| import Mathlib |
| import OAI.Geometry.CubicTorus.Profiles.CornerGrid381 |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open Set |
| namespace CubicTorus |
|
|
| theorem secondCorner_endpoint_contradiction {A h b r s : ℝ} |
| (hA0 : 0<A) (hA : A≤1) (hh : 0≤h) |
| (hb0 : 0≤b) (hb1 : b<1) (hr0 : 0≤r) (hr1 : r≤1/Real.pi) |
| (hs1 : s≤1) |
| (htop : Real.sqrt (Real.pi*r)≤A*b-h*(1/Real.pi-r)) |
| (hbottom : Real.sqrt (Real.pi*(1-s))≤A*b+h*(s-1/Real.pi)) |
| (hdrop : s-r≤A*Real.sqrt (1-b)) |
| (hbmax : 2/3≤b) (hxcap : h*A≤7/4) : False := by |
| let a : ℝ := 1 |
| let x := h*A |
| let d := Real.sqrt (1-b) |
| let w := Real.sqrt (Real.pi*r)/a |
| have ha : 0<a := by dsimp [a]; positivity |
| have hp : Real.pi≠0 := Real.pi_pos.ne' |
| have hva : 1/Real.pi=a^2/Real.pi := by norm_num [a] |
| have haA : A≤a := hA |
| have hx0 : 0≤x := mul_nonneg hh hA0.le |
| have hd0 : 0≤d := Real.sqrt_nonneg _ |
| have hd2 : d^2=1-b := Real.sq_sqrt (by linarith) |
| have hw0 : 0≤w := div_nonneg (Real.sqrt_nonneg _) ha.le |
| have haw : a*w=Real.sqrt (Real.pi*r) := by dsimp only [w]; field_simp |
| have hrw : r=a^2*w^2/Real.pi := by |
| have hz := Real.sq_sqrt (mul_nonneg Real.pi_pos.le hr0) |
| rw [← haw,mul_pow] at hz |
| apply (eq_div_iff hp).mpr |
| nlinarith only [hz] |
| have hw1 : w≤1 := by |
| rw [hva,hrw] at hr1 |
| have hz := (div_le_div_iff_of_pos_right Real.pi_pos).mp hr1 |
| have ha2 := sq_pos_of_pos ha |
| have hw2' : w^2≤1 := (mul_le_mul_iff_right₀ ha2).mp (by simpa only [mul_one] using hz) |
| nlinarith only [hw2',hw0] |
| have hw2 : 0≤1-w^2 := by nlinarith |
| have hxn : x≤h*a := mul_le_mul_of_nonneg_left haA hh |
| have hnorm : w+x/Real.pi*(1-w^2)≤b := by |
| have hleft := mul_le_mul_of_nonneg_right |
| (div_le_div_of_nonneg_right hxn Real.pi_pos.le) hw2 |
| have hleft' := mul_le_mul_of_nonneg_left (add_le_add (le_refl w) hleft) ha.le |
| have ht : a*(w+h*a/Real.pi*(1-w^2))≤A*b := by |
| have hid : a*(w+h*a/Real.pi*(1-w^2))=a*w+h*(1/Real.pi-r) := by |
| rw [hva,hrw] |
| ring |
| rw [hid,haw] |
| linarith only [htop] |
| have hright := mul_le_mul_of_nonneg_right haA hb0 |
| exact (mul_le_mul_iff_right₀ ha).mp (hleft'.trans (ht.trans hright)) |
| let f : ℝ → ℝ := fun t => t+x/Real.pi*(1-t^2) |
| have hf : Continuous f := by fun_prop |
| have hf1 : f 1=1 := by dsimp [f]; ring |
| obtain ⟨t,ht,hft⟩ := intermediate_value_Icc hw1 hf.continuousOn |
| (show b∈Icc (f w) (f 1) from ⟨hnorm,by rw [hf1]; exact hb1.le⟩) |
| have ht0 : 0≤t := hw0.trans ht.1 |
| have ht1 : t<1 := by |
| by_contra hn |
| have he : t=1 := le_antisymm ht.2 (le_of_not_gt hn) |
| rw [he,hf1] at hft |
| linarith |
| have htden : 0<1-t^2 := by nlinarith |
| have hroot : t+x/Real.pi*(1-t^2)=b := hft |
| have hxterm : 0≤x/Real.pi*(1-t^2) := by positivity |
| have htd : t+d^2≤1 := by linarith only [hroot,hxterm,hd2] |
| have hxroot : x=Real.pi*(1-t-d^2)/(1-t^2) := by |
| apply (eq_div_iff htden.ne').mpr |
| have he := congrArg (fun y : ℝ => y*Real.pi) hroot |
| field_simp at he |
| rw [show 1-t-d^2=b-t by linarith only [hd2]] |
| nlinarith only [he] |
| have hwtsq : w^2≤t^2 := (sq_le_sq₀ hw0 ht0).mpr ht.1 |
| have hrbound : r≤a^2*t^2/Real.pi := by |
| rw [hrw] |
| exact div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left hwtsq (sq_nonneg a)) Real.pi_pos.le |
| have hEt : A*b-h*(1/Real.pi-a^2*t^2/Real.pi)≤a*t := by |
| have he : A*b-h*(1/Real.pi-a^2*t^2/Real.pi)= |
| A*t+(h/Real.pi)*(A^2-a^2)*(1-t^2) := by |
| rw [hva,← hroot] |
| dsimp only [x] |
| ring |
| have hA2 : A^2≤a^2 := (sq_le_sq₀ hA0.le ha.le).mpr haA |
| have hcorr : (h/Real.pi)*(A^2-a^2)*(1-t^2)≤0 := |
| mul_nonpos_of_nonpos_of_nonneg |
| (mul_nonpos_of_nonneg_of_nonpos (div_nonneg hh Real.pi_pos.le) (sub_nonpos.mpr hA2)) htden.le |
| rw [he] |
| have hat := mul_le_mul_of_nonneg_right haA ht0 |
| linarith only [hcorr,hat] |
| have hsb : s≤a^2*t^2/Real.pi+a*d := by |
| have hAd := mul_le_mul_of_nonneg_right haA hd0 |
| linarith only [hdrop,hrbound,hAd] |
| have hbottom' : Real.sqrt (Real.pi*(1-s))≤a*t+x*d := by |
| have hdiff := mul_le_mul_of_nonneg_left hdrop hh |
| have hrr := mul_le_mul_of_nonneg_left hrbound hh |
| have hxd : h*(A*d)=x*d := by dsimp [x]; ring |
| rw [hxd] at hdiff |
| linarith only [hbottom,hdiff,hrr,hEt] |
| have hbsq : Real.pi*(1-s)≤(a*t+x*d)^2 := by |
| have hy : 0≤a*t+x*d := by positivity |
| have hh' := (sq_le_sq₀ (Real.sqrt_nonneg (Real.pi*(1-s))) hy).mpr hbottom' |
| rwa [Real.sq_sqrt (mul_nonneg Real.pi_pos.le (by linarith : 0≤1-s))] at hh' |
| have hnec : 1≤a*d+(a^2*t^2+(a*t+x*d)^2)/Real.pi := by |
| have hh' := mul_le_mul_of_nonneg_left hsb Real.pi_pos.le |
| have hcancel : Real.pi*(a^2*t^2/Real.pi+a*d)=a^2*t^2+Real.pi*a*d := by field_simp |
| rw [hcancel] at hh' |
| have hh2 : 1≤((a*d)*Real.pi+(a^2*t^2+(a*t+x*d)^2))/Real.pi := |
| (le_div_iff₀ Real.pi_pos).mpr (by nlinarith only [hh',hbsq]) |
| simpa only [add_div,mul_div_cancel_right₀ _ hp] using hh2 |
| have hd3 : d^2≤1/3 := by linarith only [hd2,hbmax] |
| have hroot' : Real.pi*(1-t-d^2)=x*(1-t^2) := by |
| exact ((eq_div_iff htden.ne').mp hxroot).symm |
| have hstrict := corner_root_strict (show CornerRootData t d x from |
| ⟨ht0,ht1,hd0,hd3,hx0,hxcap,hroot'⟩) |
| have hnec' : 1≤d+(t^2+(t+x*d)^2)/Real.pi := by |
| simpa only [a,one_mul,one_pow] using hnec |
| exact (not_lt_of_ge hnec') hstrict |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem isMinimizerAt_second_transition_corner_excluded |
| {E : Set Torus} (hE : MeasurableSet E) (hm : IsMinimizerAt (1/Real.pi) 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) |
| {c : Fin 3 → Bool} |
| (hsg : ∀ (x : Cover) (i : Fin 3), |
| quotientMap x∈frontier (openRepresentative E)∩coordinateHalf i (if c i then 0 else 1/2) →0<N x i) |
| (hw0 : 2/3≤axialFluxWeight μ 0) (hw1 : axialFluxWeight μ 0<1) |
| (ht1 : quotientMap (chamberVertex c (fun j => decide (j=0 ∨ j=1)))∉openRepresentative E) |
| (ht2 : quotientMap (chamberVertex c (fun j => decide (j=0 ∨ j=2)))∉openRepresentative E) |
| (hb1 : quotientMap (chamberVertex c (fun j => decide (j=1)))∈openRepresentative E) |
| (hb2 : quotientMap (chamberVertex c (fun j => decide (j=2)))∈openRepresentative E) : False := by |
| have hvh := secondTransition_mem_Ioo_half |
| have hV : 1/Real.pi∈Ioo (0 : ℝ) 1 := ⟨hvh.1,by linarith [hvh.2]⟩ |
| obtain ⟨H,hH⟩ := isMinimizerAt_coordinate_multiplier hV hE hm hμ |
| obtain ⟨hA,_,hA1,hh,hr,hrV,_,hs1,hb,_,htop,hbottom,hdrop⟩ := |
| isMinimizerAt_four_corner_necessary_data hvh hE hm hμ hN hp ha hs hsg hH ht1 ht2 hb1 hb2 |
| exact secondCorner_endpoint_contradiction hA hA1 hh hb hw1 hr hrV hs1 htop hbottom hdrop hw0 |
| (isMinimizerAt_second_endpoint_linear_bound hE hm hμ hH).le |
| |
| theorem isMinimizerAt_second_transition_standard {E : Set Torus} |
| (hmE : IsMinimizerAt (1/Real.pi) E) : SmallStandardRegion (1/Real.pi) E := by |
| by_contra hnstd |
| obtain ⟨F,μ,N,hN,hp,c,H,hF,hm,hμ,ha,hs,hsg,hH,hws,hw,haxis,hface⟩ := |
| isMinimizerAt_second_transition_nonstandard_star hmE hnstd |
| obtain ⟨i,hi⟩ : ∃ i : Fin 3,2/3≤axialFluxWeight μ i := by |
| by_contra! hn |
| rw [Fin.sum_univ_three] at hws |
| linarith [hn 0,hn 1,hn 2] |
| classical |
| let e : Fin 3 ≃ Fin 3 := Equiv.swap 0 i |
| have he0 : e.symm 0=i := by simp [e] |
| have he1 : e.symm 1≠i := by rw [← he0]; exact e.symm.injective.ne (by decide) |
| have he2 : e.symm 2≠i := by rw [← he0]; exact e.symm.injective.ne (by decide) |
| have hpair (j : Fin 3) : ((fun k : Fin 3 => decide (k=0 ∨ k=j)) ∘ e)= |
| (fun k => decide (k=i ∨ k=e.symm j)) := by |
| funext k |
| simp only [Function.comp_apply,← Equiv.eq_symm_apply,he0] |
| have hsingle (j : Fin 3) : ((fun k : Fin 3 => decide (k=j)) ∘ e)= |
| (fun k => decide (k=e.symm j)) := by |
| funext k |
| simp only [Function.comp_apply,← Equiv.eq_symm_apply] |
| have ht (j : Fin 3) (hj : j≠i) : |
| quotientMap (chamberVertex c (fun k => decide (k=i ∨ k=j)))∉openRepresentative F := by |
| obtain ⟨k,hk⟩ := fin3_pair_complement i j hj.symm |
| rw [hk] |
| exact fun h => hface k (subset_closure h) |
| have hv (b : Fin 3 → Bool) : |
| quotientMap (chamberVertex (c ∘ e.symm) b)∈openRepresentative (torusCoordinatePermute e ⁻¹' F) ↔ |
| quotientMap (chamberVertex c (b ∘ e))∈openRepresentative F := by |
| rw [openRepresentative_coordinatePermute hF e] |
| change torusCoordinatePermute e (quotientMap (chamberVertex (c ∘ e.symm) b))∈openRepresentative F ↔ _ |
| rw [← quotientMap_coordinatePermute,chamberVertex_coordinatePermute] |
| apply isMinimizerAt_second_transition_corner_excluded |
| ((torusCoordinatePermute e).continuous.measurable hF) (isMinimizerAt_coordinatePermute hm e) |
| (isNormalMeasure_coordinatePermute hF hμ e) |
| (coordinatePermutedField_contDiff e hN) (coordinatePermutedField_periodic e hp) |
| (permuted_normal_ae hN.continuous hp ha e) (reflection_invariant_coordinatePermute hs e) |
| (chamberSign_coordinatePermute hF hsg e) |
| · simpa [axialFluxWeight_permute,e] using hi |
| · rw [axialFluxWeight_permute] |
| exact (hw _).2 |
| · rw [hv,hpair] |
| exact ht _ he1 |
| · rw [hv,hpair] |
| exact ht _ he2 |
| · rw [hv,hsingle] |
| exact haxis _ |
| · rw [hv,hsingle] |
| exact haxis _ |
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem isoperimetricProfile_second_transition : |
| isoperimetricProfile (1/Real.pi)=1 := by |
| have hV := secondTransition_mem_Ioo_half |
| obtain ⟨E,hm⟩ := exists_isMinimizerAt (show 1/Real.pi∈Ioo (0 : ℝ) 1 from |
| ⟨hV.1,by linarith [hV.2]⟩) |
| have hs := isMinimizerAt_second_transition_standard hm |
| have hP := (hs.bv_data ⟨hV.1,hV.2.le⟩).2.2 |
| rw [candidateProfile_slab (show 1/Real.pi∈Icc TwoVolume.V₂ (1-TwoVolume.V₂) from |
| ⟨le_rfl,by change 1/Real.pi≤1-1/Real.pi; linarith [hV.2]⟩)] at hP |
| rw [isoperimetricProfile_eq_of_isMinimizerAt hm,hP,ENNReal.toReal_ofReal (by positivity)] |
| norm_num |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| def cylinderRatio (V : ℝ) : ℝ := isoperimetricProfile V/Real.sqrt V |
|
|
| lemma profile_ge_second_chord {V : ℝ} (hV : V∈Icc (0 : ℝ) (1/Real.pi)) : |
| Real.pi*V ≤ isoperimetricProfile V := by |
| have hv2 := secondTransition_mem_Ioo_half |
| have hvp : V*Real.pi≤1 := (le_div_iff₀ Real.pi_pos).mp hV.2 |
| have hh := concaveOn_isoperimetricProfile.2 |
| (show (0 : ℝ)∈Icc (0 : ℝ) 1 by constructor <;> norm_num) |
| (show 1/Real.pi∈Icc (0 : ℝ) 1 from ⟨hv2.1.le,by linarith [hv2.2]⟩) |
| (show 0≤1-V*Real.pi by linarith) (show 0≤V*Real.pi from mul_nonneg hV.1 Real.pi_pos.le) |
| (show (1-V*Real.pi)+V*Real.pi=1 by ring) |
| have he : (1-V*Real.pi)*0+V*Real.pi*(1/Real.pi)=V := by field_simp; ring |
| simp only [smul_eq_mul,isoperimetricProfile_zero,isoperimetricProfile_second_transition, |
| mul_zero,mul_one,zero_add] at hh |
| simpa only [mul_assoc,mul_one_div_cancel Real.pi_pos.ne',mul_one,mul_comm] using hh |
| |
| lemma cylinder_contact_support_deriv {V : ℝ} (hV : 0<V) |
| (hc : IsLocalMin cylinderRatio V) {C : ℝ → ℝ} |
| (hs : Reduction.UpperSupport isoperimetricProfile C V) : |
| deriv C V=isoperimetricProfile V/(2*V) := by |
| have hloc : IsLocalMin (fun W => C W/Real.sqrt W) V := by |
| filter_upwards [hc,hs.above,Ioi_mem_nhds hV] with W hW hab hpos |
| change C V/Real.sqrt V≤C W/Real.sqrt W |
| rw [hs.contact] |
| exact hW.trans (div_le_div_of_nonneg_right hab (Real.sqrt_nonneg _)) |
| have hd := (hs.smooth.differentiableAt (by norm_num)).hasDerivAt.div |
| (Real.hasDerivAt_sqrt hV.ne') (Real.sqrt_ne_zero'.mpr hV) |
| have hz := hloc.hasDerivAt_eq_zero hd |
| rw [hs.contact] at hz |
| have hq : (Real.sqrt V)^2=V := Real.sq_sqrt hV.le |
| have hn : Real.sqrt V≠0 := Real.sqrt_ne_zero'.mpr hV |
| field_simp [hn] at hz |
| apply (eq_div_iff (by positivity : 2*V≠0)).mpr |
| rw [hq] at hz |
| nlinarith only [hz] |
|
|
| theorem isMinimizerAt_cylinder_contact_multiplier {V : ℝ} (hV : V∈Ioo 0 1) |
| (hc : IsLocalMin cylinderRatio V) {E : Set Torus} |
| (hE : MeasurableSet E) (hm : IsMinimizerAt V E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {H : ℝ} (hH : ∀ D : Periodic.Variation,Periodic.normalFirstVariation μ D= |
| H*directionalFlux E D.coordinate D.function) : |
| (H/2)*V=((perimeter E).toReal/2)/2 := by |
| obtain ⟨N,C,hN,hp,ha,hs,hd,_⟩ := Periodic.profile_hessian_of_CMC hV hE hm hμ |
| (fun _ hF hpF => Periodic.weakCMC_identity hμ hH hF hpF) |
| have hh := cylinder_contact_support_deriv hV.1 hc hs |
| rw [hd,isoperimetricProfile_eq_of_isMinimizerAt hm] at hh |
| have he := (eq_div_iff (ne_of_gt (mul_pos (by norm_num : (0 : ℝ)<2) hV.1))).mp hh |
| nlinarith only [he] |
|
|
| theorem isMinimizerAt_cylinder_contact_bounds {V : ℝ} (hV : V∈Icc TwoVolume.V₁ TwoVolume.V₂) |
| (hc : IsLocalMin cylinderRatio V) {E : Set Torus} |
| (hE : MeasurableSet E) (hm : IsMinimizerAt V E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {H : ℝ} (hH : ∀ D : Periodic.Variation,Periodic.normalFirstVariation μ D= |
| H*directionalFlux E D.coordinate D.function) : |
| let A := (perimeter E).toReal/2 |
| let h := H/2 |
| 0<A ∧ A≤1 ∧ h*V=A/2 ∧ Real.pi/2≤h ∧ h*A≤Real.pi/2 := by |
| have hv1 := firstTransition_mem_Ioo_half |
| have hv2 := secondTransition_mem_Ioo_half |
| have hv : V∈Ioo (0 : ℝ) 1 := ⟨hv1.1.trans_le hV.1,lt_of_le_of_lt hV.2 (by change 1/Real.pi<1; linarith [hv2.2])⟩ |
| let A := (perimeter E).toReal/2 |
| let h := H/2 |
| have hpos : 0<A := by dsimp only [A]; rw [←isoperimetricProfile_eq_of_isMinimizerAt hm]; exact isoperimetricProfile_pos hv |
| have hA1 : A≤1 := isMinimizerAt_perimeter_half_le_one hv hm |
| have he : h*V=A/2 := isMinimizerAt_cylinder_contact_multiplier hv hc hE hm hμ hH |
| have hlow := profile_ge_second_chord (show V∈Icc (0 : ℝ) (1/Real.pi) from ⟨hv.1.le,hV.2⟩) |
| rw [isoperimetricProfile_eq_of_isMinimizerAt hm] at hlow |
| change Real.pi*V≤A at hlow |
| have hh : Real.pi/2≤h := by |
| apply (mul_le_mul_iff_left₀ hv.1).mp |
| change Real.pi/2*V≤h*V |
| rw [he] |
| linarith only [hlow] |
| have hupper := isoperimetricProfile_le_candidate ⟨hv.1.le,hv.2.le⟩ |
| rw [candidateProfile_tube hV,isoperimetricProfile_eq_of_isMinimizerAt hm] at hupper |
| change A≤2*Real.sqrt (Real.pi*V)/2 at hupper |
| have hsq := Real.sq_sqrt (show 0≤Real.pi*V from mul_nonneg Real.pi_pos.le hv.1.le) |
| have hA2 : A^2≤Real.pi*V := by nlinarith [Real.sqrt_nonneg (Real.pi*V)] |
| have hcap : h*A≤Real.pi/2 := by |
| apply (mul_le_mul_iff_left₀ hv.1).mp |
| nlinarith only [hA2,congrArg (fun x : ℝ => x*A) he] |
| exact ⟨hpos,hA1,he,hh,hcap⟩ |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Set |
| namespace CubicTorus |
|
|
| def criticalCorrection (h : ℝ) : ℝ := (49/50 : ℝ)/Real.pi*h+(51*Real.pi/200)*h⁻¹-1 |
|
|
| lemma criticalCorrection_convex : ConvexOn ℝ (Ioi 0) criticalCorrection := by |
| have hi : ConvexOn ℝ (Ioi (0 : ℝ)) (fun h : ℝ => h⁻¹) := by |
| simpa using (strictConvexOn_zpow (m := -1) (by norm_num) (by norm_num)).convexOn |
| refine ⟨convex_Ioi 0,?_⟩ |
| intro x hx y hy a b ha hb hab |
| have hh := mul_le_mul_of_nonneg_left (hi.2 hx hy ha hb hab) (show 0≤51*Real.pi/200 by positivity) |
| simp only [smul_eq_mul] at hh ⊢ |
| unfold criticalCorrection |
| nlinarith only [hh,hab] |
|
|
| lemma criticalCorrection_nonpos {h : ℝ} (hl : Real.pi/2≤h) (hu : h≤81/50) : |
| criticalCorrection h≤0 := by |
| have hp : 0<h := lt_of_lt_of_le (by positivity : 0<Real.pi/2) hl |
| have hupper : h≤51*Real.pi/98 := by linarith [Real.pi_gt_d4] |
| have hf := mul_nonpos_of_nonneg_of_nonpos (sub_nonneg.mpr hl) (sub_nonpos.mpr hupper) |
| have he : criticalCorrection h*(Real.pi*h)= |
| (49/50 : ℝ)*(h-Real.pi/2)*(h-51*Real.pi/98) := by |
| unfold criticalCorrection |
| field_simp |
| ring |
| have hn : criticalCorrection h*(Real.pi*h)≤0 := by rw [he, mul_assoc]; exact mul_nonpos_of_nonneg_of_nonpos (by norm_num) hf |
| exact (mul_le_mul_iff_left₀ (mul_pos Real.pi_pos hp)).mp (by simpa using hn) |
|
|
| lemma criticalCorrection_rational_upper {h : ℝ} (hh : 0<h) : |
| criticalCorrection h≤(49/50 : ℝ)/(3141592/1000000)*h+ |
| (51*(3141593/1000000)/200)*h⁻¹-1 := by |
| have hpl : (3141592/1000000 : ℝ)≤Real.pi := by linarith [Real.pi_gt_d6] |
| have hpu : Real.pi≤(3141593/1000000 : ℝ) := by linarith [Real.pi_lt_d6] |
| have hl := mul_le_mul_of_nonneg_right |
| (div_le_div_of_nonneg_left (by norm_num : (0 : ℝ)≤49/50) (by norm_num) hpl) hh.le |
| have hr := mul_le_mul_of_nonneg_right hpu (show 0≤(51/200 : ℝ)*h⁻¹ by positivity) |
| unfold criticalCorrection |
| nlinarith only [hl,hr] |
|
|
| lemma criticalCorrection_162 : criticalCorrection (81/50)≤-1/10000 := by |
| have hh := criticalCorrection_rational_upper (h := 81/50) (by norm_num) |
| norm_num at hh ⊢ |
| linarith |
| lemma criticalCorrection_170 : criticalCorrection (17/10)≤31/20000 := by |
| have hh := criticalCorrection_rational_upper (h := 17/10) (by norm_num) |
| norm_num at hh ⊢ |
| linarith |
| lemma criticalCorrection_190 : criticalCorrection (19/10)≤717/50000 := by |
| have hh := criticalCorrection_rational_upper (h := 19/10) (by norm_num) |
| norm_num at hh ⊢ |
| linarith |
|
|
| lemma critical_baseline_chord {a b x l r : ℝ} (ha : 0<a) (hab : a<b) |
| (hx : x∈Icc a b) |
| (hl : l≤nestingBaseline a-criticalCorrection a) |
| (hr : r≤nestingBaseline b-criticalCorrection b) : |
| min l r-(b-a)^2/(4*a^2)≤nestingBaseline x-criticalCorrection x := by |
| have hc := criticalCorrection_convex.subset (show Icc a b⊆Ioi 0 from fun y hy => ha.trans_le hy.1) (convex_Icc a b) |
| have hf : ConcaveOn ℝ (Icc a b) |
| (fun h => nestingBaseline h-criticalCorrection h-(1/a^2)*h^2) := by |
| apply ((ReusedScalar205.CubicTorus.Scalar.concaveOn_baseline_corrected (b := b) ha).sub hc).congr |
| intro y hy |
| dsimp only [Pi.sub_apply] |
| rw [nestingBaseline_eq_reused (ha.trans_le hy.1)] |
| ring |
| have hb := ReusedScalar205.CubicTorus.Scalar.semiconcave_chord_bound hab |
| (show 0≤(1 : ℝ)/a^2 by positivity) hf hx |
| have hm := min_le_min hl hr |
| have he : (1/a^2)*(b-a)^2/4=(b-a)^2/(4*a^2) := by ring |
| rw [he] at hb |
| linarith |
|
|
| lemma critical_baseline_positive {h : ℝ} (hh : h∈Icc (81/50 : ℝ) (19/10)) : |
| 0<nestingBaseline h-criticalCorrection h := by |
| have hl : (978/1000000 : ℝ)≤nestingBaseline (81/50)-criticalCorrection (81/50) := by |
| rw [nestingBaseline_eq_reused (by norm_num)] |
| linarith [ReusedScalar205.CubicTorus.Scalar.Cert.baseline_lower_162,criticalCorrection_162] |
| have hm : (3829/1000000 : ℝ)≤nestingBaseline (17/10)-criticalCorrection (17/10) := by |
| rw [nestingBaseline_eq_reused (by norm_num)] |
| linarith [ReusedScalar205.CubicTorus.Scalar.Cert.baseline_lower_170,criticalCorrection_170] |
| have hr : (8423/1000000 : ℝ)≤nestingBaseline (19/10)-criticalCorrection (19/10) := by |
| rw [nestingBaseline_eq_reused (by norm_num)] |
| linarith [ReusedScalar205.CubicTorus.Scalar.Cert.baseline_lower_190,criticalCorrection_190] |
| by_cases hmidh : h≤17/10 |
| · have hb := critical_baseline_chord (by norm_num) (by norm_num) ⟨hh.1,hmidh⟩ hl hm |
| have hp : (0 : ℝ) < min (978/1000000 : ℝ) (3829/1000000)-((17/10 : ℝ)-81/50)^2/(4*(81/50 : ℝ)^2) := by norm_num |
| exact hp.trans_le hb |
| · have hb := critical_baseline_chord (by norm_num) (by norm_num) ⟨(lt_of_not_ge hmidh).le,hh.2⟩ hm hr |
| have hp : (0 : ℝ) < min (3829/1000000 : ℝ) (8423/1000000)-((19/10 : ℝ)-17/10)^2/(4*(17/10 : ℝ)^2) := by norm_num |
| exact hp.trans_le hb |
|
|
| theorem critical_nesting_contradiction {h : ℝ} (hh : h∈Icc (Real.pi/2) (19/10)) |
| (hB : nestingBaseline h≤criticalCorrection h) |
| (hBS : h≤81/50 →nestingBaseline h+nestingGainInfimum h≤criticalCorrection h) : False := by |
| by_cases hm : h≤81/50 |
| · have hp := nesting_scalar_positive.2 h ⟨by linarith [hh.1,Real.pi_gt_three],hm⟩ |
| have hd := criticalCorrection_nonpos hh.1 hm |
| have hb := hBS hm |
| linarith |
| · have hp := critical_baseline_positive ⟨(lt_of_not_ge hm).le,hh.2⟩ |
| linarith |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma critical_three_vertex_parameters {h s A : ℝ} |
| (hh : Real.pi/2 ≤ h) (hA : 0 ≤ A) (hs : 0 ≤ s) |
| (hcap : h*A ≤ Real.pi/2) (hp : s^2 ≤ A^2/2) |
| (hd : Real.pi*(1-s) ≤ (h*s)^2) : h ≤ 19/10 ∧ Shape.t h ≤ s := by |
| have hh0 : 0<h := lt_of_lt_of_le (by positivity : 0<Real.pi/2) hh |
| have hmul := mul_le_mul_of_nonneg_left hp (sq_nonneg h) |
| have hsq := (sq_le_sq₀ (mul_nonneg hh0.le hA) (by positivity : 0 ≤ Real.pi/2)).mpr hcap |
| have hhs : 8*(h*s)^2 ≤ Real.pi^2 := by nlinarith only [hmul,hsq] |
| have hpi : Real.pi<22/7 := Real.pi_lt_d4.trans (by norm_num) |
| have hslo : 3/5 ≤ s := by |
| have hh' := mul_lt_mul_of_pos_left hpi Real.pi_pos |
| have hp0 : 0<Real.pi := Real.pi_pos |
| nlinarith only [hhs,hd,hh',hp0] |
| have hs2 : (3/5 : ℝ)^2 ≤ s^2 := (sq_le_sq₀ (by norm_num) hs).mpr hslo |
| have hm := mul_le_mul_of_nonneg_left hs2 (sq_nonneg h) |
| have hpi2 := (sq_le_sq₀ Real.pi_pos.le (by norm_num : (0 : ℝ) ≤ 22/7)).mpr hpi.le |
| exact ⟨by nlinarith only [hhs,hm,hpi2,hh0], source_root_le_of_bottom hh0 hs hd⟩ |
|
|
| lemma critical_area_correction {A h V : ℝ} (hh : 0<h) |
| (he : h*V=A/2) (hcap : h*A ≤ Real.pi/2) : |
| Nesting.lambda*h/Real.pi+A-Nesting.lambda*h*V-1 ≤ criticalCorrection h := by |
| have ha : A ≤ (Real.pi/2)*h⁻¹ := by |
| rw [←div_eq_mul_inv] |
| exact (le_div_iff₀ hh).mpr (by nlinarith only [hcap]) |
| have hmul : Nesting.lambda*h*V=Nesting.lambda*(A/2) := by rw [mul_assoc,he] |
| rw [hmul] |
| unfold Nesting.lambda criticalCorrection |
| simp only [div_eq_mul_inv] at ha ⊢ |
| nlinarith only [ha] |
|
|
| theorem isMinimizerAt_cylinder_contact_not_three_vertices {V : ℝ} |
| (hV : V∈Icc TwoVolume.V₁ TwoVolume.V₂) (hcontact : IsLocalMin cylinderRatio V) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| (hc : IsPreconnected (frontier (openRepresentative E))) |
| (hR : ∀ i : Fin 3,E=ᵐ[torusVolume]((axisReflection i).torus ⁻¹' E)) |
| (b : Fin 3 →Bool) |
| (hd : ∀ x∈upperOctant,∀ y∈upperOctant,OrientedCoordinateLE b x y → |
| quotientMap y∈openRepresentative E →quotientMap x∈openRepresentative E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) {H : ℝ} |
| (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))) |
| (i : Fin 3) (ha : (axialAreaMarginal μ i : Measure AxisCircle)≪volume) |
| (hv : quotientMap (orientedUpperVertex b (fun j ↦ if j=i then true else false))∉openRepresentative E) |
| (haxes : ∀ j : Fin 2,quotientMap (orientedUpperVertex b (fun k ↦ if k=i.succAbove j then true else false))∈openRepresentative E) : False := by |
| have hv1 := firstTransition_mem_Ioo_half |
| have hv2 := secondTransition_mem_Ioo_half |
| have hv' : V∈Ioo (0 : ℝ) 1 := |
| ⟨hv1.1.trans_le hV.1,lt_of_le_of_lt hV.2 (by change 1/Real.pi<1; linarith [hv2.2])⟩ |
| obtain ⟨hout,hin⟩ := isMinimizerAt_three_vertex_bottom hv' hmin hc hR b hd i hv haxes |
| obtain ⟨_,hdisc,_⟩ := isMinimizerAt_bottom_both_extents hv' hE hmin hR b hd hμ hCMC i ha hv hout (hin 0) (hin 1) |
| have hH := Periodic.coordinateCMC_of_global hμ hCMC |
| obtain ⟨hA0,hA1,he,hlo,hcap⟩ := isMinimizerAt_cylinder_contact_bounds hV hcontact hE hmin hμ hH |
| have hproj := isMinimizerAt_missing_neighbor_projection hv' hE hmin hR b hd hμ hH i ha hv |
| rw [he] at hproj |
| have hp : _ ≤ ((perimeter E).toReal/2)^2/2 := hproj.trans_eq (by ring) |
| obtain ⟨hhi,ht⟩ := critical_three_vertex_parameters hlo hA0.le measureReal_nonneg hcap hp hdisc |
| have hH0 : 0 ≤ H/2 := by linarith [Real.pi_pos] |
| have hcorr := critical_area_correction (lt_of_lt_of_le (by positivity : 0<Real.pi/2) hlo) he hcap |
| have hflux := isMinimizerAt_missing_neighbor_sharp_area_flux hv' hE hmin hR b hd hμ (by linarith) hH i ha hv |
| have hB : nestingBaseline (H/2) ≤ criticalCorrection (H/2) := by |
| have hb := nestingBaseline_slack hH0 ht |
| rw [show 2*(H/2)=H by ring] at hb |
| have hn := mul_nonneg (show 0 ≤ Nesting.ell by norm_num [Nesting.ell]) (sub_nonneg.mpr ht) |
| change Nesting.lambda*H*V+2*(∫ v in (0 : ℝ).. |
| volume.real (axialSlice (openRepresentative E) i (octantHeight b i 0 : AxisCircle)),sharpSectionWeight H v) ≤ (perimeter E).toReal at hflux |
| nlinarith only [hb,hn,hflux,hcorr] |
| apply critical_nesting_contradiction ⟨hlo,hhi⟩ hB |
| intro hmid |
| have hm : H/2∈Icc (51/50 : ℝ) (81/50) := ⟨by linarith [Real.pi_gt_three,hlo],hmid⟩ |
| have hg := isMinimizerAt_nesting_cubic_gain hv' hE hmin hR b hd hμ hCMC i ha hv hout (hin 0) (hin 1) hm |
| have hb := nestingBaseline_slack hH0 ht |
| have hs := nestingGainInfimum_actual_slack hH0 ht |
| rw [show 2*(H/2)=H by ring] at hb |
| dsimp only [nestingC0] at hs |
| dsimp only at hg |
| nlinarith only [hb,hs,hg,hcorr] |
| |
| end CubicTorus |
| |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
| |
| theorem isMinimizerAt_cylinder_contact_nonstandard_star |
| {V : ℝ} (hVrange : V∈Icc TwoVolume.V₁ TwoVolume.V₂) |
| (hcontact : IsLocalMin cylinderRatio V) |
| {E : Set Torus} (hmE : IsMinimizerAt V E) |
| (hnstd : ¬SmallStandardRegion V E) : |
| ∃ (F : Set Torus) (μ : FiniteMeasure NormalBundle) (N : Cover → Cover) |
| (_hN : ContDiff ℝ (⊤ : ℕ∞) N) (_hp : ∀ m x,N (x+latticeVector m)=N x) |
| (c : Fin 3 → Bool) (H : ℝ), |
| MeasurableSet F ∧ IsMinimizerAt V F ∧ IsNormalMeasure F μ ∧ |
| (∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) ∧ |
| (∀ j : Fin 3,coordinateReflection j ⁻¹' F =ᵐ[torusVolume] F) ∧ |
| (∀ (x : Cover) (j : Fin 3), |
| quotientMap x∈frontier (openRepresentative F)∩coordinateHalf j (if c j then 0 else 1/2) →0<N x j) ∧ |
| (∀ D : Periodic.Variation,Periodic.normalFirstVariation μ D= |
| H*directionalFlux F D.coordinate D.function) ∧ |
| (∑ j : Fin 3,axialFluxWeight μ j)=2 ∧ |
| (∀ j : Fin 3,0≤axialFluxWeight μ j ∧ axialFluxWeight μ j<1) ∧ |
| (∀ j : Fin 3,quotientMap (chamberVertex c (fun l => decide (l=j)))∈openRepresentative F) ∧ |
| (∀ j : Fin 3,quotientMap (chamberVertex c (fun l => decide (l≠j)))∉closure (openRepresentative F)) := by |
| have hv1 := firstTransition_mem_Ioo_half |
| have hv2 := secondTransition_mem_Ioo_half |
| have hvh : V∈Ioo (0 : ℝ) (1/2) := ⟨hv1.1.trans_le hVrange.1,hVrange.2.trans_lt hv2.2⟩ |
| have hV : V∈Ioo (0 : ℝ) 1 := ⟨hvh.1,by linarith [hvh.2]⟩ |
| obtain ⟨v,F,μ,N,hN,hp,c,H,hF,he,hm,hcon,hμ,ha,hs,hsg,hH,_,hws,hw,hvi,_⟩ := |
| isMinimizerAt_nonstandard_section_system ⟨hvh.1,hvh.2.le⟩ hmE hnstd |
| have hnF : ¬SmallStandardRegion V F := fun h => hnstd (smallStandardRegion_of_translated he h) |
| have hnstdF : ¬StandardRegionAt V F := by |
| simpa only [StandardRegionAt,ite_eq_left hvh.2.le] using hnF |
| have hR : ∀ i : Fin 3,F=ᵐ[torusVolume]((axisReflection i).torus ⁻¹' F) := by |
| intro i |
| rw [axisReflection_eq_coordinateReflection] |
| exact (hs i).symm |
| have hd := isMinimizerAt_chamber_upper_downset hV hF hm hμ hN hp ha hs hsg |
| have heq (j : Fin 3) : |
| quotientMap (orientedUpperVertex c (fun k => if k=j then true else false))= |
| quotientMap (chamberVertex c (fun k => decide (k=j))) := by |
| rw [quotient_orientedUpperVertex_eq] |
| congr 2 |
| obtain ⟨i,hi⟩ := hvi |
| have hiO : quotientMap (chamberVertex c (fun j => decide (j=i)))∈openRepresentative F := |
| mem_open_of_mem_closure_not_frontier (isOpen_openRepresentative F) hi |
| (isMinimizerAt_reflection_vertex_excluded hV hF hm hμ hN hp ha hs |
| (chamberVertex_sine_zero c _)) |
| have hcF : IsPreconnected (frontier (openRepresentative F)) := |
| (Periodic.isMinimizerAt_connected_or_standard_slab hV hm).resolve_right (fun h => hnstdF h.2.2) |
| obtain ⟨ν,K,hν,hCMC,hAC,_⟩ := isMinimizerAt_nonstandard_integrated_flux ⟨hvh.1,hvh.2.le⟩ hm hnstdF hR |
| have hno (k : Fin 3) |
| (hk : quotientMap (chamberVertex c (fun j => decide (j=k)))∉openRepresentative F) |
| (hall : ∀ j : Fin 2,quotientMap (chamberVertex c (fun l => decide (l=k.succAbove j)))∈openRepresentative F) : False := by |
| apply isMinimizerAt_cylinder_contact_not_three_vertices hVrange hcontact hF hm hcF hR c hd hν hCMC k (hAC k) |
| · rwa [heq] |
| · intro j |
| rw [heq] |
| exact hall j |
| have hall : ∀ j : Fin 3,quotientMap (chamberVertex c (fun l => decide (l=j)))∈openRepresentative F := by |
| rcases three_axes_dichotomy _ ⟨i,hiO⟩ hno with hall|⟨k,hk⟩ |
| · exact hall |
| · obtain ⟨hzero,_,hdown⟩ := isMinimizerAt_chamber_vertex_order hV hF hm hμ ha |
| (fun x hx j => (hsg x j ⟨hx.1,mem_iInter.mp hx.2 j⟩).le) hs |
| have hv := bool_vertex_two_point_ideal hzero hdown k ((hk k).mpr rfl) |
| (fun j hj => (hk j).mp hj) |
| exact (isMinimizerAt_strict_two_vertices_impossible ⟨hvh.1,hvh.2.le⟩ hF hm hμ hN hp ha hs hsg k hv).elim |
| exact ⟨F,μ,N,hN,hp,c,H,hF,hm,hμ,ha,hs,hsg,hH,hws,hw,hall, |
| isMinimizerAt_smaller_side_double_vertices_vacant hvh hF hm hμ hN hp ha hs hsg hH⟩ |
| end CubicTorus |
| |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| open Set |
| namespace CubicTorus |
| |
| theorem fourCorner_normalized_contradiction {A h b r s a V : ℝ} |
| (hA0 : 0<A) (ha : 0<a) (ha1 : a≤1) (hA : A≤a) (hva : V=a^2/Real.pi) (hh : 0≤h) |
| (hb0 : 0≤b) (hb1 : b<1) (hr0 : 0≤r) (hr1 : r≤V) |
| (hs1 : s≤1) |
| (htop : Real.sqrt (Real.pi*r)≤A*b-h*(V-r)) |
| (hbottom : Real.sqrt (Real.pi*(1-s))≤A*b+h*(s-V)) |
| (hdrop : s-r≤A*Real.sqrt (1-b)) |
| (hbmax : 2/3≤b) (hxcap : h*A≤7/4) : False := by |
| let x := h*A |
| let d := Real.sqrt (1-b) |
| let w := Real.sqrt (Real.pi*r)/a |
| have hp : Real.pi≠0 := Real.pi_pos.ne' |
| have haA : A≤a := hA |
| have hx0 : 0≤x := mul_nonneg hh hA0.le |
| have hd0 : 0≤d := Real.sqrt_nonneg _ |
| have hd2 : d^2=1-b := Real.sq_sqrt (by linarith) |
| have hw0 : 0≤w := div_nonneg (Real.sqrt_nonneg _) ha.le |
| have haw : a*w=Real.sqrt (Real.pi*r) := by dsimp only [w]; field_simp |
| have hrw : r=a^2*w^2/Real.pi := by |
| have hz := Real.sq_sqrt (mul_nonneg Real.pi_pos.le hr0) |
| rw [← haw,mul_pow] at hz |
| apply (eq_div_iff hp).mpr |
| nlinarith only [hz] |
| have hw1 : w≤1 := by |
| rw [hva,hrw] at hr1 |
| have hz := (div_le_div_iff_of_pos_right Real.pi_pos).mp hr1 |
| have ha2 := sq_pos_of_pos ha |
| have hw2' : w^2≤1 := (mul_le_mul_iff_right₀ ha2).mp (by simpa only [mul_one] using hz) |
| nlinarith only [hw2',hw0] |
| have hw2 : 0≤1-w^2 := by nlinarith |
| have hxn : x≤h*a := mul_le_mul_of_nonneg_left haA hh |
| have hnorm : w+x/Real.pi*(1-w^2)≤b := by |
| have hleft := mul_le_mul_of_nonneg_right |
| (div_le_div_of_nonneg_right hxn Real.pi_pos.le) hw2 |
| have hleft' := mul_le_mul_of_nonneg_left (add_le_add (le_refl w) hleft) ha.le |
| have ht : a*(w+h*a/Real.pi*(1-w^2))≤A*b := by |
| have hid : a*(w+h*a/Real.pi*(1-w^2))=a*w+h*(V-r) := by |
| rw [hva,hrw] |
| ring |
| rw [hid,haw] |
| linarith only [htop] |
| have hright := mul_le_mul_of_nonneg_right haA hb0 |
| exact (mul_le_mul_iff_right₀ ha).mp (hleft'.trans (ht.trans hright)) |
| let f : ℝ → ℝ := fun t => t+x/Real.pi*(1-t^2) |
| have hf : Continuous f := by fun_prop |
| have hf1 : f 1=1 := by dsimp [f]; ring |
| obtain ⟨t,ht,hft⟩ := intermediate_value_Icc hw1 hf.continuousOn |
| (show b∈Icc (f w) (f 1) from ⟨hnorm,by rw [hf1]; exact hb1.le⟩) |
| have ht0 : 0≤t := hw0.trans ht.1 |
| have ht1 : t<1 := by |
| by_contra hn |
| have he : t=1 := le_antisymm ht.2 (le_of_not_gt hn) |
| rw [he,hf1] at hft |
| linarith |
| have htden : 0<1-t^2 := by nlinarith |
| have hroot : t+x/Real.pi*(1-t^2)=b := hft |
| have hxterm : 0≤x/Real.pi*(1-t^2) := by positivity |
| have htd : t+d^2≤1 := by linarith only [hroot,hxterm,hd2] |
| have hxroot : x=Real.pi*(1-t-d^2)/(1-t^2) := by |
| apply (eq_div_iff htden.ne').mpr |
| have he := congrArg (fun y : ℝ => y*Real.pi) hroot |
| field_simp at he |
| rw [show 1-t-d^2=b-t by linarith only [hd2]] |
| nlinarith only [he] |
| have hwtsq : w^2≤t^2 := (sq_le_sq₀ hw0 ht0).mpr ht.1 |
| have hrbound : r≤a^2*t^2/Real.pi := by |
| rw [hrw] |
| exact div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left hwtsq (sq_nonneg a)) Real.pi_pos.le |
| have hEt : A*b-h*(V-a^2*t^2/Real.pi)≤a*t := by |
| have he : A*b-h*(V-a^2*t^2/Real.pi)= |
| A*t+(h/Real.pi)*(A^2-a^2)*(1-t^2) := by |
| rw [hva,← hroot] |
| dsimp only [x] |
| ring |
| have hA2 : A^2≤a^2 := (sq_le_sq₀ hA0.le ha.le).mpr haA |
| have hcorr : (h/Real.pi)*(A^2-a^2)*(1-t^2)≤0 := |
| mul_nonpos_of_nonpos_of_nonneg |
| (mul_nonpos_of_nonneg_of_nonpos (div_nonneg hh Real.pi_pos.le) (sub_nonpos.mpr hA2)) htden.le |
| rw [he] |
| have hat := mul_le_mul_of_nonneg_right haA ht0 |
| linarith only [hcorr,hat] |
| have hsb : s≤a^2*t^2/Real.pi+a*d := by |
| have hAd := mul_le_mul_of_nonneg_right haA hd0 |
| linarith only [hdrop,hrbound,hAd] |
| have hbottom' : Real.sqrt (Real.pi*(1-s))≤a*t+x*d := by |
| have hdiff := mul_le_mul_of_nonneg_left hdrop hh |
| have hrr := mul_le_mul_of_nonneg_left hrbound hh |
| have hxd : h*(A*d)=x*d := by dsimp [x]; ring |
| rw [hxd] at hdiff |
| linarith only [hbottom,hdiff,hrr,hEt] |
| have hbsq : Real.pi*(1-s)≤(a*t+x*d)^2 := by |
| have hy : 0≤a*t+x*d := by positivity |
| have hh' := (sq_le_sq₀ (Real.sqrt_nonneg (Real.pi*(1-s))) hy).mpr hbottom' |
| rwa [Real.sq_sqrt (mul_nonneg Real.pi_pos.le (by linarith : 0≤1-s))] at hh' |
| have hnec : 1≤a*d+(a^2*t^2+(a*t+x*d)^2)/Real.pi := by |
| have hh' := mul_le_mul_of_nonneg_left hsb Real.pi_pos.le |
| have hcancel : Real.pi*(a^2*t^2/Real.pi+a*d)=a^2*t^2+Real.pi*a*d := by field_simp |
| rw [hcancel] at hh' |
| have hh2 : 1≤((a*d)*Real.pi+(a^2*t^2+(a*t+x*d)^2))/Real.pi := |
| (le_div_iff₀ Real.pi_pos).mpr (by nlinarith only [hh',hbsq]) |
| simpa only [add_div,mul_div_cancel_right₀ _ hp] using hh2 |
| have hd3 : d^2≤1/3 := by linarith only [hd2,hbmax] |
| have hroot' : Real.pi*(1-t-d^2)=x*(1-t^2) := by |
| exact ((eq_div_iff htden.ne').mp hxroot).symm |
| have hstrict := corner_root_strict (show CornerRootData t d x from |
| ⟨ht0,ht1,hd0,hd3,hx0,hxcap,hroot'⟩) |
| have hnec' : 1≤d+(t^2+(t+x*d)^2)/Real.pi := by |
| have had := mul_le_mul_of_nonneg_right ha1 hd0 |
| have hat : a*t≤t := by simpa using mul_le_mul_of_nonneg_right ha1 ht0 |
| have hatsq : a^2*t^2≤t^2 := by |
| simpa only [mul_pow] using (sq_le_sq₀ (mul_nonneg ha.le ht0) ht0).mpr hat |
| have hsum : (a*t+x*d)^2≤(t+x*d)^2 := by |
| have hat0 : 0≤a*t+x*d := by positivity |
| have hxd0 : 0≤x*d := by positivity |
| exact (sq_le_sq₀ hat0 (add_nonneg ht0 hxd0)).mpr (add_le_add hat (le_refl (x*d))) |
| have hdiv := div_le_div_of_nonneg_right (add_le_add hatsq hsum) Real.pi_pos.le |
| linarith only [hnec,had,hdiv] |
| exact (not_lt_of_ge hnec') hstrict |
| |
| end CubicTorus |
| |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
| |
| theorem isMinimizerAt_cylinder_contact_corner_excluded |
| {V : ℝ} (hVrange : V∈Icc TwoVolume.V₁ TwoVolume.V₂) |
| (hcontact : IsLocalMin cylinderRatio V) |
| {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) |
| {c : Fin 3 → Bool} |
| (hsg : ∀ (x : Cover) (i : Fin 3), |
| quotientMap x∈frontier (openRepresentative E)∩coordinateHalf i (if c i then 0 else 1/2) →0<N x i) |
| (hw0 : 2/3 ≤ axialFluxWeight μ 0) (hw1 : axialFluxWeight μ 0<1) |
| (ht1 : quotientMap (chamberVertex c (fun j => decide (j=0 ∨ j=1)))∉openRepresentative E) |
| (ht2 : quotientMap (chamberVertex c (fun j => decide (j=0 ∨ j=2)))∉openRepresentative E) |
| (hb1 : quotientMap (chamberVertex c (fun j => decide (j=1)))∈openRepresentative E) |
| (hb2 : quotientMap (chamberVertex c (fun j => decide (j=2)))∈openRepresentative E) : False := by |
| have hv1 := firstTransition_mem_Ioo_half |
| have hv2 := secondTransition_mem_Ioo_half |
| have hvh : V∈Ioo (0 : ℝ) (1/2) := ⟨hv1.1.trans_le hVrange.1,hVrange.2.trans_lt hv2.2⟩ |
| have hV : V∈Ioo (0 : ℝ) 1 := ⟨hvh.1,by linarith [hvh.2]⟩ |
| obtain ⟨H,hH⟩ := isMinimizerAt_coordinate_multiplier hV hE hm hμ |
| obtain ⟨hA,hAc,hA1,hh,hr,hrV,_,hs1,hb,_,htop,hbottom,hdrop⟩ := |
| isMinimizerAt_four_corner_necessary_data hvh hE hm hμ hN hp ha hs hsg hH ht1 ht2 hb1 hb2 |
| have hpos : 0<Real.sqrt (Real.pi*V) := Real.sqrt_pos.mpr (mul_pos Real.pi_pos hvh.1) |
| have hle : Real.sqrt (Real.pi*V) ≤ 1 := by |
| apply Real.sqrt_le_one.mpr |
| simpa only [mul_comm] using (le_div_iff₀ Real.pi_pos).mp hVrange.2 |
| have hv : V=(Real.sqrt (Real.pi*V))^2/Real.pi := by |
| rw [Real.sq_sqrt (mul_nonneg Real.pi_pos.le hvh.1.le)] |
| field_simp |
| have hcap := (isMinimizerAt_cylinder_contact_bounds hVrange hcontact hE hm hμ hH).2.2.2.2 |
| rw [← hμ.1] at hcap |
| exact fourCorner_normalized_contradiction hA hpos hle hAc hv hh hb hw1 hr hrV hs1 htop hbottom hdrop hw0 |
| (hcap.trans (by linarith [Real.pi_lt_d2])) |
|
|
| theorem isMinimizerAt_cylinder_contact_standard {V : ℝ} |
| (hVrange : V∈Icc TwoVolume.V₁ TwoVolume.V₂) (hcontact : IsLocalMin cylinderRatio V) |
| {E : Set Torus} |
| (hmE : IsMinimizerAt V E) : SmallStandardRegion V E := by |
| by_contra hnstd |
| obtain ⟨F,μ,N,hN,hp,c,H,hF,hm,hμ,ha,hs,hsg,hH,hws,hw,haxis,hface⟩ := |
| isMinimizerAt_cylinder_contact_nonstandard_star hVrange hcontact hmE hnstd |
| obtain ⟨i,hi⟩ : ∃ i : Fin 3,2/3 ≤ axialFluxWeight μ i := by |
| by_contra! hn |
| rw [Fin.sum_univ_three] at hws |
| linarith [hn 0,hn 1,hn 2] |
| classical |
| let e : Fin 3 ≃ Fin 3 := Equiv.swap 0 i |
| have he0 : e.symm 0=i := by simp [e] |
| have he1 : e.symm 1≠i := by rw [← he0]; exact e.symm.injective.ne (by decide) |
| have he2 : e.symm 2≠i := by rw [← he0]; exact e.symm.injective.ne (by decide) |
| have hpair (j : Fin 3) : ((fun k : Fin 3 => decide (k=0 ∨ k=j)) ∘ e)= |
| (fun k => decide (k=i ∨ k=e.symm j)) := by |
| funext k |
| simp only [Function.comp_apply,← Equiv.eq_symm_apply,he0] |
| have hsingle (j : Fin 3) : ((fun k : Fin 3 => decide (k=j)) ∘ e)= |
| (fun k => decide (k=e.symm j)) := by |
| funext k |
| simp only [Function.comp_apply,← Equiv.eq_symm_apply] |
| have ht (j : Fin 3) (hj : j≠i) : |
| quotientMap (chamberVertex c (fun k => decide (k=i ∨ k=j)))∉openRepresentative F := by |
| obtain ⟨k,hk⟩ := fin3_pair_complement i j hj.symm |
| rw [hk] |
| exact fun h => hface k (subset_closure h) |
| have hv (b : Fin 3 → Bool) : |
| quotientMap (chamberVertex (c ∘ e.symm) b)∈openRepresentative (torusCoordinatePermute e ⁻¹' F) ↔ |
| quotientMap (chamberVertex c (b ∘ e))∈openRepresentative F := by |
| rw [openRepresentative_coordinatePermute hF e] |
| change torusCoordinatePermute e (quotientMap (chamberVertex (c ∘ e.symm) b))∈openRepresentative F ↔ _ |
| rw [← quotientMap_coordinatePermute,chamberVertex_coordinatePermute] |
| apply isMinimizerAt_cylinder_contact_corner_excluded hVrange hcontact |
| ((torusCoordinatePermute e).continuous.measurable hF) (isMinimizerAt_coordinatePermute hm e) |
| (isNormalMeasure_coordinatePermute hF hμ e) |
| (coordinatePermutedField_contDiff e hN) (coordinatePermutedField_periodic e hp) |
| (permuted_normal_ae hN.continuous hp ha e) (reflection_invariant_coordinatePermute hs e) |
| (chamberSign_coordinatePermute hF hsg e) |
| · simpa [axialFluxWeight_permute,e] using hi |
| · rw [axialFluxWeight_permute] |
| exact (hw _).2 |
| · rw [hv,hpair] |
| exact ht _ he1 |
| · rw [hv,hpair] |
| exact ht _ he2 |
| · rw [hv,hsingle] |
| exact haxis _ |
| · rw [hv,hsingle] |
| exact haxis _ |
| end CubicTorus |
| |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
| |
| lemma cylinder_interval_interior {V : ℝ} (hV : V∈Icc TwoVolume.V₁ TwoVolume.V₂) : |
| V∈Ioo (0 : ℝ) (1/2) := |
| ⟨firstTransition_mem_Ioo_half.1.trans_le hV.1,hV.2.trans_lt secondTransition_mem_Ioo_half.2⟩ |
| |
| lemma profile_eq_tube_of_small_standard {V : ℝ} (hV : V∈Icc TwoVolume.V₁ TwoVolume.V₂) |
| {E : Set Torus} (hm : IsMinimizerAt V E) (hs : SmallStandardRegion V E) : |
| isoperimetricProfile V=Real.sqrt (Real.pi*V) := by |
| have hv := cylinder_interval_interior hV |
| have hP := (hs.bv_data ⟨hv.1,hv.2.le⟩).2.2 |
| rw [candidateProfile_tube hV] at hP |
| rw [isoperimetricProfile_eq_of_isMinimizerAt hm,hP,ENNReal.toReal_ofReal (by positivity)] |
| ring |
| |
| lemma cylinderRatio_eq_of_profile {V : ℝ} (hV : 0<V) |
| (he : isoperimetricProfile V=Real.sqrt (Real.pi*V)) : |
| cylinderRatio V=Real.sqrt Real.pi := by |
| rw [cylinderRatio,he,Real.sqrt_mul Real.pi_pos.le,mul_div_cancel_right₀ _ (Real.sqrt_ne_zero'.mpr hV)] |
|
|
| lemma cylinderRatio_first : cylinderRatio TwoVolume.V₁=Real.sqrt Real.pi := by |
| have hv := firstTransition_mem_Ioo_half |
| obtain ⟨E,hm⟩ := exists_isMinimizerAt (show TwoVolume.V₁∈Ioo (0 : ℝ) 1 from ⟨hv.1,by change 4*Real.pi/81<1; linarith [hv.2]⟩) |
| apply cylinderRatio_eq_of_profile hv.1 |
| apply profile_eq_tube_of_small_standard ⟨le_rfl,TwoVolume.transitions.2.1.le⟩ hm |
| exact isMinimizerAt_first_transition_standard hm |
|
|
| lemma cylinderRatio_second : cylinderRatio TwoVolume.V₂=Real.sqrt Real.pi := by |
| have hv := secondTransition_mem_Ioo_half |
| obtain ⟨E,hm⟩ := exists_isMinimizerAt (show TwoVolume.V₂∈Ioo (0 : ℝ) 1 from ⟨hv.1,by change 1/Real.pi<1; linarith [hv.2]⟩) |
| apply cylinderRatio_eq_of_profile hv.1 |
| apply profile_eq_tube_of_small_standard ⟨TwoVolume.transitions.2.1.le,le_rfl⟩ hm |
| exact isMinimizerAt_second_transition_standard hm |
|
|
| lemma cylinderRatio_contact {V : ℝ} (hV : V∈Icc TwoVolume.V₁ TwoVolume.V₂) |
| (hc : IsLocalMin cylinderRatio V) : cylinderRatio V=Real.sqrt Real.pi := by |
| have hv := cylinder_interval_interior hV |
| obtain ⟨E,hm⟩ := exists_isMinimizerAt (show V∈Ioo (0 : ℝ) 1 from ⟨hv.1,by linarith [hv.2]⟩) |
| exact cylinderRatio_eq_of_profile hv.1 (profile_eq_tube_of_small_standard hV hm |
| (isMinimizerAt_cylinder_contact_standard hV hc hm)) |
|
|
| theorem isoperimetricProfile_cylinder_interval {V : ℝ} |
| (hV : V∈Icc TwoVolume.V₁ TwoVolume.V₂) : |
| isoperimetricProfile V=Real.sqrt (Real.pi*V) := by |
| have hv := cylinder_interval_interior hV |
| have hc : ContinuousOn cylinderRatio (Icc TwoVolume.V₁ TwoVolume.V₂) := by |
| apply (continuousOn_isoperimetricProfile.mono ?_).div Real.continuous_sqrt.continuousOn |
| · intro W hW |
| exact Real.sqrt_ne_zero'.mpr (cylinder_interval_interior hW).1 |
| · intro W hW |
| have hh := cylinder_interval_interior hW |
| exact ⟨hh.1.le,by linarith [hh.2]⟩ |
| obtain ⟨W,hW,hmin⟩ := isCompact_Icc.exists_isMinOn |
| (show (Icc TwoVolume.V₁ TwoVolume.V₂).Nonempty from ⟨V,hV⟩) hc |
| have heW : cylinderRatio W=Real.sqrt Real.pi := by |
| rcases eq_or_lt_of_le hW.1 with he|he |
| · rw [←he,cylinderRatio_first] |
| rcases eq_or_lt_of_le hW.2 with he'|he' |
| · rw [he',cylinderRatio_second] |
| exact cylinderRatio_contact hW (hmin.isLocalMin (Icc_mem_nhds he he')) |
| have hr := hmin hV |
| rw [heW] at hr |
| change Real.sqrt Real.pi ≤ isoperimetricProfile V/Real.sqrt V at hr |
| have hlow := (le_div_iff₀ (Real.sqrt_pos.mpr hv.1)).mp hr |
| rw [←Real.sqrt_mul Real.pi_pos.le] at hlow |
| apply le_antisymm _ hlow |
| have hu := isoperimetricProfile_le_candidate (show V∈Icc (0 : ℝ) 1 from ⟨hv.1.le,by linarith [hv.2]⟩) |
| rw [candidateProfile_tube hV] at hu |
| linarith only [hu] |
| |
| lemma isoperimetricProfile_cylinder_contact {V : ℝ} (hV : V∈Ioo TwoVolume.V₁ TwoVolume.V₂) : |
| IsLocalMin cylinderRatio V := by |
| have he := cylinderRatio_eq_of_profile (cylinder_interval_interior ⟨hV.1.le,hV.2.le⟩).1 |
| (isoperimetricProfile_cylinder_interval ⟨hV.1.le,hV.2.le⟩) |
| filter_upwards [Ioo_mem_nhds hV.1 hV.2] with W hW |
| rw [he,cylinderRatio_eq_of_profile (cylinder_interval_interior ⟨hW.1.le,hW.2.le⟩).1 |
| (isoperimetricProfile_cylinder_interval ⟨hW.1.le,hW.2.le⟩)] |
| |
| theorem isMinimizerAt_cylinder_interval_standard {V : ℝ} |
| (hV : V∈Icc TwoVolume.V₁ TwoVolume.V₂) {E : Set Torus} (hm : IsMinimizerAt V E) : |
| SmallStandardRegion V E := by |
| rcases eq_or_lt_of_le hV.1 with he|he |
| · subst V |
| exact isMinimizerAt_first_transition_standard hm |
| rcases eq_or_lt_of_le hV.2 with he'|he' |
| · subst V |
| exact isMinimizerAt_second_transition_standard hm |
| exact isMinimizerAt_cylinder_contact_standard hV (isoperimetricProfile_cylinder_contact ⟨he,he'⟩) hm |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma TwoVolume.sphereArea_three_halves {V : ℝ} (hV : 0≤V) : |
| TwoVolume.sphereArea V^(3/2 : ℝ)=(9*Real.pi/2)^(1/2 : ℝ)*V := by |
| unfold TwoVolume.sphereArea |
| rw [Real.mul_rpow (by positivity) (Real.rpow_nonneg hV _), |
| ← Real.rpow_mul (by positivity : 0≤9*Real.pi/2),← Real.rpow_mul hV] |
| norm_num |
|
|
| theorem isoperimetricProfile_sphere_interval {V : ℝ} (hV : V∈Icc 0 TwoVolume.V₁) : |
| isoperimetricProfile V=TwoVolume.sphereArea V := by |
| have hv1 := TwoVolume.transitions.1 |
| have hv1h : TwoVolume.V₁<1/2 := TwoVolume.transitions.2.1.trans TwoVolume.transitions.2.2 |
| have hv1' : TwoVolume.V₁≤1 := by linarith |
| have hlimit : isoperimetricProfile TwoVolume.V₁=TwoVolume.sphereArea TwoVolume.V₁ := by |
| rw [isoperimetricProfile_cylinder_interval ⟨le_rfl,TwoVolume.transitions.2.1.le⟩, |
| TwoVolume.sphereArea_eq_tubeArea_V₁] |
| rfl |
| have hb : 0≤V/TwoVolume.V₁ := div_nonneg hV.1 hv1.le |
| have hb1 : V/TwoVolume.V₁≤1 := (div_le_one hv1).mpr hV.2 |
| have hh := concaveOn_isoperimetricProfile_three_halves.2 |
| (show (0 : ℝ)∈Icc (0 : ℝ) 1 by constructor <;> norm_num) |
| (show TwoVolume.V₁∈Icc (0 : ℝ) 1 from ⟨hv1.le,hv1'⟩) |
| (show 0≤1-V/TwoVolume.V₁ by linarith) hb |
| (show (1-V/TwoVolume.V₁)+V/TwoVolume.V₁=1 by ring) |
| have hcenter : (1-V/TwoVolume.V₁)*0+(V/TwoVolume.V₁)*TwoVolume.V₁=V := by |
| field_simp |
| ring |
| simp only [smul_eq_mul] at hh |
| rw [hcenter] at hh |
| simp only [isoperimetricProfile_zero,Real.zero_rpow (by norm_num : (3/2 : ℝ)≠0), |
| mul_zero,zero_add,hlimit,TwoVolume.sphereArea_three_halves hv1.le] at hh |
| have hl : TwoVolume.sphereArea V^(3/2 : ℝ) ≤ isoperimetricProfile V^(3/2 : ℝ) := by |
| rw [TwoVolume.sphereArea_three_halves hV.1] |
| calc |
| (9*Real.pi/2)^(1/2 : ℝ)*V = (V/TwoVolume.V₁)*((9*Real.pi/2)^(1/2 : ℝ)*TwoVolume.V₁) := by |
| field_simp |
| _ ≤ isoperimetricProfile V^(3/2 : ℝ) := hh |
| have hl' := (Real.rpow_le_rpow_iff (TwoVolume.sphereArea_nonneg hV.1) |
| (isoperimetricProfile_nonneg V) (by norm_num : (0 : ℝ)<3/2)).mp hl |
| have hu := isoperimetricProfile_le_candidate (show V∈Icc (0 : ℝ) 1 from ⟨hV.1,hV.2.trans hv1'⟩) |
| rw [candidateProfile_sphere hV,unit_sphereArea] at hu |
| apply le_antisymm _ hl' |
| linarith only [hu] |
| |
| theorem isMinimizerAt_sphere_open_standard {V : ℝ} (hV : V∈Ioo 0 TwoVolume.V₁) |
| {E : Set Torus} (hm : IsMinimizerAt V E) : SmallStandardRegion V E := by |
| have hv1h : TwoVolume.V₁<1/2 := TwoVolume.transitions.2.1.trans TwoVolume.transitions.2.2 |
| have hvh : V∈Ioo (0 : ℝ) (1/2) := ⟨hV.1,hV.2.trans hv1h⟩ |
| have hv : V∈Ioo (0 : ℝ) 1 := ⟨hV.1,by linarith [hvh.2]⟩ |
| rcases Periodic.isMinimizerAt_connected_or_standard_slab hv hm with hc|hs |
| · let F := toMeasurable torusVolume E |
| have hF : MeasurableSet F := measurableSet_toMeasurable _ _ |
| have hFE : F=ᵐ[torusVolume]E := hm.1.1.toMeasurable_ae_eq |
| have hmF := (isMinimizerAt_congr_ae hFE).mpr hm |
| have hcF : IsPreconnected (frontier (openRepresentative F)) := by rwa [openRepresentative_congr_ae hFE] |
| have he : ∀ᶠ W in 𝓝 V,isoperimetricProfile W^(3/2 : ℝ)=(9*Real.pi/2)^(1/2 : ℝ)*W+0 := by |
| filter_upwards [Ioo_mem_nhds hV.1 hV.2] with W hW |
| rw [isoperimetricProfile_sphere_interval ⟨hW.1.le,hW.2.le⟩, |
| TwoVolume.sphereArea_three_halves hW.1.le,add_zero] |
| have hh := Periodic.isMinimizerAt_power_affine_standard_ball hvh hF hmF hcF he |
| have hstd : SmallStandardRegion V F := Or.inl hh |
| exact (smallStandardRegion_congr_ae hFE V).mp hstd |
| · have hs' : SmallStandardRegion V E := by |
| simpa only [StandardRegionAt,ite_eq_left hvh.2.le] using hs.2.2 |
| exact hs' |
| |
| end CubicTorus |
| |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
| |
| theorem isoperimetricProfile_slab_interval {V : ℝ} |
| (hV : V∈Icc TwoVolume.V₂ (1-TwoVolume.V₂)) : isoperimetricProfile V=1 := by |
| have hv2 := secondTransition_mem_Ioo_half |
| have hh := isoperimetricProfile_chord hv2.1.le |
| (show 1-TwoVolume.V₂≤1 by change 1-1/Real.pi≤1; linarith [hv2.1]) |
| (show TwoVolume.V₂<1-TwoVolume.V₂ by change 1/Real.pi<1-1/Real.pi; linarith [hv2.2]) V hV |
| rw [isoperimetricProfile_compl (show TwoVolume.V₂∈Icc (0 : ℝ) 1 from |
| ⟨hv2.1.le,by change 1/Real.pi≤1; linarith [hv2.2]⟩)] at hh |
| change isoperimetricProfile (1/Real.pi)+_ ≤ isoperimetricProfile V at hh |
| simp only [TwoVolume.V₂,sub_self,mul_zero,zero_div,add_zero,isoperimetricProfile_second_transition] at hh |
| have hv : V∈Icc (0 : ℝ) 1 := ⟨hv2.1.le.trans hV.1,hV.2.trans (by change 1-1/Real.pi≤1; linarith [hv2.1])⟩ |
| have hu := isoperimetricProfile_le_candidate hv |
| rw [candidateProfile_slab hV] at hu |
| exact le_antisymm (by linarith only [hu]) hh |
| |
| lemma isoperimetricProfile_slab_local_min {V : ℝ} |
| (hV : V∈Ioo TwoVolume.V₂ (1-TwoVolume.V₂)) : IsLocalMin isoperimetricProfile V := by |
| filter_upwards [Ioo_mem_nhds hV.1 hV.2] with W hW |
| rw [isoperimetricProfile_slab_interval ⟨hV.1.le,hV.2.le⟩, |
| isoperimetricProfile_slab_interval ⟨hW.1.le,hW.2.le⟩] |
| |
| theorem isMinimizerAt_small_slab_standard {V : ℝ} (hV : V∈Icc TwoVolume.V₂ (1/2)) |
| {E : Set Torus} (hm : IsMinimizerAt V E) : SmallStandardRegion V E := by |
| rcases eq_or_lt_of_le hV.1 with he|he |
| · subst V |
| exact isMinimizerAt_second_transition_standard hm |
| have hv2 := secondTransition_mem_Ioo_half |
| have hv : V∈Ioo (0 : ℝ) 1 := ⟨hv2.1.trans_le hV.1,by linarith [hV.2]⟩ |
| have hc := isoperimetricProfile_slab_local_min (show V∈Ioo TwoVolume.V₂ (1-TwoVolume.V₂) from |
| ⟨he,by change V<1-1/Real.pi; linarith [hV.2,hv2.2]⟩) |
| have hh := Periodic.isMinimizerAt_profile_local_min_slab hv hm hc |
| simpa only [StandardRegionAt,ite_eq_left hV.2] using hh.2.2 |
| |
| end CubicTorus |
| |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
| |
| theorem isMinimizerAt_small_volume_standard {V : ℝ} (hV : V ∈ Ioc 0 (1/2 : ℝ)) |
| {E : Set Torus} (hm : IsMinimizerAt V E) : SmallStandardRegion V E := by |
| by_cases hfirst : V < TwoVolume.V₁ |
| · exact isMinimizerAt_sphere_open_standard ⟨hV.1,hfirst⟩ hm |
| · by_cases hsecond : V ≤ TwoVolume.V₂ |
| · exact isMinimizerAt_cylinder_interval_standard ⟨le_of_not_gt hfirst,hsecond⟩ hm |
| · exact isMinimizerAt_small_slab_standard ⟨(lt_of_not_ge hsecond).le,hV.2⟩ hm |
| |
| theorem isMinimizerAt_all_volume_standard {V : ℝ} (hV : V ∈ Ioo 0 1) |
| {E : Set Torus} (hm : IsMinimizerAt V E) : StandardRegionAt V E := by |
| unfold StandardRegionAt |
| split_ifs with hv |
| · exact isMinimizerAt_small_volume_standard ⟨hV.1,hv⟩ hm |
| · have hvc : 1-V ∈ Ioc (0 : ℝ) (1/2) := by constructor <;> linarith [hV.2] |
| exact isMinimizerAt_small_volume_standard hvc |
| ((isMinimizerAt_compl ⟨hV.1.le,hV.2.le⟩).2 hm) |
| |
| theorem perimeter_eq_candidate_of_minimizer {V : ℝ} (hV : V ∈ Ioo 0 1) |
| {E : Set Torus} (hm : IsMinimizerAt V E) : |
| perimeter E = ENNReal.ofReal (candidateProfile V) := |
| ((isMinimizerAt_all_volume_standard hV hm).bv_data hV).2.2 |
| |
| theorem StandardRegionAt.isMinimizerAt {V : ℝ} (hV : V ∈ Ioo 0 1) |
| {E : Set Torus} (hs : StandardRegionAt V E) : IsMinimizerAt V E := by |
| obtain ⟨F,hmF⟩ := exists_isMinimizerAt hV |
| obtain ⟨hE,heV,heP⟩ := hs.bv_data hV |
| have hfP := perimeter_eq_candidate_of_minimizer hV hmF |
| refine ⟨hE,heV,?_⟩ |
| intro G hG hgV |
| rw [heP,←hfP] |
| exact hmF.2.2 G hG hgV |
| |
| theorem unit_cubic_isoperimetric (V : ℝ) (hV : V ∈ Ioo 0 1) : |
| (∃ E : Set Torus, IsMinimizerAt V E) ∧ |
| (∀ E : Set Torus, IsMinimizerAt V E → |
| perimeter E = ENNReal.ofReal (candidateProfile V)) ∧ |
| (∀ E : Set Torus, IsMinimizerAt V E ↔ StandardRegionAt V E) := by |
| refine ⟨exists_isMinimizerAt hV,?_,?_⟩ |
| · intro E hm |
| exact perimeter_eq_candidate_of_minimizer hV hm |
| · intro E |
| exact ⟨isMinimizerAt_all_volume_standard hV,StandardRegionAt.isMinimizerAt hV⟩ |
| |
| end CubicTorus |
| |
| end |
| end |
| |
| end OAI |
| |
