NormalSpace M] [SigmaCompactSpace M] |
| [MeasurableSpace M] [BorelSpace M] |
| {μ : Measure M} [μ.OuterRegular] |
|
|
| omit [BorelSpace M] in |
|
|
| theorem exists_small_measure_smooth_cutoff {C H : Set M} |
| (hC : IsCompact C) (hnull : μ C=0) (hH : IsClosed H) (hdis : Disjoint C H) |
| {δ : ℝ} (hδ : 0<δ) : |
| ∃ ζ : M → ℝ, ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ ζ ∧ HasCompactSupport ζ ∧ |
| (∀ x, ζ x ∈ Icc (0:ℝ) 1) ∧ (∀ x ∈ H, ζ x=0) ∧ |
| (∀ᶠ x in 𝓝ˢ C, ζ x=1) ∧ μ (tsupport ζ) < ENNReal.ofReal δ := by |
| let : LocallyCompactSpace M := ChartedSpace.locallyCompactSpace E M |
| obtain ⟨U,hCU,hU,hμU⟩ := C.exists_isOpen_lt_of_lt (μ := μ) (ENNReal.ofReal δ) |
| (by simpa only [hnull] using (ENNReal.ofReal_pos.mpr hδ)) |
| have hCV : C ⊆ U ∩ Hᶜ := fun x hx => ⟨hCU hx,disjoint_left.mp hdis hx⟩ |
| obtain ⟨D,hD,hCD,hDV⟩ := exists_compact_between hC (hU.inter hH.isOpen_compl) hCV |
| obtain ⟨ζ,hζ1,hζ0,hζ⟩ := exists_contMDiffMap_one_nhds_of_subset_interior |
| (n := (⊤ : ℕ∞)) 𝓘(ℝ,E) hC.isClosed hCD |
| have hsupport : tsupport (ζ : M → ℝ) ⊆ D := |
| closure_minimal (fun x hx => by by_contra hn; exact hx (hζ0 x hn)) hD.isClosed |
| refine ⟨ζ,ζ.contMDiff,HasCompactSupport.of_support_subset_isCompact hD |
| ((subset_tsupport ζ).trans hsupport),hζ,?_,hζ1,?_⟩ |
| · intro x hx |
| exact hζ0 x (fun hxd => (hDV hxd).2 hx) |
| · exact (measure_mono (hsupport.trans (hDV.trans inter_subset_left))).trans_lt hμU |
|
|
|
|
| end |
|
|
| section |
| open scoped ContDiff Manifold Topology ENNReal |
| open Set Function Manifold MeasureTheory |
| variable {E P : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| [NormedAddCommGroup P] [NormedSpace ℝ P] |
| {M : Type*} [TopologicalSpace M] [ChartedSpace E M] |
| [IsManifold 𝓘(ℝ,E) ∞ M] [T2Space M] [NormalSpace M] [SigmaCompactSpace M] |
| [MeasurableSpace M] [BorelSpace M] |
| {μ : Measure M} [μ.OuterRegular] [IsFiniteMeasureOnCompacts μ] |
|
|
| theorem exists_uniform_small_area_density {Y : Set P} {K C H : Set M} |
| (hY : IsCompact Y) (hK : IsCompact K) (hC : IsCompact C) (hnull : μ C=0) |
| (hH : IsClosed H) (hdis : Disjoint C H) |
| {a : P × M → ℝ} (ha : ContMDiff (𝓘(ℝ,P).prod 𝓘(ℝ,E)) 𝓘(ℝ,ℝ) ∞ a) |
| (hapos : ∀ p ∈ Y, ∀ x ∈ K, 0<a (p,x)) {ε : ℝ} (hε : 0<ε) : |
| ∃ (e : ℝ) (b : P × M → ℝ), 0<e ∧ |
| ContMDiff (𝓘(ℝ,P).prod 𝓘(ℝ,E)) 𝓘(ℝ,ℝ) ∞ b ∧ |
| (∀ p ∈ Y, ∀ x ∈ K, 0<b (p,x)) ∧ |
| (∃ U : Set M, IsOpen U ∧ C ⊆ U ∧ ∀ p x, x ∈ U → b (p,x)=a (p,x)) ∧ |
| (∀ p x, x ∈ H → b (p,x)=e) ∧ |
| (∀ p ∈ Y, (∫ x in K, b (p,x) ∂μ)<ε) := by |
| obtain ⟨A₀,hA₀⟩ := (hY.prod hK).bddAbove_image ha.continuous.continuousOn |
| let A : ℝ := max A₀ 1 |
| have hA1 : 1≤A := le_max_right _ _ |
| have hA : 0<A := lt_of_lt_of_le zero_lt_one hA1 |
| have hbound : ∀ p ∈ Y, ∀ x ∈ K, a (p,x)≤A := by |
| intro p hp x hx |
| exact (hA₀ ⟨(p,x),⟨hp,hx⟩,rfl⟩).trans (le_max_left _ _) |
| let δ : ℝ := ε/(4*A) |
| have hδ : 0<δ := div_pos hε (by positivity) |
| have hAδ : A*δ=ε/4 := by dsimp [δ]; field_simp |
| obtain ⟨ζ,hζ,hζc,hζrange,hζH,hζC,hζμ⟩ := |
| exists_small_measure_smooth_cutoff (E := E) hC hnull hH hdis hδ |
| obtain ⟨U,hU,hCU,hUζ⟩ := mem_nhdsSet_iff_exists.mp hζC |
| let e : ℝ := ε/(4*((μ K).toReal+1)) |
| have hμK : 0≤(μ K).toReal := ENNReal.toReal_nonneg |
| have he : 0<e := div_pos hε (by positivity) |
| have hem : e*((μ K).toReal+1)=ε/4 := by |
| dsimp [e] |
| field_simp |
| have hesmall : e*(μ K).toReal<ε/4 := by nlinarith |
| let b : P × M → ℝ := fun q => ζ q.2*a q+(1-ζ q.2)*e |
| have hb : ContMDiff (𝓘(ℝ,P).prod 𝓘(ℝ,E)) 𝓘(ℝ,ℝ) ∞ b := |
| ((hζ.comp contMDiff_snd).mul ha).add |
| ((contMDiff_const.sub (hζ.comp contMDiff_snd)).mul contMDiff_const) |
| refine ⟨e,b,he,hb,?_,⟨U,hU,hCU,?_⟩,?_,?_⟩ |
| · intro p hp x hx |
| have hc := hζrange x |
| have hap := hapos p hp x hx |
| change 0<ζ x*a (p,x)+(1-ζ x)*e |
| by_cases hz : ζ x=0 |
| · simp only [hz,zero_mul,sub_zero,one_mul,zero_add]; exact he |
| · have hzp : 0<ζ x := lt_of_le_of_ne hc.1 (Ne.symm hz) |
| exact add_pos_of_pos_of_nonneg (mul_pos hzp hap) (mul_nonneg (sub_nonneg.mpr hc.2) he.le) |
| · intro p x hx |
| change ζ x*a (p,x)+(1-ζ x)*e=a (p,x) |
| rw [hUζ hx]; ring |
| · intro p x hx |
| change ζ x*a (p,x)+(1-ζ x)*e=e |
| rw [hζH x hx]; ring |
| · intro p hp |
| have hζsmall : (∫ x in K, ζ x ∂μ)<δ := by |
| apply (ENNReal.ofReal_lt_ofReal_iff hδ).mp |
| apply lt_of_le_of_lt _ hζμ |
| apply le_trans (integral_le_measure (μ := μ.restrict K) |
| (s := tsupport ζ) (fun x _ => (hζrange x).2) |
| (fun x hx => le_of_eq (image_eq_zero_of_notMem_tsupport hx))) |
| exact Measure.restrict_apply_le _ _ |
| have hζint : IntegrableOn ζ K μ := hζ.continuous.continuousOn.integrableOn_compact hK |
| have hbint : IntegrableOn (fun x => b (p,x)) K μ := |
| (hb.continuous.comp (continuous_const.prodMk continuous_id)).continuousOn.integrableOn_compact hK |
| have hcint : IntegrableOn (fun _ : M => e) K μ := |
| continuousOn_const.integrableOn_compact hK |
| have hle : (∫ x in K, b (p,x) ∂μ) ≤ ∫ x in K, (A*ζ x+e) ∂μ := by |
| apply setIntegral_mono_on hbint ((hζint.const_mul A).add hcint) hK.measurableSet |
| intro x hx |
| have hmul := mul_le_mul_of_nonneg_left (hbound p hp x hx) (hζrange x).1 |
| dsimp [b] |
| nlinarith [(hζrange x).1] |
| have heq : (∫ x in K, (A*ζ x+e) ∂μ) = A*(∫ x in K, ζ x ∂μ)+e*(μ K).toReal := by |
| rw [integral_add (hζint.const_mul A) hcint,integral_const_mul,setIntegral_const] |
| change A*(∫ x in K, ζ x ∂μ)+(μ K).toReal*e = _ |
| ring |
| rw [heq] at hle |
| have hh := mul_lt_mul_of_pos_left hζsmall hA |
| nlinarith |
|
|
|
|
| end |
|
|
| section |
|
|
| open scoped ContDiff Manifold Topology ENNReal |
| open Set Function Manifold MeasureTheory |
| variable {P : Type} [NormedAddCommGroup P] [NormedSpace ℝ P] |
| {M : Type*} [TopologicalSpace M] [ChartedSpace Plane M] |
| [IsManifold 𝓘(ℝ,Plane) ∞ M] [T2Space M] [NormalSpace M] [SigmaCompactSpace M] |
| [TopologicalSpace.PseudoMetrizableSpace M] [MeasurableSpace M] [BorelSpace M] |
| {I : Type*} [Fintype I] {K : Set M} |
|
|
| theorem exists_uniform_small_area_form |
| (B : I → SurfaceCoordinateBox M) |
| (ρ : SmoothPartitionOfUnity I 𝓘(ℝ,Plane) M K) |
| (hρ : ρ.IsSubordinate (fun i => (B i).carrier)) |
| {Λ : ManifoldTwoForm Plane M} (hΛ : SmoothTwoForm Λ) |
| (hpos : ∀ c y,y∈(extChartAt 𝓘(ℝ,Plane) c).target→ |
| 0<chartTwoForm Λ c y (1,0) (0,1)) |
| {Y : Set P} {C H : Set M} (hY : IsCompact Y) (hK : IsCompact K) |
| (hC : IsCompact C) |
| (hnull : ∀ i,volume ((extChartAt 𝓘(ℝ,Plane) (B i).center).target ∩ |
| (extChartAt 𝓘(ℝ,Plane) (B i).center).symm ⁻¹' C)=0) |
| (hH : IsClosed H) (hdis : Disjoint C H) |
| {a : P × M → ℝ} (ha : ContMDiff (𝓘(ℝ,P).prod 𝓘(ℝ,Plane)) 𝓘(ℝ,ℝ) ∞ a) |
| (hapos : ∀ p∈Y,∀ x∈K,0<a (p,x)) {ε : ℝ} (hε : 0<ε) : |
| ∃ (e : ℝ) (b : P × M → ℝ), 0<e ∧ |
| ContMDiff (𝓘(ℝ,P).prod 𝓘(ℝ,Plane)) 𝓘(ℝ,ℝ) ∞ b ∧ |
| SmoothTwoFormFamily (fun p x => b (p,x) • Λ x) ∧ |
| (∀ p∈Y,∀ c y,y∈(extChartAt 𝓘(ℝ,Plane) c).target→ |
| (extChartAt 𝓘(ℝ,Plane) c).symm y∈K→ |
| 0<chartTwoForm (fun x => b (p,x) • Λ x) c y (1,0) (0,1)) ∧ |
| (∃ U : Set M,IsOpen U ∧ C⊆U ∧ ∀ p x,x∈U→b (p,x) • Λ x=a (p,x) • Λ x) ∧ |
| (∀ p x,x∈H→b (p,x) • Λ x=e • Λ x) ∧ |
| (∀ p∈Y,restrictedPartitionFormMass B ρ K (fun x => b (p,x) • Λ x)<ε) := by |
| let μ := partitionAreaMeasure B ρ Λ |
| let := partitionAreaMeasure_finite B ρ hρ hΛ |
| have hn : μ C=0 := partitionAreaMeasure_null B ρ Λ hC.measurableSet hnull |
| obtain ⟨e,b,he,hb,hbp,hU,hH',hsmall⟩ := |
| exists_uniform_small_area_density (μ := μ) hY hK hC hn hH hdis ha hapos hε |
| refine ⟨e,b,he,hb,(SmoothTwoFormFamily.const hΛ).coefficient_smul hb,?_,?_,?_,?_⟩ |
| · intro p hp c y hy hk |
| rw [chartTwoForm_spatial_smul] |
| exact mul_pos (hbp p hp _ hk) (hpos c y hy) |
| · obtain ⟨U,hU,hCU,hUa⟩ := hU |
| exact ⟨U,hU,hCU,fun p x hx => congrArg (fun r : ℝ => r • Λ x) (hUa p x hx)⟩ |
| · intro p x hx |
| rw [hH' p x hx] |
| · intro p hp |
| have hbs : ContMDiff 𝓘(ℝ,Plane) 𝓘(ℝ,ℝ) ∞ (fun x => b (p,x)) := |
| hb.comp (contMDiff_const.prodMk contMDiff_id) |
| rw [← integral_partitionAreaMeasure_restrict B ρ hρ hΛ (fun c y hy => (hpos c y hy).le) |
| hK.measurableSet hbs] |
| exact hsmall p hp |
| |
| |
| |
| theorem exists_uniform_small_area_coefficient |
| (B : I → SurfaceCoordinateBox M) |
| (ρ : SmoothPartitionOfUnity I 𝓘(ℝ,Plane) M K) |
| (hρ : ρ.IsSubordinate (fun i => (B i).carrier)) |
| {Λ : ManifoldTwoForm Plane M} (hΛ : SmoothTwoForm Λ) |
| (hpos : ∀ c y,y∈(extChartAt 𝓘(ℝ,Plane) c).target→ |
| 0≤chartTwoForm Λ c y (1,0) (0,1)) |
| {Y : Set P} {C H : Set M} (hY : IsCompact Y) (hK : IsCompact K) |
| (hC : IsCompact C) |
| (hnull : ∀ i,volume ((extChartAt 𝓘(ℝ,Plane) (B i).center).target ∩ |
| (extChartAt 𝓘(ℝ,Plane) (B i).center).symm ⁻¹' C)=0) |
| (hH : IsClosed H) (hdis : Disjoint C H) |
| {a : P × M → ℝ} (ha : ContMDiff (𝓘(ℝ,P).prod 𝓘(ℝ,Plane)) 𝓘(ℝ,ℝ) ∞ a) |
| (hapos : ∀ p∈Y,∀ x∈K,0<a (p,x)) {ε : ℝ} (hε : 0<ε) : |
| ∃ (e : ℝ) (b : P × M → ℝ), 0<e ∧ |
| ContMDiff (𝓘(ℝ,P).prod 𝓘(ℝ,Plane)) 𝓘(ℝ,ℝ) ∞ b ∧ |
| (∀ p∈Y,∀ x∈K,0<b (p,x)) ∧ |
| (∃ U : Set M,IsOpen U ∧ C⊆U ∧ ∀ p x,x∈U→b (p,x)=a (p,x)) ∧ |
| (∀ p x,x∈H→b (p,x)=e) ∧ |
| (∀ p∈Y,restrictedPartitionFormMass B ρ K (fun x => b (p,x) • Λ x)<ε) := by |
| let μ := partitionAreaMeasure B ρ Λ |
| let := partitionAreaMeasure_finite B ρ hρ hΛ |
| have hn : μ C=0 := partitionAreaMeasure_null B ρ Λ hC.measurableSet hnull |
| obtain ⟨e,b,he,hb,hbp,hU,hH',hsmall⟩ := |
| exists_uniform_small_area_density (μ := μ) hY hK hC hn hH hdis ha hapos hε |
| refine ⟨e,b,he,hb,hbp,hU,hH',?_⟩ |
| intro p hp |
| have hbs : ContMDiff 𝓘(ℝ,Plane) 𝓘(ℝ,ℝ) ∞ (fun x => b (p,x)) := |
| hb.comp (contMDiff_const.prodMk contMDiff_id) |
| rw [← integral_partitionAreaMeasure_restrict B ρ hρ hΛ hpos hK.measurableSet hbs] |
| exact hsmall p hp |
|
|
|
|
| end |
|
|
|
|
| open scoped ContDiff Manifold Topology |
| open Set Function Manifold MeasureTheory |
| variable {P : Type} [NormedAddCommGroup P] [NormedSpace ℝ P] [FiniteDimensional ℝ P] |
| {M : Type*} [TopologicalSpace M] [ChartedSpace Plane M] |
| [IsManifold 𝓘(ℝ,Plane) ∞ M] [T2Space M] [NormalSpace M] [SigmaCompactSpace M] |
| [TopologicalSpace.PseudoMetrizableSpace M] [MeasurableSpace M] [BorelSpace M] |
| {I : Type*} [Fintype I] {K T Z : Set M} |
|
|
| theorem exists_surface_area_redistribution |
| (hK : IsCompact K) (hc : IsPreconnected (interior K)) (hne : (interior K).Nonempty) |
| (hor : PositivePlaneTransitions M) |
| (B : I → SurfaceCoordinateBox M) (ρ : SmoothPartitionOfUnity I 𝓘(ℝ,Plane) M K) |
| (hρ : ρ.IsSubordinate (fun i => (B i).carrier)) |
| (hnull : ∀ i,volume ((extChartAt 𝓘(ℝ,Plane) (B i).center).target ∩ |
| (extChartAt 𝓘(ℝ,Plane) (B i).center).symm ⁻¹' frontier K)=0) |
| (hZ : IsClosed Z) (hdis : Disjoint (frontier K) Z) |
| {Λ : ManifoldTwoForm Plane M} (hΛ : SmoothTwoForm Λ) |
| (hsΛ : ∀ x u v,Λ x u v= -Λ x v u) |
| (hpΛ : ∀ c y,y∈(extChartAt 𝓘(ℝ,Plane) c).target→ |
| 0<chartTwoForm Λ c y (1,0) (0,1)) |
| {a : P × M → ℝ} (ha : ContMDiff (𝓘(ℝ,P).prod 𝓘(ℝ,Plane)) 𝓘(ℝ,ℝ) ∞ a) |
| {Y : Set P} (hY : IsCompact Y) (hapos : ∀ p∈Y,∀ x∈K,0<a (p,x)) |
| {χ : ManifoldTwoForm Plane M} (hχ : SmoothTwoForm χ) |
| (hsχ : ∀ x u v,χ x u v= -χ x v u) |
| (hpχ : ∀ c y,y∈(extChartAt 𝓘(ℝ,Plane) c).target→ |
| 0≤chartTwoForm χ c y (1,0) (0,1)) |
| (hT : IsCompact T) (hTK : T⊆interior K) (hzχ : ∀ x,x∉T→χ x=0) |
| (hmχ : partitionFormMass B ρ χ=1) |
| {A : P → ℝ} {ε : ℝ} (hε : 0<ε) |
| (hA : ∀ p∈Y,2*ε<A p) |
| (herr : ∀ p∈Y,|restrictedPartitionFormMass B ρ K (fun x => a (p,x) • Λ x)-A p|<ε) : |
| ∃ (e : ℝ) (b : P × M → ℝ) (H : P → ℝ) (Ξ : P → ManifoldTwoForm Plane M), |
| 0<e ∧ ContMDiff (𝓘(ℝ,P).prod 𝓘(ℝ,Plane)) 𝓘(ℝ,ℝ) ∞ b ∧ |
| ContDiff ℝ ∞ H ∧ SmoothTwoFormFamily Ξ ∧ |
| (∀ p,H p=restrictedPartitionFormMass B ρ K (fun x => a (p,x) • Λ x)- |
| restrictedPartitionFormMass B ρ K (fun x => b (p,x) • Λ x)) ∧ |
| (∀ p∈Y,0<H p ∧ |H p-A p|<2*ε) ∧ |
| (∀ p x,x∈K→a (p,x) • Λ x+Ξ p x=b (p,x) • Λ x+H p • χ x) ∧ |
| (∀ p∈Y,∀ c y,y∈(extChartAt 𝓘(ℝ,Plane) c).target→ |
| (extChartAt 𝓘(ℝ,Plane) c).symm y∈K→ |
| 0<chartTwoForm (fun x => a (p,x) • Λ x+Ξ p x) c y (1,0) (0,1)) ∧ |
| (∀ p x,x∈Z→b (p,x)=e) ∧ |
| HasInteriorPrimitiveFamily (interior K) Ξ ∧ |
| (∀ p∈Y,∀ x∈K,0<b (p,x)) ∧ |
| (∀ p∈Y,restrictedPartitionFormMass B ρ K (fun x => b (p,x) • Λ x)<ε) ∧ |
| (∃ U : Set M,IsOpen U ∧ frontier K⊆U ∧ ∀ p x,x∈U→b (p,x)=a (p,x)) := by |
| have hpΛ' := fun c y hy => (hpΛ c y hy).le |
| obtain ⟨e,b,he,hb,hbp,⟨V,hV,hCV,hba⟩,hbZ,hbm⟩ := |
| exists_uniform_small_area_coefficient B ρ hρ hΛ hpΛ' hY hK (hK.of_isClosed_subset isClosed_frontier |
| (by simpa only [hK.isClosed.closure_eq] using (frontier_subset_closure : frontier K ⊆ closure K))) |
| hnull hZ hdis ha hapos hε |
| obtain ⟨L,d,hL,hLK,hd,hde,hdz⟩ := |
| exists_interior_coefficient_difference hK hV hCV ha hb |
| (fun p x hx => (hba p x hx).symm) |
| let D : P → ManifoldTwoForm Plane M := fun p x => d (p,x) • Λ x |
| have hD : SmoothTwoFormFamily D := (SmoothTwoFormFamily.const hΛ).coefficient_smul hd |
| have hsD : ∀ p x u v,D p x u v= -D p x v u := by |
| intro p x u v |
| change d (p,x)*Λ x u v= -(d (p,x)*Λ x v u) |
| rw [hsΛ x u v]; ring |
| have hzD : ∀ p x,x∉L→D p x=0 := by |
| intro p x hx |
| change d (p,x) • Λ x=0 |
| rw [hdz p x hx] |
| apply ContinuousLinearMap.ext |
| intro u |
| apply ContinuousLinearMap.ext |
| intro v |
| exact zero_mul _ |
| obtain ⟨hH,hΞ,_,hprimitive⟩ := surface_interior_redistribution hK hc hne hor B ρ hρ |
| hD hsD hL hLK hzD hχ hsχ hT hTK hzχ hmχ |
| let H : P → ℝ := fun p => partitionFormMass B ρ (D p) |
| let Ξ := redistributedDifference B ρ D χ |
| have hmass (p : P) : H p=restrictedPartitionFormMass B ρ K (fun x => a (p,x) • Λ x)- |
| restrictedPartitionFormMass B ρ K (fun x => b (p,x) • Λ x) := |
| interior_difference_mass B ρ hρ hK.measurableSet hΛ hpΛ' ha hb hd |
| (hLK.trans interior_subset) hdz hde p |
| have hbound (p : P) (hp : p∈Y) : 0<H p ∧ |H p-A p|<2*ε := by |
| have hm0 : 0≤restrictedPartitionFormMass B ρ K (fun x => b (p,x) • Λ x) := |
| restricted_form_mass_nonneg B ρ hρ hK.measurableSet hΛ hpΛ' |
| (hb.comp (contMDiff_const.prodMk contMDiff_id)) (fun x hx => (hbp p hp x hx).le) |
| have herr' := abs_lt.mp (herr p hp) |
| have hh := hmass p |
| have hhsmall := hbm p hp |
| have hlarge := hA p hp |
| constructor |
| · linarith |
| · exact abs_lt.mpr ⟨by linarith,by linarith⟩ |
| have hformula (p : P) (x : M) (hx : x∈K) : |
| a (p,x) • Λ x+Ξ p x=b (p,x) • Λ x+H p • χ x := by |
| apply ContinuousLinearMap.ext |
| intro u |
| apply ContinuousLinearMap.ext |
| intro v |
| change a (p,x)*Λ x u v+(H p*χ x u v-d (p,x)*Λ x u v)= |
| b (p,x)*Λ x u v+H p*χ x u v |
| rw [hde p x hx]; ring |
| refine ⟨e,b,H,Ξ,he,hb,hH,hΞ,hmass,hbound,hformula,?_,hbZ,hprimitive,hbp,hbm,⟨V,hV,hCV,hba⟩⟩ |
| intro p hp c y hy hk |
| have heq : chartTwoForm (fun x => a (p,x) • Λ x+Ξ p x) c y= |
| chartTwoForm (fun x => b (p,x) • Λ x+H p • χ x) c y := by |
| dsimp only [chartTwoForm] |
| rw [hformula p _ hk] |
| rw [heq] |
| change 0<chartTwoForm ((fun x => b (p,x) • Λ x)+H p • χ) c y (1,0) (0,1) |
| rw [chartTwoForm_add,chartTwoForm_spatial_smul,chartTwoForm_smul] |
| change 0<b (p,(extChartAt 𝓘(ℝ,Plane) c).symm y)*chartTwoForm Λ c y (1,0) (0,1)+ |
| H p*chartTwoForm χ c y (1,0) (0,1) |
| exact add_pos_of_pos_of_nonneg (mul_pos (hbp p hp _ hk) (hpΛ c y hy)) |
| (mul_nonneg (hbound p hp).1.le (hpχ c y hy)) |
|
|
|
|
|
|
| end PackingSufficiencySupport.Hamiltonian |
| end |
|
|
| end OAI |
|
|
