import OAI.Analysis.HotSpots.Eigenfunction namespace OAI section ActualInteriorNodal noncomputable section NodalVariationalInteriorLayer open Set MeasureTheory Filter Metric open scoped ContDiff InnerProductSpace Topology namespace StrictHotSpots.InteriorNodal open StrictHotSpots.Nodal lemma minimizer_zero_of_open_zero {Ω : Set Plane} {u : Plane → ℝ} {g : Plane → Plane} (hΩ : IsOpen Ω) (hc : IsPreconnected Ω) (hb : Bornology.IsBounded Ω) (hv : 0 < (volume Ω).toReal) (hu : HasH1Gradient Ω u g) (hm : (∫ x in Ω, u x) = 0) (he : (∫ x in Ω, ‖g x‖^2) = firstPositiveNeumannValue Ω * (∫ x in Ω, (u x)^2)) (hz : ∃ a ∈ Ω, ∃ ε > 0, ∀ᵐ x ∂volume, x ∈ ball a ε → u x = 0) : u =ᵐ[volume.restrict Ω] 0 := by let : IsFiniteMeasure (volume.restrict Ω) := isFiniteMeasure_restrict.mpr hb.measure_lt_top.ne have hi : IntegrableOn u Ω := hu.1.integrable (by norm_num) have hp := planeWeakEquation_of_euler hΩ hu (fun v k hvk => minimizer_euler hb hv hu hvk hm he) exact (ae_restrict_iff' hΩ.measurableSet).mpr (planeWeak_uniqueContinuation hΩ hc hi hp hz) lemma integral_indicator_pos {Ω U : Set Plane} {u : Plane → ℝ} (_hΩ : IsOpen Ω) (hb : Bornology.IsBounded Ω) (hU : IsOpen U) (hs : U ⊆ Ω) (hne : U.Nonempty) (hc : ContinuousOn u Ω) (hu : MemLp u 2 (volume.restrict Ω)) (hp : ∀ x ∈ U, 0 < u x) : 0 < ∫ x in Ω, U.indicator u x := by let : IsFiniteMeasure (volume.restrict Ω) := isFiniteMeasure_restrict.mpr hb.measure_lt_top.ne have hi := (hu.indicator hU.measurableSet).integrable (by norm_num : 1 ≤ (2:ENNReal)) have hn (x : Plane) : 0 ≤ U.indicator u x := by by_cases hx : x ∈ U · simpa only [indicator_of_mem hx] using (hp x hx).le · simp only [indicator_of_notMem hx,le_refl] have hnn : 0 ≤ ∫ x in Ω, U.indicator u x := integral_nonneg hn by_contra hf have hz : (∫ x in Ω, U.indicator u x) = 0 := le_antisymm (le_of_not_gt hf) hnn have hae := (integral_eq_zero_iff_of_nonneg hn hi).mp hz have he : u =ᵐ[volume.restrict U] 0 := by have hh := ae_restrict_of_ae_restrict_of_subset hs hae filter_upwards [hh,ae_restrict_mem hU.measurableSet] with x hx hxm simpa only [indicator_of_mem hxm,Pi.zero_apply] using hx have hh := Measure.eqOn_open_of_ae_eq he hU (hc.mono hs) continuousOn_const obtain ⟨x,hx⟩ := hne have hp' := hp x hx simp only [hh hx,Pi.zero_apply,lt_self_iff_false] at hp' lemma restriction_energy {Ω U : Set Plane} {u : Plane → ℝ} (hu : InInteriorNeumannEigenspace Ω u) (hH : HasH1Gradient Ω (U.indicator u) (U.indicator (gradient u))) : (∫ x in Ω, ‖U.indicator (gradient u) x‖^2) = firstPositiveNeumannValue Ω * (∫ x in Ω, (U.indicator u x)^2) := by have he := hu.2.2.2 _ _ hH convert he using 1 · apply integral_congr_ae exact Eventually.of_forall fun x => by by_cases hx : x ∈ U · simp only [indicator_of_mem hx,real_inner_self_eq_norm_sq] · simp only [indicator_of_notMem hx,norm_zero,zero_pow (by norm_num : (2:ℕ)≠0),inner_zero_right] · congr 1 apply integral_congr_ae exact Eventually.of_forall fun x => by by_cases hx : x ∈ U · simp only [indicator_of_mem hx,pow_two] · simp only [indicator_of_notMem hx,zero_pow (by norm_num : (2:ℕ)≠0),mul_zero] lemma disjoint_indicator_norm_sq {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {U V : Set Plane} (hd : Disjoint U V) (g : Plane → E) (a b : ℝ) (x : Plane) : ‖a • U.indicator g x + b • V.indicator g x‖^2 = a^2 * ‖U.indicator g x‖^2 + b^2 * ‖V.indicator g x‖^2 := by by_cases hxU : x ∈ U · have hxV : x ∉ V := fun hxV => Set.disjoint_left.mp hd hxU hxV simp only [indicator_of_mem hxU,indicator_of_notMem hxV,smul_zero,add_zero,norm_zero, zero_pow (by norm_num : (2:ℕ)≠0),mul_zero,norm_smul,mul_pow,Real.norm_eq_abs,sq_abs] · simp only [indicator_of_notMem hxU,smul_zero,zero_add,norm_zero, zero_pow (by norm_num : (2:ℕ)≠0),mul_zero,norm_smul,mul_pow,Real.norm_eq_abs,sq_abs] lemma disjoint_restriction_combination_energy {Ω U V : Set Plane} {u : Plane → ℝ} (hu : InInteriorNeumannEigenspace Ω u) (hd : Disjoint U V) (hU : HasH1Gradient Ω (U.indicator u) (U.indicator (gradient u))) (hV : HasH1Gradient Ω (V.indicator u) (V.indicator (gradient u))) (a b : ℝ) : (∫ x in Ω, ‖a • U.indicator (gradient u) x + b • V.indicator (gradient u) x‖^2) = firstPositiveNeumannValue Ω * (∫ x in Ω, (a * U.indicator u x + b * V.indicator u x)^2) := by have ha : (∫ x in Ω, ‖a • U.indicator (gradient u) x + b • V.indicator (gradient u) x‖^2) = a^2 * (∫ x in Ω, ‖U.indicator (gradient u) x‖^2) + b^2 * (∫ x in Ω, ‖V.indicator (gradient u) x‖^2) := by simp_rw [disjoint_indicator_norm_sq hd] rw [integral_add (hU.2.1.norm.integrable_sq.const_mul _) (hV.2.1.norm.integrable_sq.const_mul _), integral_const_mul,integral_const_mul] have hb : (∫ x in Ω, (a * U.indicator u x + b * V.indicator u x)^2) = a^2 * (∫ x in Ω, (U.indicator u x)^2) + b^2 * (∫ x in Ω, (V.indicator u x)^2) := by have he (x : Plane) := disjoint_indicator_norm_sq hd u a b x simp only [smul_eq_mul,Real.norm_eq_abs,sq_abs] at he simp_rw [he] rw [integral_add (hU.1.integrable_sq.const_mul _) (hV.1.integrable_sq.const_mul _), integral_const_mul,integral_const_mul] rw [ha,hb,restriction_energy hu hU,restriction_energy hu hV] ring end StrictHotSpots.InteriorNodal end NodalVariationalInteriorLayer noncomputable section NodalConnectedInteriorLayer open Set MeasureTheory Filter Metric open scoped ContDiff InnerProductSpace Topology namespace StrictHotSpots.InteriorNodal open StrictHotSpots.Nodal lemma gradient_neg_fun (u : Plane → ℝ) : gradient (-u) = -gradient u := by funext x simp only [gradient,Pi.neg_apply,fderiv_neg,map_neg] lemma eigenfunction_neg {Ω : Set Plane} {u : Plane → ℝ} (hu : InInteriorNeumannEigenspace Ω u) : InInteriorNeumannEigenspace Ω (-u) := by refine ⟨hu.1.neg,?_,?_,?_⟩ · simpa only [gradient_neg_fun,neg_one_smul] using hu.2.1.smul (-1) · simp only [Pi.neg_apply,integral_neg,hu.2.2.1,neg_zero] · intro v g hv simp only [gradient_neg_fun,Pi.neg_apply,inner_neg_left,neg_mul,integral_neg] rw [hu.2.2.2 v g hv] ring lemma eigenfunction_has_negative {Ω : Set Plane} {u : Plane → ℝ} (hΩ : AdmissibleDomain Ω) (hu : InInteriorNeumannEigenspace Ω u) (hne : ∃ x ∈ Ω, u x ≠ 0) : ∃ x ∈ Ω, u x < 0 := by let : IsFiniteMeasure (volume.restrict Ω) := isFiniteMeasure_restrict.mpr hΩ.2.2.1.measure_lt_top.ne by_contra hn push Not at hn have hae : 0 ≤ᵐ[volume.restrict Ω] u := (ae_restrict_iff' hΩ.2.1.measurableSet).mpr (Eventually.of_forall hn) have hz := (integral_eq_zero_iff_of_nonneg_ae hae (hu.2.1.1.integrable (by norm_num))).mp hu.2.2.1 have hall := Measure.eqOn_open_of_ae_eq hz hΩ.2.1 hu.1.continuousOn continuousOn_const obtain ⟨x,hx,hux⟩ := hne exact hux (hall hx) lemma eigenfunction_has_positive {Ω : Set Plane} {u : Plane → ℝ} (hΩ : AdmissibleDomain Ω) (hu : InInteriorNeumannEigenspace Ω u) (hne : ∃ x ∈ Ω, u x ≠ 0) : ∃ x ∈ Ω, 0 < u x := by have hn : ∃ x ∈ Ω, (-u) x ≠ 0 := by obtain ⟨x,hx,hux⟩ := hne exact ⟨x,hx,by simpa only [Pi.neg_apply,neg_ne_zero] using hux⟩ obtain ⟨x,hx,hux⟩ := eigenfunction_has_negative hΩ (eigenfunction_neg hu) hn exact ⟨x,hx,by simpa only [Pi.neg_apply,neg_lt_zero] using hux⟩ lemma no_two_positive_regions {Ω U V : Set Plane} {u : Plane → ℝ} (hΩ : AdmissibleDomain Ω) (hu : InInteriorNeumannEigenspace Ω u) (hU : IsOpen U) (hV : IsOpen V) (hsU : U ⊆ Ω) (hsV : V ⊆ Ω) (hnU : U.Nonempty) (hnV : V.Nonempty) (hd : Disjoint U V) (hpU : ∀ x ∈ U, 0 < u x) (hpV : ∀ x ∈ V, 0 < u x) (hhU : HasH1Gradient Ω (U.indicator u) (U.indicator (gradient u))) (hhV : HasH1Gradient Ω (V.indicator u) (V.indicator (gradient u))) (hneg : ∃ x ∈ Ω, u x < 0) : False := by let : IsFiniteMeasure (volume.restrict Ω) := isFiniteMeasure_restrict.mpr hΩ.2.2.1.measure_lt_top.ne let A := ∫ x in Ω, U.indicator u x let B := ∫ x in Ω, V.indicator u x have hB : 0 < B := integral_indicator_pos hΩ.2.1 hΩ.2.2.1 hV hsV hnV hu.1.continuousOn hu.2.1.1 hpV let w := B • U.indicator u + (-A) • V.indicator u let g := B • U.indicator (gradient u) + (-A) • V.indicator (gradient u) have hw : HasH1Gradient Ω w g := (hhU.smul B).add (hhV.smul (-A)) have hm : (∫ x in Ω, w x) = 0 := by change (∫ x in Ω, B * U.indicator u x + (-A) * V.indicator u x) = 0 rw [integral_add ((hhU.1.integrable (by norm_num)).const_mul B) ((hhV.1.integrable (by norm_num)).const_mul (-A)),integral_const_mul,integral_const_mul] change B*A+(-A)*B=0 ring have he : (∫ x in Ω, ‖g x‖^2) = firstPositiveNeumannValue Ω * (∫ x in Ω, (w x)^2) := disjoint_restriction_combination_energy hu hd hhU hhV B (-A) have hz : ∃ a ∈ Ω, ∃ ε > 0, ∀ᵐ x ∂volume, x ∈ ball a ε → w x = 0 := by obtain ⟨a,ha,han⟩ := hneg have haU : {x | u x < 0} ∈ 𝓝 a := (hu.1.continuousOn a ha).continuousAt (hΩ.2.1.mem_nhds ha) |>.eventually (gt_mem_nhds han) obtain ⟨ε,hε,he⟩ := Metric.mem_nhds_iff.mp haU refine ⟨a,ha,ε,hε,Eventually.of_forall fun x hx => ?_⟩ have hn : u x < 0 := he hx have hxu : x ∉ U := fun hxu => (not_lt_of_ge (hpU x hxu).le) hn have hxv : x ∉ V := fun hxv => (not_lt_of_ge (hpV x hxv).le) hn simp only [w,Pi.add_apply,Pi.smul_apply,indicator_of_notMem hxu,indicator_of_notMem hxv,smul_zero,add_zero] have hv : 0 < (volume Ω).toReal := ENNReal.toReal_pos (ne_of_gt (hΩ.2.1.measure_pos volume hΩ.1)) hΩ.2.2.1.measure_lt_top.ne have hall := minimizer_zero_of_open_zero hΩ.2.1 hΩ.2.2.2.1.isPathConnected.isConnected.isPreconnected hΩ.2.2.1 hv hw hm he hz have hvu : (fun x => B * u x) =ᵐ[volume.restrict U] 0 := by filter_upwards [ae_restrict_of_ae_restrict_of_subset hsU hall,ae_restrict_mem hU.measurableSet] with x hx hxm have hxv : x ∉ V := fun hxv => Set.disjoint_left.mp hd hxm hxv simpa only [w,Pi.add_apply,Pi.smul_apply,indicator_of_mem hxm,indicator_of_notMem hxv, smul_zero,add_zero,smul_eq_mul,mul_zero] using hx have hcont : ContinuousOn (fun x => B*u x) U := continuousOn_const.mul (hu.1.continuousOn.mono hsU) have hh := Measure.eqOn_open_of_ae_eq hvu hU hcont continuousOn_const obtain ⟨x,hx⟩ := hnU have hp := mul_pos hB (hpU x hx) have hzero : B*u x = 0 := hh hx linarith lemma positive_set_connected {Ω : Set Plane} {u : Plane → ℝ} (hΩ : AdmissibleDomain Ω) (hu : InInteriorNeumannEigenspace Ω u) (hne : ∃ x ∈ Ω, u x ≠ 0) : IsConnected (Ω ∩ {x | 0 < u x}) := by let P := Ω ∩ {x | 0 < u x} have hopen : IsOpen P := positive_set_open hΩ.2.1 hu.1.continuousOn obtain ⟨a,ha,hua⟩ := eigenfunction_has_positive hΩ hu hne have hap : a ∈ P := ⟨ha,hua⟩ have heq : P = connectedComponentIn P a := by apply subset_antisymm _ (connectedComponentIn_subset P a) intro b hb by_contra hba let U := connectedComponentIn P a let V := connectedComponentIn P b have hd : Disjoint U V := by apply Set.disjoint_left.mpr intro x hxU hxV have heq : U = V := (connectedComponentIn_eq hxU).trans (connectedComponentIn_eq hxV).symm have hbV : b ∈ V := mem_connectedComponentIn hb have hbU : b ∈ U := heq.symm ▸ hbV exact hba hbU exact no_two_positive_regions hΩ hu hopen.connectedComponentIn hopen.connectedComponentIn ((connectedComponentIn_subset P a).trans inter_subset_left) ((connectedComponentIn_subset P b).trans inter_subset_left) ⟨a,mem_connectedComponentIn hap⟩ ⟨b,mem_connectedComponentIn hb⟩ hd (fun x hx => (connectedComponentIn_subset P a hx).2) (fun x hx => (connectedComponentIn_subset P b hx).2) (positive_component_hasH1Gradient hΩ.2.1 hu.1 hu.2.1 a) (positive_component_hasH1Gradient hΩ.2.1 hu.1 hu.2.1 b) (eigenfunction_has_negative hΩ hu hne) rw [show Ω ∩ {x | 0 < u x} = P from rfl,heq] exact isConnected_connectedComponentIn_iff.mpr hap lemma negative_set_connected {Ω : Set Plane} {u : Plane → ℝ} (hΩ : AdmissibleDomain Ω) (hu : InInteriorNeumannEigenspace Ω u) (hne : ∃ x ∈ Ω, u x ≠ 0) : IsConnected (Ω ∩ {x | u x < 0}) := by have hn : ∃ x ∈ Ω, (-u) x ≠ 0 := by obtain ⟨x,hx,hux⟩ := hne exact ⟨x,hx,by simpa only [Pi.neg_apply,neg_ne_zero] using hux⟩ simpa only [Pi.neg_apply,neg_pos] using positive_set_connected hΩ (eigenfunction_neg hu) hn end StrictHotSpots.InteriorNodal end NodalConnectedInteriorLayer end ActualInteriorNodal section ActualInteriorBoundarySign noncomputable section open Set Filter MeasureTheory open scoped Topology ContDiff InnerProductSpace ENNReal namespace StrictHotSpots.InteriorBoundarySign open StrictHotSpots.BoundarySign variable {Ω : Set Plane} {u : Plane → ℝ} lemma gap_test (ho : IsOpen Ω) (hu : InInteriorNeumannEigenspace Ω u) (hc : ContinuousOn u (closure Ω)) (hbound : Bornology.IsBounded Ω) (hb : ∀ x ∈ frontier Ω, u x ≤ 0) {lam : ℝ} (hg : ∀ v : H10 ho, lam * ‖H10.value ho v‖^2 ≤ ‖H10.grad ho v‖^2) (n : ℕ) : lam * (∫ x in Ω, (Nodal.cutoff n (u x))^2) ≤ ∫ x in Ω, (deriv (Nodal.cutoff n) (u x))^2 * ‖gradient u x‖^2 := by let f := test_interior ho hu.1 hc hbound hb n have hh := hg (smoothTestToH10 ho f) change lam * ‖(HasH1Gradient.test ho f.property.1 f.property.2.1).1.toLp f.val‖^2 ≤ ‖(HasH1Gradient.test ho f.property.1 f.property.2.1).2.1.toLp (gradient f.val)‖^2 at hh rw [← real_inner_self_eq_norm_sq, ← real_inner_self_eq_norm_sq, L2Inner.toLp_pair,L2Inner.toLp_pair] at hh have hval : (∫ x in Ω, inner ℝ (f.val x) (f.val x)) = ∫ x in Ω, (Nodal.cutoff n (u x))^2 := by apply setIntegral_congr_fun ho.measurableSet intro x hx simp only [f,test_interior,cut,indicator_of_mem hx,RCLike.inner_apply,conj_trivial,pow_two] have hgrad : (∫ x in Ω, inner ℝ (gradient f.val x) (gradient f.val x)) = ∫ x in Ω, (deriv (Nodal.cutoff n) (u x))^2 * ‖gradient u x‖^2 := by apply setIntegral_congr_fun ho.measurableSet intro x hx dsimp only rw [real_inner_self_eq_norm_sq] change ‖gradient (cut Ω u n) x‖^2 = _ rw [cut_gradient_interior ho hu.1 n hx,norm_smul,mul_pow,Real.norm_eq_abs,sq_abs] rwa [hval,hgrad] at hh lemma equation_test (ho : IsOpen Ω) (hu : InInteriorNeumannEigenspace Ω u) (hc : ContinuousOn u (closure Ω)) (hbound : Bornology.IsBounded Ω) (hb : ∀ x ∈ frontier Ω, u x ≤ 0) (n : ℕ) : (∫ x in Ω, deriv (Nodal.cutoff n) (u x) * ‖gradient u x‖^2) = firstPositiveNeumannValue Ω * (∫ x in Ω, u x * Nodal.cutoff n (u x)) := by let f := test_interior ho hu.1 hc hbound hb n have hh := hu.2.2.2 f.val (gradient f.val) (HasH1Gradient.test ho f.property.1 f.property.2.1) convert hh using 1 · apply setIntegral_congr_fun ho.measurableSet intro x hx change _ = inner ℝ (gradient u x) (gradient (cut Ω u n) x) rw [cut_gradient_interior ho hu.1 n hx,inner_smul_right,real_inner_self_eq_norm_sq] · congr 1 apply setIntegral_congr_fun ho.measurableSet intro x hx simp only [f,test_interior,cut,indicator_of_mem hx] lemma positivePart_square (t : ℝ) : (if 0 < t then t^2 else 0) = (max t 0)^2 := by by_cases h : 0 < t · rw [ite_eq_left h,max_eq_left h.le] · rw [ite_eq_right h,max_eq_right (le_of_not_gt h),zero_pow (by decide : 2 ≠ 0)] lemma positivePart_memLp (hu : MemLp u 2 (volume.restrict Ω)) : MemLp (fun x => max (u x) 0) 2 (volume.restrict Ω) := by apply hu.mono (((continuous_id.max continuous_const : Continuous (fun t : ℝ => max t 0))).comp_aestronglyMeasurable hu.aestronglyMeasurable) exact Eventually.of_forall fun x => by change ‖max (u x) 0‖ ≤ ‖u x‖ by_cases h : 0 ≤ u x · rw [max_eq_left h] · rw [max_eq_right (le_of_not_ge h),norm_zero] exact norm_nonneg _ lemma nonpos_of_boundary_nonpos (ho : IsOpen Ω) (hne : Ω.Nonempty) (hbound : Bornology.IsBounded Ω) (hu : InInteriorNeumannEigenspace Ω u) (hc : ContinuousOn u (closure Ω)) (hb : ∀ x ∈ frontier Ω, u x ≤ 0) : ∀ x ∈ Ω, u x ≤ 0 := by obtain ⟨lam,_hl,hgap,hDir⟩ := H10.subcritical ho hne hbound let M := ∫ x in Ω, if 0 < u x then (u x)^2 else 0 let E := ∫ x in Ω, if 0 < u x then ‖gradient u x‖^2 else 0 have hIg : Integrable (fun x => ‖gradient u x‖^2) (volume.restrict Ω) := hu.2.1.2.1.integrable_norm_pow (by norm_num) have hM := Nodal.cutoff_square_integral hu.2.1.1 have hE := Nodal.cutoff_deriv_square_integral hu.2.1.1.aestronglyMeasurable hIg (fun x => sq_nonneg _) have hi : lam * M ≤ E := le_of_tendsto_of_tendsto (hM.const_mul lam) hE (Eventually.of_forall (gap_test ho hu hc hbound hb hDir)) have hD := Nodal.cutoff_deriv_integral hu.2.1.1.aestronglyMeasurable hIg (fun x => sq_nonneg _) have hP := Nodal.cutoff_product_integral hu.2.1.1 have he : E = firstPositiveNeumannValue Ω * M := by apply tendsto_nhds_unique hD apply (hP.const_mul (firstPositiveNeumannValue Ω)).congr intro n exact (equation_test ho hu hc hbound hb n).symm have hMn : 0 ≤ M := integral_nonneg fun x => by split_ifs <;> positivity have hMz : M = 0 := by nlinarith have hMz' : (∫ x in Ω, (max (u x) 0)^2) = 0 := by simpa only [M,positivePart_square] using hMz have hz := (integral_eq_zero_iff_of_nonneg (fun x => sq_nonneg _) (positivePart_memLp hu.2.1.1).integrable_sq).mp hMz' have hae : (fun x => max (u x) 0) =ᵐ[volume.restrict Ω] 0 := by filter_upwards [hz] with x hx exact eq_zero_of_pow_eq_zero hx have hall := Measure.eqOn_open_of_ae_eq hae ho ((continuous_id.max continuous_const : Continuous (fun t : ℝ => max t 0)).comp_continuousOn hu.1.continuousOn) continuousOn_const intro x hx exact (le_max_left (u x) 0).trans (le_of_eq (hall hx)) lemma zero_of_boundary_nonpos (ho : IsOpen Ω) (hne : Ω.Nonempty) (hbound : Bornology.IsBounded Ω) (hu : InInteriorNeumannEigenspace Ω u) (hc : ContinuousOn u (closure Ω)) (hb : ∀ x ∈ frontier Ω, u x ≤ 0) : ∀ x ∈ Ω, u x = 0 := by let : IsFiniteMeasure (volume.restrict Ω) := isFiniteMeasure_restrict.mpr hbound.measure_lt_top.ne have hnp := nonpos_of_boundary_nonpos ho hne hbound hu hc hb have hn : 0 ≤ᵐ[volume.restrict Ω] -u := by filter_upwards [ae_restrict_mem ho.measurableSet] with x hx simpa only [Pi.neg_apply,Pi.zero_apply,neg_nonneg] using hnp x hx have hm : (∫ x in Ω, (-u) x) = 0 := by simp only [Pi.neg_apply,integral_neg,hu.2.2.1,neg_zero] have hz := (integral_eq_zero_iff_of_nonneg_ae hn (hu.2.1.1.neg.integrable (by norm_num))).mp hm have hae : u =ᵐ[volume.restrict Ω] 0 := by filter_upwards [hz] with x hx simpa only [Pi.neg_apply,Pi.zero_apply,neg_eq_zero] using hx exact fun x hx => Measure.eqOn_open_of_ae_eq hae ho hu.1.continuousOn continuousOn_const hx lemma exists_boundary_pos (ho : IsOpen Ω) (hbound : Bornology.IsBounded Ω) (hu : InInteriorNeumannEigenspace Ω u) (hc : ContinuousOn u (closure Ω)) (hne : ∃ x ∈ Ω, u x ≠ 0) : ∃ x ∈ frontier Ω, 0 < u x := by by_contra hn push Not at hn obtain ⟨x,hx,hux⟩ := hne exact hux (zero_of_boundary_nonpos ho ⟨x,hx⟩ hbound hu hc hn x hx) lemma eigen_neg (hu : InInteriorNeumannEigenspace Ω u) : InInteriorNeumannEigenspace Ω (-u) := by have hg : gradient (-u) = -gradient u := by funext x simp only [gradient,Pi.neg_apply,fderiv_neg,map_neg] refine ⟨hu.1.neg,?_,?_,?_⟩ · simpa only [hg,neg_one_smul] using hu.2.1.smul (-1) · simp only [Pi.neg_apply,integral_neg,hu.2.2.1,neg_zero] · intro v g hv simp only [hg,Pi.neg_apply,inner_neg_left,neg_mul,integral_neg] rw [hu.2.2.2 v g hv] ring lemma exists_boundary_neg (ho : IsOpen Ω) (hbound : Bornology.IsBounded Ω) (hu : InInteriorNeumannEigenspace Ω u) (hc : ContinuousOn u (closure Ω)) (hne : ∃ x ∈ Ω, u x ≠ 0) : ∃ x ∈ frontier Ω, u x < 0 := by have hn : ∃ x ∈ Ω, (-u) x ≠ 0 := by obtain ⟨x,hx,hux⟩ := hne exact ⟨x,hx,by simpa only [Pi.neg_apply,neg_ne_zero] using hux⟩ obtain ⟨x,hx,hux⟩ := exists_boundary_pos ho hbound (eigen_neg hu) hc.neg hn exact ⟨x,hx,by simpa only [Pi.neg_apply,neg_pos] using hux⟩ end StrictHotSpots.InteriorBoundarySign end end ActualInteriorBoundarySign section MultiplicityComplete noncomputable section section PositiveWeightLayer open Set Filter MeasureTheory Function open scoped Topology InnerProductSpace namespace StrictHotSpots.PositiveWeight variable {X E : Type*} [TopologicalSpace X] [CompactSpace X] [Nonempty X] [MeasurableSpace X] [OpensMeasurableSpace X] [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] (μ : Measure X) [IsFiniteMeasure μ] omit [Nonempty X] in lemma continuous_integrable {G : Type*} [NormedAddCommGroup G] (f : C(X,G)) : Integrable f μ := f.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) omit [Nonempty X] [FiniteDimensional ℝ E] in lemma product_integrable (F : C(X,E)) (f : C(X,ℝ)) : Integrable (fun s => f s • F s) μ := (f.continuous.smul F.continuous).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) def momentLinear (F : C(X,E)) : C(X,ℝ) →ₗ[ℝ] E where toFun f := ∫ s, f s • F s ∂μ map_add' f g := by simp only [ContinuousMap.add_apply,add_smul] exact integral_add (product_integrable μ F f) (product_integrable μ F g) map_smul' a f := by simp only [ContinuousMap.smul_apply,smul_eq_mul,mul_smul,integral_smul,RingHom.id_apply] def moment (F : C(X,E)) : C(X,ℝ) →L[ℝ] E := (momentLinear μ F).mkContinuous (‖F‖ * μ.real univ) (fun f => by have hh := norm_integral_le_of_norm_le_const (μ := μ) (Eventually.of_forall fun s => show ‖f s • F s‖ ≤ ‖f‖*‖F‖ by rw [norm_smul] exact mul_le_mul (f.norm_coe_le_norm s) (F.norm_coe_le_norm s) (norm_nonneg _) (norm_nonneg _)) change ‖∫ s, f s • F s ∂μ‖ ≤ _ nlinarith only [hh]) omit [Nonempty X] [FiniteDimensional ℝ E] in lemma moment_apply (F : C(X,E)) (f : C(X,ℝ)) : moment μ F f = ∫ s, f s • F s ∂μ := rfl def probe (F : C(X,E)) : E →ₗ[ℝ] C(X,ℝ) where toFun a := ⟨fun s => inner ℝ a (F s),continuous_const.inner F.continuous⟩ map_add' a b := by ext s change inner ℝ (a+b) (F s) = inner ℝ a (F s) + inner ℝ b (F s) exact inner_add_left _ _ _ map_smul' c a := by ext s change inner ℝ (c • a) (F s) = c * inner ℝ a (F s) simp only [inner_smul_left,conj_trivial] omit [Nonempty X] in lemma dual_moment (F : C(X,E)) (f : C(X,ℝ)) (l : E →L[ℝ] ℝ) : l (moment μ F f) = ∫ s, f s * l (F s) ∂μ := by rw [moment_apply,← l.integral_comp_comm (product_integrable μ F f)] congr 1 funext s exact l.map_smul (f s) (F s) omit [Nonempty X] in lemma integral_sq_pos [μ.IsOpenPosMeasure] (f : C(X,ℝ)) (hf : ∃ x, f x ≠ 0) : 0 < ∫ s, (f s)^2 ∂μ := by have hn : 0 ≤ ∫ s, (f s)^2 ∂μ := integral_nonneg fun _ => sq_nonneg _ by_contra hp have hz : (∫ s, (f s)^2 ∂μ) = 0 := le_antisymm (le_of_not_gt hp) hn have hi : Integrable (fun s => (f s)^2) μ := (f.continuous.pow 2).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) have ha := (integral_eq_zero_iff_of_nonneg (fun _ => sq_nonneg _) hi).mp hz have hae : f =ᵐ[μ.restrict univ] (0 : X → ℝ) := by rw [Measure.restrict_univ] filter_upwards [ha] with x hx exact eq_zero_of_pow_eq_zero hx have he := Measure.eqOn_open_of_ae_eq hae isOpen_univ f.continuous.continuousOn continuousOn_const obtain ⟨x,hx⟩ := hf exact hx (he (mem_univ x)) omit [Nonempty X] in lemma moment_surjective [μ.IsOpenPosMeasure] (F : C(X,E)) (hF : ∀ a : E, a ≠ 0 → ∃ s, 0 < inner ℝ a (F s)) : Surjective (moment μ F) := by let G : E →ₗ[ℝ] E := (moment μ F).toLinearMap.comp (probe F) have hz (a : E) (ha : G a = 0) : a = 0 := by by_contra hn obtain ⟨s,hs⟩ := hF a hn have hp := integral_sq_pos μ (probe F a) ⟨s,ne_of_gt hs⟩ have he := dual_moment μ F (probe F a) (innerSL ℝ a) have hv : (∫ s, (probe F a s)^2 ∂μ) = 0 := by change inner ℝ a (G a) = ∫ s, inner ℝ a (F s) * inner ℝ a (F s) ∂μ at he rw [ha,inner_zero_right] at he simpa only [probe,LinearMap.coe_mk,AddHom.coe_mk,ContinuousMap.coe_mk,pow_two] using he.symm linarith have hi : Injective G := by intro a b hab apply sub_eq_zero.mp apply hz rw [map_sub,hab,sub_self] intro a obtain ⟨b,hb⟩ := G.surjective_of_injective hi a exact ⟨probe F b,hb⟩ def positive : Set C(X,ℝ) := {f | ∀ s, 0 < f s} omit [CompactSpace X] [Nonempty X] [MeasurableSpace X] [OpensMeasurableSpace X] in lemma positive_convex : Convex ℝ (positive (X := X)) := by intro f hf g hg a b ha hb hab s change 0 < a * f s + b * g s by_cases hza : a = 0 · have hbo : b = 1 := by linarith simpa [hza,hbo] using hg s · exact add_pos_of_pos_of_nonneg (mul_pos (lt_of_le_of_ne ha (Ne.symm hza)) (hf s)) (mul_nonneg hb (hg s).le) omit [MeasurableSpace X] [OpensMeasurableSpace X] in lemma positive_open : IsOpen (positive (X := X)) := by rw [Metric.isOpen_iff] intro f hf obtain ⟨x,_,hx⟩ := isCompact_univ.exists_isMinOn (univ_nonempty : (univ : Set X).Nonempty) f.continuous.continuousOn refine ⟨f x,hf x,?_⟩ intro g hg s have hd : |g s-f s| < f x := by rw [← Real.dist_eq] exact (ContinuousMap.dist_apply_le_dist s).trans_lt hg have hm : f x ≤ f s := hx (mem_univ s) have hl := (abs_lt.mp hd).1 linarith theorem exists_positive_annihilator [μ.IsOpenPosMeasure] (F : C(X,E)) (hF : ∀ a : E, a ≠ 0 → ∃ s, 0 < inner ℝ a (F s)) : ∃ w : C(X,ℝ), (∀ s, 0 < w s) ∧ (∫ s, w s • F s ∂μ) = 0 := by let T := moment μ F have hT : Surjective T := moment_surjective μ F hF have ho : IsOpen (T '' positive) := T.isOpenMap hT _ positive_open have hc : Convex ℝ (T '' positive) := positive_convex.linear_image T.toLinearMap have hz : (0:E) ∈ T '' positive := by by_contra hn obtain ⟨l,hl⟩ := geometric_hahn_banach_open_point hc ho hn let one : C(X,ℝ) := ContinuousMap.const X 1 have hone : one ∈ positive := fun _ => zero_lt_one have hnegative (f : C(X,ℝ)) (hf : f ∈ positive) : l (T f) < 0 := by simpa only [map_zero] using hl (T f) ⟨f,hf,rfl⟩ have hl0 : l ≠ 0 := by intro he have h := hnegative one hone rw [he] at h exact (lt_irrefl (0:ℝ)) h let a : E := (InnerProductSpace.toDual ℝ E).symm l have ha : a ≠ 0 := by intro hz have he := congrArg (InnerProductSpace.toDual ℝ E) hz simp only [a,LinearIsometryEquiv.apply_symm_apply,map_zero] at he exact hl0 he obtain ⟨s,hs⟩ := hF a ha have hla : ∀ x, l (F x) = inner ℝ a (F x) := by intro x exact congrArg (fun k : E →L[ℝ] ℝ => k (F x)) ((InnerProductSpace.toDual ℝ E).apply_symm_apply l).symm let q : C(X,ℝ) := ⟨fun x => max (l (F x)) 0,(l.continuous.comp F.continuous).max continuous_const⟩ have hqnonneg (x : X) : 0 ≤ q x := le_max_right _ _ have hqpos : 0 < q s := lt_of_lt_of_le (by rwa [hla s]) (le_max_left _ _) let A := ∫ x, (q x)^2 ∂μ have hA : 0 < A := integral_sq_pos μ q ⟨s,ne_of_gt hqpos⟩ have he : l (T q) = A := by rw [show T q = moment μ F q from rfl,dual_moment] apply integral_congr_ae exact Eventually.of_forall fun x => by change max (l (F x)) 0 * l (F x) = (max (l (F x)) 0)^2 by_cases h : 0 ≤ l (F x) · rw [max_eq_left h,pow_two] · rw [max_eq_right (le_of_not_ge h),zero_mul,zero_pow (by decide : 2 ≠ 0)] let B := l (T one) let c := (|B|+1)/A have hcpos : 0 < c := div_pos (by positivity) hA let f := one + c • q have hf : f ∈ positive := by intro x change 0 < 1 + c * q x linarith [mul_nonneg hcpos.le (hqnonneg x)] have hbad := hnegative f hf have hval : l (T f) = B + (|B|+1) := by simp only [f,map_add,map_smul,smul_eq_mul,he] change B + c*A = B+(|B|+1) dsimp only [c] rw [div_mul_cancel₀ _ hA.ne'] rw [hval] at hbad linarith [neg_abs_le B] obtain ⟨w,hw,he⟩ := hz exact ⟨w,hw,he⟩ end StrictHotSpots.PositiveWeight end PositiveWeightLayer section CirclePositiveWeightLayer open Set MeasureTheory open scoped Topology InnerProductSpace namespace StrictHotSpots.PositiveWeight local instance : Fact (0 < 2*Real.pi) := ⟨Real.two_pi_pos⟩ variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] theorem exists_circle_weight (F : C(Circle,E)) (hF : ∀ a : E, a ≠ 0 → ∃ s, 0 < inner ℝ a (F s)) : ∃ w : ℝ → ℝ, Continuous w ∧ Function.Periodic w (2*Real.pi) ∧ (∀ θ, 0 < w θ) ∧ ∀ a : E, (∫ θ in (0:ℝ)..2*Real.pi, w θ*inner ℝ a (F (Circle.exp θ))) = 0 := by let G : C(AddCircle (2*Real.pi),E) := F.comp ⟨AddCircle.homeomorphCircle',AddCircle.homeomorphCircle'.continuous⟩ have hG (a : E) (ha : a ≠ 0) : ∃ s, 0 < inner ℝ a (G s) := by obtain ⟨s,hs⟩ := hF a ha refine ⟨AddCircle.homeomorphCircle'.symm s,?_⟩ change 0 < inner ℝ a (F (AddCircle.homeomorphCircle' (AddCircle.homeomorphCircle'.symm s))) rwa [Homeomorph.apply_symm_apply] obtain ⟨w,hw,he⟩ := exists_positive_annihilator (volume : Measure (AddCircle (2*Real.pi))) G hG let W : ℝ → ℝ := fun θ => w (θ : AddCircle (2*Real.pi)) refine ⟨W,w.continuous.comp (AddCircle.continuous_mk' (2*Real.pi)),?_,fun θ => hw _,?_⟩ · intro θ exact congrArg w (AddCircle.coe_add_period (2*Real.pi) θ) · intro a have hh := dual_moment (volume : Measure (AddCircle (2*Real.pi))) G w (innerSL ℝ a) rw [moment_apply,he,map_zero] at hh have hI := AddCircle.intervalIntegral_preimage (2*Real.pi) 0 (fun s : AddCircle (2*Real.pi) => w s*inner ℝ a (G s)) simp only [zero_add] at hI simpa only [W,G,ContinuousMap.comp_apply,ContinuousMap.coe_mk, AddCircle.homeomorphCircle'_apply_mk] using hI.trans hh.symm end StrictHotSpots.PositiveWeight end CirclePositiveWeightLayer section PlanarCrossingLayer open Set Filter _root_.Complex _root_.OAI.Complex Metric open scoped unitInterval Topology ComplexConjugate namespace StrictHotSpots.PlanarTopology private lemma boundary_halfPlane {x b : ℂ} (hx : ‖x‖ ≤ 1) (hb : ‖b‖ = 1) (hne : x ≠ b) : 0 < ((x-b)*(-conj b)).re := by have hx' : normSq x ≤ 1 := by rw [normSq_eq_norm_sq]; nlinarith [norm_nonneg x] have hb' : normSq b = 1 := by rw [normSq_eq_norm_sq,hb]; norm_num have hd : 0 < normSq (x-b) := normSq_pos.mpr (sub_ne_zero.mpr hne) simp only [normSq_apply,sub_re,sub_im] at hx' hb' hd simp only [mul_re,sub_re,sub_im,neg_re,neg_im,conj_re,conj_im] nlinarith private lemma log_difference {P : C(unitInterval,ℂ)} {b : ℂ} (hP : ∀ s, ‖P s‖ ≤ 1) (hb : ‖b‖ = 1) (hne : ∀ s, P s ≠ b) {l : C(unitInterval,ℂ)} (hl : ∀ s, exp (l s) = P s-b) : l 1-l 0 = log ((P 1-b)*(-conj b))-log ((P 0-b)*(-conj b)) := by let w : unitInterval → ℂ := fun s => (P s-b)*(-conj b) have hw : Continuous w := by dsimp [w]; fun_prop have hwpos (s) : 0 < (w s).re := boundary_halfPlane (hP s) hb (hne s) have hws (s) : w s ∈ slitPlane := mem_slitPlane_iff.mpr (Or.inl (hwpos s)) have hwn (s) : w s ≠ 0 := fun h => by simpa [h] using hwpos s have hc : Continuous (fun s => log (w s)-log (w 0)+l 0) := ((hw.clog hws).sub continuous_const).add continuous_const have he (s) : exp (log (w s)-log (w 0)+l 0) = exp (l s) := by rw [exp_add,exp_sub,exp_log (hwn s),exp_log (hwn 0),hl s,hl 0] dsimp [w] have hb0 : -conj b ≠ 0 := neg_ne_zero.mpr ((map_ne_zero (starRingEnd ℂ)).mpr (by intro h; simp [h] at hb)) field_simp [sub_ne_zero.mpr (hne 0),neg_ne_zero.mp hb0] have eqfun : (fun s => log (w s)-log (w 0)+l 0) = l := by exact isCoveringMap_exp.eq_of_comp_eq hc l.continuous (funext fun s => Subtype.ext (he s)) 0 (by simp) have heq := congrFun eqfun 1 dsimp [w] at heq linear_combination -heq private lemma square_log {P N : C(unitInterval,ℂ)} (hne : ∀ s t, P s ≠ N t) : ∃ L : C(unitInterval × unitInterval,ℂ), ∀ s t, exp (L (s,t)) = P s-N t := by let : ContractibleSpace unitInterval := (convex_Icc (0:ℝ) 1).contractibleSpace ⟨0,by simp⟩ let : LocallyPathConnectedSpace unitInterval := (convex_Icc (0:ℝ) 1).locallyPathConnectedSpace let f : C(unitInterval × unitInterval,{z : ℂ // z ≠ 0}) := ⟨fun st => ⟨P st.1-N st.2,sub_ne_zero.mpr (hne st.1 st.2)⟩, ((P.continuous.comp continuous_fst).sub (N.continuous.comp continuous_snd)).subtype_mk _⟩ obtain ⟨L,hL,-⟩ := isCoveringMap_exp.existsUnique_continuousMap_lifts f (0,0) (log (f (0,0))) (Subtype.ext (exp_log (f (0,0)).property)) exact ⟨L,fun s t => congrArg Subtype.val (congrFun hL.2 (s,t))⟩ theorem disk_paths_cross {P N : C(unitInterval,ℂ)} (hP : ∀ s, ‖P s‖ ≤ 1) (hN : ∀ t, ‖N t‖ ≤ 1) (hP0 : P 0 = -1) (hP1 : P 1 = 1) (hN0 : ‖N 0‖ = 1) (hN1 : ‖N 1‖ = 1) (hup : 0 < (N 0).im) (hdn : (N 1).im < 0) : ∃ s t, P s = N t := by by_contra h push Not at h obtain ⟨L,hL⟩ := square_log h let D : unitInterval → ℂ := fun t => L (1,t)-L (0,t) have hD : Continuous D := by dsimp [D]; fun_prop have hed (t : unitInterval) (ht : ‖N t‖ = 1) : D t = log ((1-N t)*(-conj (N t)))-log ((-1-N t)*(-conj (N t))) := by have hh := log_difference hP ht (fun s => h s t) (l := ⟨fun s => L (s,t),by fun_prop⟩) (fun s => hL s t) simpa only [ContinuousMap.coe_mk,hP0,hP1] using hh have him1 (z : ℂ) : ((1-z)*(-conj z)).im = z.im := by simp only [mul_im,sub_re,sub_im,one_re,one_im,neg_re,neg_im,conj_re,conj_im] ring have him0 (z : ℂ) : ((-1-z)*(-conj z)).im = -z.im := by simp only [mul_im,sub_re,sub_im,one_re,one_im,neg_re,neg_im,conj_re,conj_im] ring have hp : 0 < (D 0).im := by rw [hed 0 hN0,sub_im,log_im,log_im] have ha := arg_nonneg_iff.mpr (by rw [him1]; exact hup.le) have hb := arg_neg_iff.mpr (by rw [him0]; exact neg_neg_of_pos hup) linarith have hn : (D 1).im < 0 := by rw [hed 1 hN1,sub_im,log_im,log_im] have ha := arg_neg_iff.mpr (by rw [him1]; exact hdn) have hb := arg_nonneg_iff.mpr (by rw [him0]; exact neg_nonneg.mpr hdn.le) linarith obtain ⟨t,ht⟩ := intermediate_value_univ (1 : unitInterval) 0 (continuous_im.comp hD) ⟨hn.le,hp.le⟩ have he : (1-N t) = exp (D t) * (-1-N t) := by have hd : exp (D t) = (1-N t)/(-1-N t) := by dsimp only [D] rw [exp_sub,hL,hL,hP1,hP0] exact ((div_eq_iff (sub_ne_zero.mpr (hP0 ▸ h 0 t))).mp hd.symm) have hex : exp (D t) = (Real.exp (D t).re : ℂ) := by apply Complex.ext <;> simp [exp_re,exp_im,show (D t).im = 0 from ht] rw [hex] at he have hv : 0 < normSq (-1-N t) := normSq_pos.mpr (sub_ne_zero.mpr (hP0 ▸ h 0 t)) have hx : normSq (N t) ≤ 1 := by rw [normSq_eq_norm_sq] nlinarith [hN t,norm_nonneg (N t)] have hpv := mul_pos (Real.exp_pos (D t).re) hv have hr := congrArg re (congrArg (fun z : ℂ => z*conj (-1-N t)) he) simp only [mul_re,mul_im,sub_re,sub_im,one_re,one_im,neg_re,neg_im,conj_re,conj_im, ofReal_re,ofReal_im,zero_mul,add_zero,sub_zero] at hr simp only [normSq_apply,sub_re,sub_im,one_re,one_im,neg_re,neg_im] at hx hpv nlinarith private lemma arg_product_rightHalfPlane {u v : ℂ} (hu : 0 < u.re) (hv : 0 < v.re) : arg (u*conj v) = arg u-arg v := by have h₁ := neg_pi_div_two_lt_arg_iff.mpr (Or.inl hu) have h₂ := arg_lt_pi_div_two_iff.mpr (Or.inl hu) have h₃ := neg_pi_div_two_lt_arg_iff.mpr (Or.inl hv) have h₄ := arg_lt_pi_div_two_iff.mpr (Or.inl hv) have hvπ : arg v ≠ Real.pi := by linarith [Real.pi_pos] have hc : arg (conj v) = -arg v := by rw [arg_conj,ite_eq_right hvπ] rw [arg_mul (by intro h; simp [h] at hu) (by intro h; have := congrArg re h; simp only [conj_re,zero_re] at this; linarith) (by rw [hc]; constructor <;> linarith),hc,sub_eq_add_neg] private lemma log_difference_im {P : C(unitInterval,ℂ)} {b : ℂ} (hP : ∀ s, ‖P s‖ ≤ 1) (hb : ‖b‖ = 1) (hne : ∀ s, P s ≠ b) {l : C(unitInterval,ℂ)} (hl : ∀ s, exp (l s) = P s-b) : (l 1-l 0).im = arg ((P 1-b)*conj (P 0-b)) := by rw [log_difference hP hb hne hl,sub_im,log_im,log_im, ← arg_product_rightHalfPlane (boundary_halfPlane (hP 1) hb (hne 1)) (boundary_halfPlane (hP 0) hb (hne 0))] congr 1 have hbc : (-conj b)*conj (-conj b) = 1 := by rw [mul_conj,normSq_eq_norm_sq,norm_neg,norm_conj,hb] norm_num rw [map_mul] calc (P 1-b)*(-conj b)*(conj (P 0-b)*conj (-conj b)) = ((P 1-b)*conj (P 0-b))*((-conj b)*conj (-conj b)) := by ring _ = _ := by rw [hbc,mul_one] private lemma no_positive_chord_ratio {a c z : ℂ} {k : ℝ} (ha : ‖a‖ = 1) (hc : ‖c‖ = 1) (hz : ‖z‖ ≤ 1) (hac : a ≠ c) (haz : a ≠ z) (hk : 0 < k) (he : c-z = (k : ℂ)*(a-z)) : False := by have hna : normSq a = 1 := by rw [normSq_eq_norm_sq,ha]; norm_num have hnc : normSq c = 1 := by rw [normSq_eq_norm_sq,hc]; norm_num have hnz : normSq z ≤ 1 := by rw [normSq_eq_norm_sq]; nlinarith [norm_nonneg z] have hnaz : 0 < normSq (a-z) := normSq_pos.mpr (sub_ne_zero.mpr haz) have hce : c = z+(k : ℂ)*(a-z) := by linear_combination he have hid : normSq c-normSq a = (k-1)*(normSq a-normSq z+k*normSq (a-z)) := by rw [hce] simp only [normSq_apply,add_re,add_im,mul_re,mul_im,sub_re,sub_im,ofReal_re,ofReal_im] ring rw [hna,hnc,sub_self] at hid have hp : 0 < 1-normSq z+k*normSq (a-z) := add_pos_of_nonneg_of_pos (sub_nonneg.mpr hnz) (mul_pos hk hnaz) have he1 : k = 1 := by have hh := (mul_eq_zero.mp hid.symm).resolve_right hp.ne' linarith rw [he1,ofReal_one,one_mul] at he exact hac (sub_left_injective he).symm theorem disk_paths_cross_chord {P N : C(unitInterval,ℂ)} (hP : ∀ s, ‖P s‖ ≤ 1) (hN : ∀ t, ‖N t‖ ≤ 1) (hP0 : ‖P 0‖ = 1) (hP1 : ‖P 1‖ = 1) (hneP : P 0 ≠ P 1) (hN0 : ‖N 0‖ = 1) (hN1 : ‖N 1‖ = 1) (hup : 0 < ((P 1-N 0)*conj (P 0-N 0)).im) (hdn : ((P 1-N 1)*conj (P 0-N 1)).im < 0) : ∃ s t, P s = N t := by by_contra h push Not at h obtain ⟨L,hL⟩ := square_log h let D : unitInterval → ℂ := fun t => L (1,t)-L (0,t) have hD : Continuous D := by dsimp [D]; fun_prop have hed (t : unitInterval) (ht : ‖N t‖ = 1) : (D t).im = arg ((P 1-N t)*conj (P 0-N t)) := log_difference_im hP ht (fun s => h s t) (l := ⟨fun s => L (s,t),by fun_prop⟩) (fun s => hL s t) have hp : 0 < (D 0).im := by rw [hed 0 hN0] have hge := arg_nonneg_iff.mpr hup.le have hn : arg ((P 1-N 0)*conj (P 0-N 0)) ≠ 0 := by intro ht exact hup.ne' (arg_eq_zero_iff.mp ht).2 exact hge.lt_of_ne' hn have hn : (D 1).im < 0 := by rw [hed 1 hN1]; exact arg_neg_iff.mpr hdn obtain ⟨t,ht⟩ := intermediate_value_univ (1 : unitInterval) 0 (continuous_im.comp hD) ⟨hn.le,hp.le⟩ have he : (P 1-N t) = exp (D t)*(P 0-N t) := by have hd : exp (D t) = (P 1-N t)/(P 0-N t) := by dsimp only [D]; rw [exp_sub,hL,hL] exact (div_eq_iff (sub_ne_zero.mpr (h 0 t))).mp hd.symm have hex : exp (D t) = (Real.exp (D t).re : ℂ) := by apply Complex.ext <;> simp [exp_re,exp_im,show (D t).im = 0 from ht] rw [hex] at he exact no_positive_chord_ratio hP0 hP1 (hN t) hneP (h 0 t) (Real.exp_pos _) he end StrictHotSpots.PlanarTopology end PlanarCrossingLayer section DiskSignTopologyLayer open Set Filter Metric _root_.Complex _root_.OAI.Complex open scoped unitInterval Topology ComplexConjugate namespace StrictHotSpots.PlanarTopology private lemma disk_positive_open {u : ℂ → ℝ} (hu : ContinuousOn u (closedBall 0 1)) : IsOpen (ball (0 : ℂ) 1 ∩ {z | 0 < u z}) := by exact (hu.mono ball_subset_closedBall).isOpen_inter_preimage isOpen_ball isOpen_Ioi lemma join_positive_interior {u : ℂ → ℝ} (hu : ContinuousOn u (closedBall 0 1)) {a : ℂ} (ha : a ∈ closedBall (0 : ℂ) 1) (hua : 0 < u a) : ∃ a' ∈ ball (0 : ℂ) 1 ∩ {z | 0 < u z}, JoinedIn (closedBall (0 : ℂ) 1 ∩ {z | 0 < u z}) a a' := by obtain ⟨ε,hε,hεu⟩ := (Metric.continuousWithinAt_iff.mp (hu a ha)) (u a) hua have hpos : closedBall (0 : ℂ) 1 ∩ ball a ε ⊆ {z | 0 < u z} := by intro z hz have hd := hεu hz.1 hz.2 rw [Real.dist_eq] at hd have := (abs_lt.mp hd).1 change 0 < u z linarith have hacl : a ∈ closure (ball (0 : ℂ) 1) := by rwa [closure_ball (0 : ℂ) one_ne_zero] obtain ⟨a',ha',hda'⟩ := Metric.mem_closure_iff.mp hacl ε hε have hsa : a ∈ closedBall (0 : ℂ) 1 ∩ ball a ε := ⟨ha,mem_ball_self hε⟩ have hsa' : a' ∈ closedBall (0 : ℂ) 1 ∩ ball a ε := ⟨ball_subset_closedBall ha',by simpa only [mem_ball,dist_comm] using hda'⟩ refine ⟨a',⟨ha',hpos hsa'⟩,?_⟩ exact (((convex_closedBall (0 : ℂ) 1).inter (convex_ball a ε)).isPathConnected ⟨a,hsa⟩ |>.joinedIn a hsa a' hsa').mono (fun z hz => ⟨hz.1,hpos hz⟩) lemma join_positive_closedDisk {u : ℂ → ℝ} (hu : ContinuousOn u (closedBall 0 1)) (hc : IsConnected (ball (0 : ℂ) 1 ∩ {z | 0 < u z})) {a c : ℂ} (ha : a ∈ closedBall (0 : ℂ) 1) (hc' : c ∈ closedBall (0 : ℂ) 1) (hua : 0 < u a) (huc : 0 < u c) : JoinedIn (closedBall (0 : ℂ) 1 ∩ {z | 0 < u z}) a c := by obtain ⟨a',ha',hja⟩ := join_positive_interior hu ha hua obtain ⟨c',hc'',hjc⟩ := join_positive_interior hu hc' huc have hpath := (disk_positive_open hu).isConnected_iff_isPathConnected.mp hc have hj := (hpath.joinedIn a' ha' c' hc'').mono (show ball (0 : ℂ) 1 ∩ {z | 0 < u z} ⊆ closedBall (0 : ℂ) 1 ∩ {z | 0 < u z} from fun z hz => ⟨ball_subset_closedBall hz.1,hz.2⟩) exact (hja.trans hj).trans hjc.symm theorem no_alternating_boundary_signs {u : ℂ → ℝ} (hu : ContinuousOn u (closedBall 0 1)) (hp : IsConnected (ball (0 : ℂ) 1 ∩ {z | 0 < u z})) (hn : IsConnected (ball (0 : ℂ) 1 ∩ {z | u z < 0})) {a b c d : ℂ} (ha : ‖a‖ = 1) (hb : ‖b‖ = 1) (hc : ‖c‖ = 1) (hd : ‖d‖ = 1) (hac : a ≠ c) (hup : 0 < ((c-b)*conj (a-b)).im) (hdn : ((c-d)*conj (a-d)).im < 0) (hua : 0 < u a) (hub : u b < 0) (huc : 0 < u c) (hud : u d < 0) : False := by have hacl : a ∈ closedBall (0 : ℂ) 1 := by simp only [mem_closedBall,dist_zero_right,ha,le_refl] have hccl : c ∈ closedBall (0 : ℂ) 1 := by simp only [mem_closedBall,dist_zero_right,hc,le_refl] have hbcl : b ∈ closedBall (0 : ℂ) 1 := by simp only [mem_closedBall,dist_zero_right,hb,le_refl] have hdcl : d ∈ closedBall (0 : ℂ) 1 := by simp only [mem_closedBall,dist_zero_right,hd,le_refl] have hP := join_positive_closedDisk hu hp hacl hccl hua huc have hneg : IsConnected (ball (0 : ℂ) 1 ∩ {z | 0 < -u z}) := by simpa only [neg_pos] using hn have hN := join_positive_closedDisk hu.neg hneg hbcl hdcl (neg_pos.mpr hub) (neg_pos.mpr hud) let P : C(unitInterval,ℂ) := hP.somePath.toContinuousMap let N : C(unitInterval,ℂ) := hN.somePath.toContinuousMap have hPs (s) : ‖P s‖ ≤ 1 := by have hh := (hP.somePath_mem s).1 simpa only [P,N,Path.coe_toContinuousMap,mem_closedBall,dist_zero_right] using hh have hNs (s) : ‖N s‖ ≤ 1 := by have hh := (hN.somePath_mem s).1 simpa only [P,N,Path.coe_toContinuousMap,mem_closedBall,dist_zero_right] using hh have hP0 : P 0 = a := hP.somePath.source have hP1 : P 1 = c := hP.somePath.target have hN0 : N 0 = b := hN.somePath.source have hN1 : N 1 = d := hN.somePath.target obtain ⟨s,t,hst⟩ := disk_paths_cross_chord hPs hNs (hP0.symm ▸ ha) (hP1.symm ▸ hc) (by simpa only [hP0,hP1] using hac) (hN0.symm ▸ hb) (hN1.symm ▸ hd) (by simpa only [hP0,hP1,hN0] using hup) (by simpa only [hP0,hP1,hN1] using hdn) have hpos : 0 < u (P s) := (hP.somePath_mem s).2 have hneg : 0 < -u (N t) := (hN.somePath_mem t).2 rw [hst] at hpos linarith end StrictHotSpots.PlanarTopology end DiskSignTopologyLayer section CircleSignMomentsLayer open Set MeasureTheory namespace StrictHotSpots.CircleSigns lemma separating_cosine (a b x : ℝ) : Real.cos (x-(a+b)/2)-Real.cos ((b-a)/2) = 2*Real.sin ((x-a)/2)*Real.sin ((b-x)/2) := by rw [Real.cos_sub_cos] have h₁ : ((x-(a+b)/2)+(b-a)/2)/2 = (x-a)/2 := by ring have h₂ : ((x-(a+b)/2)-(b-a)/2)/2 = -((b-x)/2) := by ring rw [h₁,h₂,Real.sin_neg] ring lemma separating_cosine_pos {α a b x : ℝ} (ha : α < a) (hb : b < α+2*Real.pi) (hx : x ∈ Ioo a b) : 0 < Real.cos (x-(a+b)/2)-Real.cos ((b-a)/2) := by rw [separating_cosine] have h₁ : 0 < Real.sin ((x-a)/2) := Real.sin_pos_of_pos_of_lt_pi (by linarith [hx.1]) (by linarith [hx.2]) have h₂ : 0 < Real.sin ((b-x)/2) := Real.sin_pos_of_pos_of_lt_pi (by linarith [hx.2]) (by linarith [hx.1]) positivity lemma separating_cosine_nonpos {α a b x : ℝ} (ha : α < a) (hab : a < b) (hb : b < α+2*Real.pi) (hx : x ∈ Icc α (α+2*Real.pi)) (hout : x ≤ a ∨ b ≤ x) : Real.cos (x-(a+b)/2)-Real.cos ((b-a)/2) ≤ 0 := by rw [separating_cosine] rcases hout with h | h · have h₁ : Real.sin ((x-a)/2) ≤ 0 := Real.sin_nonpos_of_nonpos_of_neg_pi_le (by linarith) (by linarith [hx.1]) have h₂ : 0 ≤ Real.sin ((b-x)/2) := Real.sin_nonneg_of_nonneg_of_le_pi (by linarith) (by linarith [hx.1]) exact mul_nonpos_of_nonpos_of_nonneg (mul_nonpos_of_nonneg_of_nonpos (by norm_num) h₁) h₂ · have h₁ : 0 ≤ Real.sin ((x-a)/2) := Real.sin_nonneg_of_nonneg_of_le_pi (by linarith) (by linarith [hx.2]) have h₂ : Real.sin ((b-x)/2) ≤ 0 := Real.sin_nonpos_of_nonpos_of_neg_pi_le (by linarith) (by linarith [hx.2]) exact mul_nonpos_of_nonneg_of_nonpos (mul_nonneg (by norm_num) h₁) h₂ lemma positive_points_sandwich {h : ℝ → ℝ} (hc : Continuous h) {α x : ℝ} (hα : h α < 0) (hβ : h (α+2*Real.pi) < 0) (hx : x ∈ Icc α (α+2*Real.pi)) (hh : 0 < h x) : ∃ y z, y ∈ Icc α (α+2*Real.pi) ∧ z ∈ Icc α (α+2*Real.pi) ∧ 0 < h y ∧ 0 < h z ∧ y < x ∧ x < z := by have hαx : α < x := lt_of_le_of_ne hx.1 (by rintro rfl; linarith) have hxβ : x < α+2*Real.pi := lt_of_le_of_ne hx.2 (by intro he; rw [he] at hh; linarith) have ho : IsOpen (Ioo α (α+2*Real.pi) ∩ h ⁻¹' Ioi 0) := isOpen_Ioo.inter (isOpen_Ioi.preimage hc) obtain ⟨ε,hε,hball⟩ := Metric.isOpen_iff.mp ho x ⟨⟨hαx,hxβ⟩,hh⟩ have hy : x-ε/2 ∈ Ioo α (α+2*Real.pi) ∩ h ⁻¹' Ioi 0 := by apply hball rw [Metric.mem_ball,Real.dist_eq] have : x-ε/2-x = -(ε/2) := by ring rw [this,abs_neg,abs_of_pos (by positivity)] linarith have hz : x+ε/2 ∈ Ioo α (α+2*Real.pi) ∩ h ⁻¹' Ioi 0 := by apply hball rw [Metric.mem_ball,Real.dist_eq] have : x+ε/2-x = ε/2 := by ring rw [this,abs_of_pos (by positivity)] linarith exact ⟨_,_,⟨hy.1.1.le,hy.1.2.le⟩,⟨hz.1.1.le,hz.1.2.le⟩,hy.2,hz.2,by linarith,by linarith⟩ theorem moment_sign_changes {h w : ℝ → ℝ} (hc : Continuous h) (wc : Continuous w) (α : ℝ) (wp : ∀ x ∈ Icc α (α+2*Real.pi), 0 < w x) (hα : h α < 0) (hβ : h (α+2*Real.pi) < 0) (hp : ∃ x ∈ Icc α (α+2*Real.pi), 0 < h x) (h0 : (∫ x in α..α+2*Real.pi, w x*h x) = 0) (hcos : (∫ x in α..α+2*Real.pi, w x*h x*Real.cos x) = 0) (hsin : (∫ x in α..α+2*Real.pi, w x*h x*Real.sin x) = 0) : ∃ x y z, x ∈ Ioo α (α+2*Real.pi) ∧ z ∈ Ioo α (α+2*Real.pi) ∧ x < y ∧ y < z ∧ 0 < h x ∧ h y < 0 ∧ 0 < h z := by by_contra! hn let P : Set ℝ := Icc α (α+2*Real.pi) ∩ h ⁻¹' Ioi 0 have hP : P.Nonempty := hp have hb : BddBelow P := ⟨α,fun x hx => hx.1.1⟩ have ht : BddAbove P := ⟨α+2*Real.pi,fun x hx => hx.1.2⟩ let a := sInf P let b := sSup P have hcl : closure P ⊆ Icc α (α+2*Real.pi) ∩ h ⁻¹' Ici 0 := by apply closure_minimal · intro x hx refine ⟨hx.1,?_⟩ change 0 ≤ h x exact le_of_lt hx.2 · exact isClosed_Icc.inter (isClosed_Ici.preimage hc) have ham : a ∈ Icc α (α+2*Real.pi) ∩ h ⁻¹' Ici 0 := hcl (csInf_mem_closure hP hb) have hbm : b ∈ Icc α (α+2*Real.pi) ∩ h ⁻¹' Ici 0 := hcl (csSup_mem_closure hP ht) have ha : α < a := lt_of_le_of_ne ham.1.1 (by intro he; rw [← he] at ham; exact (not_lt_of_ge ham.2) hα) have hβb : b < α+2*Real.pi := lt_of_le_of_ne hbm.1.2 (by intro he; rw [he] at hbm; exact (not_lt_of_ge hbm.2) hβ) have hstrict (x : ℝ) (hx : x ∈ P) : a < x ∧ x < b := by obtain ⟨y,z,hy,hz,hpy,hpz,hyx,hxz⟩ := positive_points_sandwich hc hα hβ hx.1 hx.2 exact ⟨lt_of_le_of_lt (csInf_le hb ⟨hy,hpy⟩) hyx, lt_of_lt_of_le hxz (le_csSup ht ⟨hz,hpz⟩)⟩ let p := hP.some have hp : p ∈ P := hP.some_mem have hsp := hstrict p hp have hab : a < b := hsp.1.trans hsp.2 have hmid (x : ℝ) (hx : x ∈ Ioo a b) : 0 ≤ h x := by by_contra! hh obtain ⟨y,hy,hyx⟩ := exists_lt_of_csInf_lt hP hx.1 obtain ⟨z,hz,hxz⟩ := exists_lt_of_lt_csSup hP hx.2 have hsy := hstrict y hy have hsz := hstrict z hz exact (not_lt_of_ge (hn y x z ⟨ha.trans hsy.1,hsy.2.trans hβb⟩ ⟨ha.trans hsz.1,hsz.2.trans hβb⟩ hyx hxz hy.2 hh)) hz.2 have hout (x : ℝ) (hx : x ∈ Icc α (α+2*Real.pi)) (hh : x ≤ a ∨ b ≤ x) : h x ≤ 0 := by by_contra! hp have hs := hstrict x ⟨hx,hp⟩ rcases hh with h | h <;> linarith [hs.1,hs.2] let F : ℝ → ℝ := fun x => Real.cos (x-(a+b)/2)-Real.cos ((b-a)/2) have hF : Continuous F := by dsimp [F]; fun_prop have hnon (x : ℝ) (hx : x ∈ Icc α (α+2*Real.pi)) : 0 ≤ w x*h x*F x := by by_cases hi : x ∈ Ioo a b · exact mul_nonneg (mul_nonneg (wp x hx).le (hmid x hi)) (separating_cosine_pos ha hβb hi).le · have he : x ≤ a ∨ b ≤ x := by simpa only [mem_Ioo,not_and_or,not_lt] using hi exact mul_nonneg_of_nonpos_of_nonpos (mul_nonpos_of_nonneg_of_nonpos (wp x hx).le (hout x hx he)) (separating_cosine_nonpos ha hab hβb hx he) have hpos : 0 < ∫ x in α..α+2*Real.pi, w x*h x*F x := by apply intervalIntegral.integral_pos (by linarith [Real.pi_pos]) ((wc.mul hc).mul hF).continuousOn · intro x hx exact hnon x ⟨hx.1.le,hx.2⟩ · exact ⟨p,hp.1,mul_pos (mul_pos (wp p hp.1) hp.2) (separating_cosine_pos ha hβb hsp)⟩ have heq (x : ℝ) : w x*h x*F x = Real.cos ((a+b)/2)*(w x*h x*Real.cos x) + Real.sin ((a+b)/2)*(w x*h x*Real.sin x) - Real.cos ((b-a)/2)*(w x*h x) := by dsimp [F] rw [Real.cos_sub] ring have he : (∫ x in α..α+2*Real.pi, w x*h x*F x) = 0 := by simp_rw [heq] rw [intervalIntegral.integral_sub,intervalIntegral.integral_add, intervalIntegral.integral_const_mul,intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul,hcos,hsin,h0] · ring all_goals apply Continuous.intervalIntegrable; fun_prop linarith end StrictHotSpots.CircleSigns end CircleSignMomentsLayer section ChordAnglesLayer open _root_.Complex _root_.OAI.Complex Set Metric open scoped ComplexConjugate namespace StrictHotSpots.PlanarTopology lemma sine_triple_identity (a b c : ℝ) : Real.sin (c-a) - Real.sin (c-b) + Real.sin (a-b) = 4*Real.sin ((c-b)/2)*Real.sin ((a-b)/2)*Real.sin ((c-a)/2) := by have h₂ : Real.sin (c-a) = 2*Real.sin ((c-a)/2)*Real.cos ((c-a)/2) := by simpa only [show 2*((c-a)/2) = c-a by ring] using Real.sin_two_mul ((c-a)/2) have h₃ := Real.sin_sub_sin (c-b) (a-b) have he : ((c-b)-(a-b))/2 = (c-a)/2 := by ring rw [he] at h₃ have h₄ := Real.cos_sub_cos ((c-a)/2) (((c-b)+(a-b))/2) have he₁ : ((c-a)/2 + ((c-b)+(a-b))/2)/2 = (c-b)/2 := by ring have he₂ : ((c-a)/2 - ((c-b)+(a-b))/2)/2 = -((a-b)/2) := by ring rw [he₁,he₂,Real.sin_neg] at h₄ calc _ = 2*Real.sin ((c-a)/2)*(Real.cos ((c-a)/2)-Real.cos (((c-b)+(a-b))/2)) := by linarith [h₃] _ = _ := by rw [h₄]; ring lemma chord_exp_det (a b c : ℝ) : (((Circle.exp c : ℂ)-(Circle.exp b : ℂ))* conj ((Circle.exp a : ℂ)-(Circle.exp b : ℂ))).im = 4*Real.sin ((c-b)/2)*Real.sin ((a-b)/2)*Real.sin ((c-a)/2) := by rw [← sine_triple_identity] simp only [Circle.coe_exp,exp_ofReal_mul_I_re,exp_ofReal_mul_I_im, mul_im,sub_re,sub_im,conj_re,conj_im,Real.sin_sub] ring lemma angles_no_alternation {u : ℂ → ℝ} (hu : ContinuousOn u (closedBall 0 1)) (hp : IsConnected (ball (0 : ℂ) 1 ∩ {z | 0 < u z})) (hn : IsConnected (ball (0 : ℂ) 1 ∩ {z | u z < 0})) {α x y z : ℝ} (hαx : α < x) (hxy : x < y) (hyz : y < z) (hzβ : z < α+2*Real.pi) (hα : u (Circle.exp α) < 0) (hx : 0 < u (Circle.exp x)) (hy : u (Circle.exp y) < 0) (hz : 0 < u (Circle.exp z)) : False := by have hzx : (Circle.exp z : ℂ) ≠ (Circle.exp x : ℂ) := by intro he have hc : Circle.exp z = Circle.exp x := Subtype.ext he have := Circle.exp_injOn_Ico (a := α) (b := α+2*Real.pi) (by linarith) ⟨by linarith,hzβ⟩ ⟨hαx.le,by linarith⟩ hc linarith have hup : 0 < (((Circle.exp x : ℂ)-(Circle.exp y : ℂ))* conj ((Circle.exp z : ℂ)-(Circle.exp y : ℂ))).im := by rw [chord_exp_det] have hs₁ : Real.sin ((x-y)/2) < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) (by linarith) have hs₂ : 0 < Real.sin ((z-y)/2) := Real.sin_pos_of_pos_of_lt_pi (by linarith) (by linarith) have hs₃ : Real.sin ((x-z)/2) < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) (by linarith) exact mul_pos_of_neg_of_neg (mul_neg_of_neg_of_pos (mul_neg_of_pos_of_neg (by norm_num) hs₁) hs₂) hs₃ have hdn : (((Circle.exp x : ℂ)-(Circle.exp α : ℂ))* conj ((Circle.exp z : ℂ)-(Circle.exp α : ℂ))).im < 0 := by rw [chord_exp_det] have hs₁ : 0 < Real.sin ((x-α)/2) := Real.sin_pos_of_pos_of_lt_pi (by linarith) (by linarith) have hs₂ : 0 < Real.sin ((z-α)/2) := Real.sin_pos_of_pos_of_lt_pi (by linarith) (by linarith) have hs₃ : Real.sin ((x-z)/2) < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) (by linarith) exact mul_neg_of_pos_of_neg (mul_pos (mul_pos (by norm_num) hs₁) hs₂) hs₃ exact no_alternating_boundary_signs hu hp hn (Circle.exp z).norm_coe (Circle.exp y).norm_coe (Circle.exp x).norm_coe (Circle.exp α).norm_coe hzx hup hdn hz hy hx hα end StrictHotSpots.PlanarTopology end ChordAnglesLayer section DiskBoundaryMomentsLayer open Set MeasureTheory Metric namespace StrictHotSpots.CircleSigns lemma periodic_weighted_signs {h w : ℝ → ℝ} (hc : Continuous h) (wc : Continuous w) (hp : Function.Periodic h (2*Real.pi)) (wp : Function.Periodic w (2*Real.pi)) (hw : ∀ x, 0 < w x) (hne : ∃ x, h x ≠ 0) (hi : (∫ x in (0:ℝ)..2*Real.pi, w x*h x) = 0) : (∃ x, h x < 0) ∧ (∃ x, 0 < h x) := by have hneg {f : ℝ → ℝ} (fc : Continuous f) (fp : Function.Periodic f (2*Real.pi)) (fn : ∃ x, f x ≠ 0) (fi : (∫ x in (0:ℝ)..2*Real.pi, w x*f x) = 0) : ∃ x, f x < 0 := by by_contra! hh obtain ⟨x,hx⟩ := fn have hxx : 0 < f x := lt_of_le_of_ne (hh x) hx.symm have hi' : 0 < ∫ y in x..x+2*Real.pi, w y*f y := by apply intervalIntegral.integral_pos (by linarith [Real.pi_pos]) (wc.mul fc).continuousOn · intro y _ exact mul_nonneg (hw y).le (hh y) · exact ⟨x,⟨le_rfl,by linarith [Real.pi_pos]⟩,mul_pos (hw x) hxx⟩ have he := (wp.mul fp).intervalIntegral_add_eq x 0 simp only [zero_add,Pi.mul_apply] at he rw [fi] at he linarith constructor · exact hneg hc hp hne hi · obtain ⟨x,hx⟩ := hneg hc.neg (fun x => by simp only [Pi.neg_apply,hp x]) (by simpa using hne) (by simpa only [Pi.neg_apply,mul_neg,intervalIntegral.integral_neg,neg_zero] using congrArg Neg.neg hi) exact ⟨x,neg_lt_zero.mp hx⟩ lemma circle_trace_continuous {u : ℂ → ℝ} (hu : ContinuousOn u (closedBall 0 1)) : Continuous (fun θ : ℝ => u (Circle.exp θ : ℂ)) := by apply hu.comp_continuous · fun_prop · intro θ simp only [mem_closedBall,dist_zero_right,(Circle.exp θ).norm_coe,le_refl] theorem disk_moments_nonzero {u : ℂ → ℝ} (hu : ContinuousOn u (closedBall 0 1)) (hpos : IsConnected (ball (0 : ℂ) 1 ∩ {z | 0 < u z})) (hneg : IsConnected (ball (0 : ℂ) 1 ∩ {z | u z < 0})) {w : ℝ → ℝ} (wc : Continuous w) (wp : Function.Periodic w (2*Real.pi)) (hw : ∀ θ, 0 < w θ) (hne : ∃ θ, u (Circle.exp θ : ℂ) ≠ 0) (hi : (∫ θ in (0:ℝ)..2*Real.pi, w θ*u (Circle.exp θ : ℂ)) = 0) : (∫ θ in (0:ℝ)..2*Real.pi, w θ*u (Circle.exp θ : ℂ)*Real.cos θ) ≠ 0 ∨ (∫ θ in (0:ℝ)..2*Real.pi, w θ*u (Circle.exp θ : ℂ)*Real.sin θ) ≠ 0 := by by_contra! hm let h : ℝ → ℝ := fun θ => u (Circle.exp θ : ℂ) have hc : Continuous h := circle_trace_continuous hu have hp : Function.Periodic h (2*Real.pi) := by intro θ exact congrArg (fun z : Circle => u (z : ℂ)) (Circle.exp_add_two_pi θ) obtain ⟨⟨α,hα⟩,⟨p,hp'⟩⟩ := periodic_weighted_signs hc wc hp wp hw hne hi have hα' : h (α+2*Real.pi) < 0 := by rw [hp]; exact hα have hposI : ∃ x ∈ Icc α (α+2*Real.pi), 0 < h x := by have hr := hp.image_Icc Real.two_pi_pos α have hx : h p ∈ Set.range h := ⟨p,rfl⟩ rw [← hr] at hx obtain ⟨x,hx,hex⟩ := hx exact ⟨x,hx,hex ▸ hp'⟩ have hi' : (∫ x in α..α+2*Real.pi, w x*h x) = 0 := by have he := (wp.mul hp).intervalIntegral_add_eq α 0 simp only [zero_add,Pi.mul_apply] at he exact he.trans hi have hcos : (∫ x in α..α+2*Real.pi, w x*h x*Real.cos x) = 0 := by have he := ((wp.mul hp).mul Real.cos_periodic).intervalIntegral_add_eq α 0 simp only [zero_add,Pi.mul_apply] at he exact he.trans hm.1 have hsin : (∫ x in α..α+2*Real.pi, w x*h x*Real.sin x) = 0 := by have he := ((wp.mul hp).mul Real.sin_periodic).intervalIntegral_add_eq α 0 simp only [zero_add,Pi.mul_apply] at he exact he.trans hm.2 obtain ⟨x,y,z,hx,hz,hxy,hyz,hpx,hny,hpz⟩ := moment_sign_changes hc wc α (fun x _ => hw x) hα hα' hposI hi' hcos hsin exact PlanarTopology.angles_no_alternation hu hpos hneg hx.1 hxy hyz hz.2 hα hpx hny hpz end StrictHotSpots.CircleSigns end DiskBoundaryMomentsLayer section DiskMultiplicityLayer open Set MeasureTheory Metric Function Module open scoped Topology InnerProductSpace namespace StrictHotSpots.CircleSigns variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] def weightedMoment (F : ℂ → E) (w q : ℝ → ℝ) (hc : Continuous (fun θ => w θ*q θ)) (hF : ContinuousOn F (closedBall 0 1)) : E →ₗ[ℝ] ℝ where toFun a := ∫ θ in (0:ℝ)..2*Real.pi, w θ*q θ*inner ℝ a (F (Circle.exp θ : ℂ)) map_add' a b := by simp only [inner_add_left,mul_add] have hcont : Continuous (fun θ : ℝ => F (Circle.exp θ : ℂ)) := hF.comp_continuous (by fun_prop) (fun θ => by simp [mem_closedBall,dist_zero_right]) exact intervalIntegral.integral_add ((hc.mul (continuous_const.inner hcont)).intervalIntegrable _ _) ((hc.mul (continuous_const.inner hcont)).intervalIntegrable _ _) map_smul' c a := by simp only [inner_smul_left,conj_trivial,RingHom.id_apply,smul_eq_mul] simp_rw [show ∀ θ, w θ * q θ * (c * inner ℝ a (F (Circle.exp θ : ℂ))) = c * (w θ*q θ*inner ℝ a (F (Circle.exp θ : ℂ))) from fun _ => by ring] exact intervalIntegral.integral_const_mul _ _ theorem disk_family_finrank_le_two (F : ℂ → E) (hc : ContinuousOn F (closedBall 0 1)) (hb : ∀ a : E, a ≠ 0 → ∃ s : Circle, 0 < inner ℝ a (F s)) (hp : ∀ a : E, a ≠ 0 → IsConnected (ball (0:ℂ) 1 ∩ {z | 0 < inner ℝ a (F z)})) (hn : ∀ a : E, a ≠ 0 → IsConnected (ball (0:ℂ) 1 ∩ {z | inner ℝ a (F z) < 0})) : finrank ℝ E ≤ 2 := by let G : C(Circle,E) := ⟨fun s => F s, hc.comp_continuous (by fun_prop) (fun s => by simpa only [mem_closedBall,dist_zero_right,Circle.norm_coe] using (le_refl (1:ℝ)))⟩ obtain ⟨w,wc,wp,hw,hzero⟩ := PositiveWeight.exists_circle_weight G hb let T : E →ₗ[ℝ] ℝ×ℝ := (weightedMoment F w Real.cos (wc.mul Real.continuous_cos) hc).prod (weightedMoment F w Real.sin (wc.mul Real.continuous_sin) hc) have hker (a : E) (ha : T a = 0) : a = 0 := by by_contra haz obtain ⟨s,hs⟩ := hb a haz have hne : ∃ θ, inner ℝ a (F (Circle.exp θ : ℂ)) ≠ 0 := by exact ⟨(s:ℂ).arg,by simpa only [Circle.exp_arg] using ne_of_gt hs⟩ have hcont : ContinuousOn (fun z => inner ℝ a (F z)) (closedBall 0 1) := continuousOn_const.inner hc have hh := disk_moments_nonzero hcont (hp a haz) (hn a haz) wc wp hw hne (hzero a) have hreal := congrArg Prod.fst ha have himag := congrArg Prod.snd ha change (∫ θ in (0:ℝ)..2*Real.pi, w θ*Real.cos θ*inner ℝ a (F (Circle.exp θ : ℂ))) = 0 at hreal change (∫ θ in (0:ℝ)..2*Real.pi, w θ*Real.sin θ*inner ℝ a (F (Circle.exp θ : ℂ))) = 0 at himag have hre : (∫ θ in (0:ℝ)..2*Real.pi, w θ*inner ℝ a (F (Circle.exp θ : ℂ))*Real.cos θ) = 0 := by simpa only [mul_right_comm (w _) (Real.cos _) _] using hreal have him : (∫ θ in (0:ℝ)..2*Real.pi, w θ*inner ℝ a (F (Circle.exp θ : ℂ))*Real.sin θ) = 0 := by simpa only [mul_right_comm (w _) (Real.sin _) _] using himag exact hh.elim (fun h => h hre) (fun h => h him) have hi : Injective T := by intro a b he apply sub_eq_zero.mp apply hker rw [map_sub,he,sub_self] simpa only [finrank_prod,finrank_self] using T.finrank_le_finrank_of_injective hi end StrictHotSpots.CircleSigns end DiskMultiplicityLayer end end MultiplicityComplete section ClosedEigenMultiplicity noncomputable section open Set Metric Function Module open scoped Topology InnerProductSpace namespace StrictHotSpots open PlaneGreen DiskH10 def closedScalarEval {Ω : Set Plane} (ho : IsOpen Ω) (hb : Bornology.IsBounded Ω) (x : closure Ω) : closedEigenSpace Ω ho hb →ₗ[ℝ] ℝ where toFun J := J.val.1 x map_add' _ _ := rfl map_smul' _ _ := rfl theorem closedEigenSpace_family_card {Ω : Set Plane} (c : Conformal.ClosedDiskChart Ω) (hΩ : AdmissibleDomain Ω) {n : ℕ} (f : Fin n → closedEigenSpace Ω hΩ.2.1 hΩ.2.2.1) (hf : LinearIndependent ℝ f) : n ≤ 2 := by classical choose u Y hu huc hY he hjet using fun i => (f i).property let J (a : EuclideanSpace ℝ (Fin n)) : closedEigenSpace Ω hΩ.2.1 hΩ.2.2.1 := ∑ i, a i • f i have hjnz (a : EuclideanSpace ℝ (Fin n)) (ha : a ≠ 0) : J a ≠ 0 := by intro hz have hh := Fintype.linearIndependent_iff.mp hf (fun i => a i) hz exact ha (by ext i; exact hh i) have hcombo (a : EuclideanSpace ℝ (Fin n)) (x : closure Ω) : (J a).val.1 x = ∑ i, a i * u i x := by change closedScalarEval hΩ.2.1 hΩ.2.2.1 x (∑ i, a i • f i) = _ rw [map_sum] apply Finset.sum_congr rfl intro i _ rw [map_smul] change a i * (f i).val.1 x = a i * u i x rw [← hjet i] rfl let F (z : ℂ) : EuclideanSpace ℝ (Fin n) := WithLp.toLp 2 (fun i => u i (c.F (complexIso z))) have hc : ContinuousOn F (closedBall 0 1) := by apply (PiLp.continuous_toLp 2 (fun _ : Fin n => ℝ)).comp_continuousOn apply continuousOn_pi.mpr intro i exact (huc i).comp (c.lipschitz.continuous.comp complexIso.continuous).continuousOn (fun _ hz => c.complex_maps_closed hz) have hval (a : EuclideanSpace ℝ (Fin n)) {v : Plane → ℝ} {Z : Plane → Plane} (hv : closedEigenJet Ω v Z = (J a).val) (z : ℂ) (hz : z ∈ closedBall 0 1) : inner ℝ a (F z) = v (c.F (complexIso z)) := by let x : closure Ω := ⟨c.F (complexIso z),c.complex_maps_closed hz⟩ calc inner ℝ a (F z) = ∑ i, a i * u i x := by simp only [F,PiLp.inner_apply,RCLike.inner_apply,conj_trivial] apply Finset.sum_congr rfl intro i _ exact mul_comm _ _ _ = (J a).val.1 x := (hcombo a x).symm _ = v (c.F (complexIso z)) := by rw [← hv]; rfl have hdim := CircleSigns.disk_family_finrank_le_two F hc ?_ ?_ ?_ · simpa only [finrank_euclideanSpace, Fintype.card_fin] using hdim · intro a ha obtain ⟨v,Z,hv,hvc,hZ,hvZ,hvjet⟩ := (J a).property have hn : ∃ x ∈ Ω, v x ≠ 0 := closedEigenJet_nonzero_interior hΩ.2.1 hvc hZ hvZ (fun hz => hjnz a ha (Subtype.ext (hvjet.symm.trans hz))) obtain ⟨x,hx,hvx⟩ := InteriorBoundarySign.exists_boundary_pos hΩ.2.1 hΩ.2.2.1 hv hvc hn let s : Circle := c.circleBoundaryHomeomorph.symm ⟨x,hx⟩ have hs : c.F (complexIso (s:ℂ)) = x := congrArg Subtype.val (c.circleBoundaryHomeomorph.apply_symm_apply ⟨x,hx⟩) refine ⟨s,?_⟩ rw [hval a hvjet s (by simp only [mem_closedBall_zero_iff,Circle.norm_coe,le_refl]),hs] exact hvx · intro a ha obtain ⟨v,Z,hv,hvc,hZ,hvZ,hvjet⟩ := (J a).property have hn : ∃ x ∈ Ω, v x ≠ 0 := closedEigenJet_nonzero_interior hΩ.2.1 hvc hZ hvZ (fun hz => hjnz a ha (Subtype.ext (hvjet.symm.trans hz))) have hcon := c.connected_complex_pull (P := Ioi 0) (InteriorNodal.positive_set_connected hΩ hv hn) convert hcon using 1 ext z constructor · rintro ⟨hz,hp⟩ refine ⟨hz,?_⟩ simpa only [mem_ofPred_eq,mem_preimage,mem_Ioi,mem_Iio,← hval a hvjet z (ball_subset_closedBall hz)] using hp · rintro ⟨hz,hp⟩ refine ⟨hz,?_⟩ change (0:ℝ) < inner ℝ a (F z) rw [hval a hvjet z (ball_subset_closedBall hz)] exact hp · intro a ha obtain ⟨v,Z,hv,hvc,hZ,hvZ,hvjet⟩ := (J a).property have hn : ∃ x ∈ Ω, v x ≠ 0 := closedEigenJet_nonzero_interior hΩ.2.1 hvc hZ hvZ (fun hz => hjnz a ha (Subtype.ext (hvjet.symm.trans hz))) have hcon := c.connected_complex_pull (P := Iio 0) (InteriorNodal.negative_set_connected hΩ hv hn) convert hcon using 1 ext z constructor · rintro ⟨hz,hp⟩ refine ⟨hz,?_⟩ simpa only [mem_ofPred_eq,mem_preimage,mem_Ioi,mem_Iio,← hval a hvjet z (ball_subset_closedBall hz)] using hp · rintro ⟨hz,hp⟩ refine ⟨hz,?_⟩ change inner ℝ a (F z) < (0:ℝ) rw [hval a hvjet z (ball_subset_closedBall hz)] exact hp theorem closedEigenSpace_finrank_le_two {Ω : Set Plane} (c : Conformal.ClosedDiskChart Ω) (hΩ : AdmissibleDomain Ω) : Module.Finite ℝ (closedEigenSpace Ω hΩ.2.1 hΩ.2.2.1) ∧ Module.finrank ℝ (closedEigenSpace Ω hΩ.2.1 hΩ.2.2.1) ≤ 2 := by have hr : Module.rank ℝ (closedEigenSpace Ω hΩ.2.1 hΩ.2.2.1) < 3 := by by_contra hn obtain ⟨f,hf⟩ := exists_linearIndependent_of_le_rank (n := 3) (le_of_not_gt hn) have hh := closedEigenSpace_family_card c hΩ f hf norm_num at hh have hfinite : Module.Finite ℝ (closedEigenSpace Ω hΩ.2.1 hΩ.2.2.1) := Module.rank_lt_aleph0_iff.mp (hr.trans (Cardinal.natCast_lt_aleph0 (n := 3))) let : Module.Finite ℝ (closedEigenSpace Ω hΩ.2.1 hΩ.2.2.1) := hfinite refine ⟨hfinite,?_⟩ have he := Module.finrank_eq_rank ℝ (closedEigenSpace Ω hΩ.2.1 hΩ.2.2.1) rw [← he] at hr exact Nat.le_of_lt_succ (by exact_mod_cast hr) end StrictHotSpots end end ClosedEigenMultiplicity section ClosedBoundaryZeroJet noncomputable section open Set Filter Metric open scoped Topology ContDiff InnerProductSpace namespace StrictHotSpots.Conformal.ClosedDiskChart open DiskH10 PlaneGreen variable {Ω : Set Plane} (c : ClosedDiskChart Ω) include c theorem hasFDerivWithinAt_zero_of_closedGradient_zero {v : Plane → ℝ} (hv : ContDiffOn ℝ ∞ v Ω) (hvc : ContinuousOn v (closure Ω)) {Y : Plane → Plane} (hY : ContinuousOn Y (closure Ω)) (he : EqOn (gradient v) Y Ω) {p : Plane} (hp : p ∈ closure Ω) (hz : Y p = 0) : HasFDerivWithinAt v (0 : Plane →L[ℝ] ℝ) (closure Ω) p := by have hGp := c.inverse_maps_closed hp have hYG : Tendsto (fun z => Y (c.F z)) (𝓝[disk] (c.G p)) (𝓝 (0 : Plane)) := by have hc := (hY.comp c.continuous_closed c.maps_closed (c.G p) hGp).mono subset_closure change Tendsto (Y ∘ c.F) (𝓝[disk] c.G p) (𝓝 (Y (c.F (c.G p)))) at hc rw [c.right_inverse p hp,hz] at hc exact hc have hlim : Tendsto (fun z => fderiv ℝ (c.pullDatum v) z) (𝓝[disk] (c.G p)) (𝓝 (0 : Plane →L[ℝ] ℝ)) := by apply tendsto_zero_iff_norm_tendsto_zero.mpr apply squeeze_zero' (g := fun z => c.bound * ‖Y (c.F z)‖) (Eventually.of_forall fun _ => norm_nonneg _) · filter_upwards [self_mem_nhdsWithin] with z hzD have hn : ‖fderiv ℝ (c.pullDatum v) z‖ = ‖gradient (c.pullDatum v) z‖ := by simp only [gradient,LinearIsometryEquiv.norm_map] rw [hn,c.pullDatum_grad_norm (hv.differentiableOn (by simp)) hzD, he (c.diskMap.map_source hzD),c.diskMap_agrees hzD] exact mul_le_mul_of_nonneg_right (c.bounded_deriv z (c.source_eq.symm ▸ hzD)) (norm_nonneg _) · simpa only [norm_zero,mul_zero] using tendsto_const_nhds.mul hYG.norm have hD : HasFDerivWithinAt (c.pullDatum v) (0 : Plane →L[ℝ] ℝ) (closure disk) (c.G p) := hasFDerivWithinAt_closure_of_tendsto_fderiv ((c.pullDatum_smooth hv).differentiableOn (by simp)) (convex_ball (0:Plane) 1) diskOpen (fun z hzD => (c.pullDatum_continuous hvc z hzD).mono subset_closure) hlim have ht : Tendsto c.G (𝓝[closure Ω] p) (𝓝[closure disk] (c.G p)) := c.inverse_lipschitz.continuous.continuousWithinAt.tendsto_nhdsWithin c.inverse_maps_closed have hsmall := (hasFDerivWithinAt_iff_isLittleO.mp hD).comp_tendsto ht have hbig : (fun x => c.G x-c.G p) =O[𝓝[closure Ω] p] (fun x => x-p) := Asymptotics.IsBigO.of_bound c.InvLip (Eventually.of_forall (fun x => c.inverse_lipschitz.norm_sub_le x p)) have hh := hsmall.trans_isBigO hbig apply hasFDerivWithinAt_iff_isLittleO.mpr apply hh.congr' _ (Filter.EventuallyEq.refl _ _) filter_upwards [self_mem_nhdsWithin] with x hx simp only [Function.comp_apply,zero_apply,sub_zero,pullDatum, c.right_inverse x hx,c.right_inverse p hp] end StrictHotSpots.Conformal.ClosedDiskChart end end ClosedBoundaryZeroJet end OAI