| import Mathlib |
| import OAI.Geometry.CubicTorus.Curvature.LinearCMCEquation |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem LinearCMCEquation.local_gradient_energy {g : Plane → ℝ} {K : ℝ≥0} |
| (hg : LipschitzWith K g) {p B : Plane → Plane} (hp : Continuous p) (hpb : ∀ y,‖p y‖≤1) |
| {LA LB : ℝ≥0} (hA : LipschitzWith LA (fun y ↦ cmcJacobian (p y))) (hB : LipschitzWith LB B) |
| {M : ℝ} (hBb : ∀ y,‖B y‖≤M) {R r s : ℝ} |
| (hgc : ContDiffOn ℝ 1 g (Metric.ball 0 R)) (hs : 0<s) (hsr : s<r) |
| (hw : LinearCMCEquation g p B (Metric.ball 0 R)) : |
| ∃ C : ℝ,0≤C ∧ ∀ a : Plane,r+‖a‖≤R → |
| (∫ y in Metric.ball 0 s,‖gradient g (y+a)-gradient g y‖^2)≤C*‖a‖^2 := by |
| let b : ContDiffBump (0 : Plane) := ⟨s,(s+r)/2,hs,by linarith⟩ |
| have hbs : tsupport b⊆Metric.ball 0 r := by |
| rw [b.tsupport_eq] |
| exact Metric.closedBall_subset_ball (by dsimp [b]; linarith) |
| have hi := compact_derivative_square_integrable (b.contDiff (n := 1)) b.hasCompactSupport |
| have hb2 : Integrable (fun y ↦ b y^2) := by |
| have hbc : Continuous (fun y : Plane ↦ b y * b y) := b.continuous.mul b.continuous |
| have hbsq : HasCompactSupport (fun y : Plane ↦ b y * b y) := b.hasCompactSupport.mul_right |
| have hi : Integrable (fun y : Plane ↦ b y * b y) (volume : Measure Plane) := hbc.integrable_of_hasCompactSupport hbsq |
| exact hi.congr (Eventually.of_forall fun y ↦ by ring) |
| let D : ℝ := (LB:ℝ)+(LA:ℝ)*(K:ℝ) |
| let C : ℝ := 4096*(K:ℝ)^2*(∫ y,‖fderiv ℝ b y‖^2)+256*D^2*(∫ y,b y^2) |
| have hC : 0≤C := by |
| dsimp only [C] |
| exact add_nonneg (mul_nonneg (by positivity) (integral_nonneg fun _ ↦ sq_nonneg _)) |
| (mul_nonneg (by positivity) (integral_nonneg fun _ ↦ sq_nonneg _)) |
| refine ⟨C,hC,fun a ha ↦ ?_⟩ |
| obtain ⟨hAi,hBi⟩ := cmc_cutoff_energies_integrable hg (b.contDiff (n := 1)) b.hasCompactSupport a |
| have hbound : (∫ y,(g (y+a)-g y)^2*‖fderiv ℝ b y‖^2)≤ |
| ((K:ℝ)*‖a‖)^2*(∫ y,‖fderiv ℝ b y‖^2) := by |
| rw [← integral_const_mul] |
| apply integral_mono hBi (hi.const_mul _) |
| intro y |
| have hh : (g (y+a)-g y)^2≤((K:ℝ)*‖a‖)^2 := by |
| simpa only [Real.norm_eq_abs,sq_abs] using |
| pow_le_pow_left₀ (norm_nonneg _) (height_difference_norm_le hg y a) 2 |
| nlinarith [mul_le_mul_of_nonneg_right hh (sq_nonneg ‖fderiv ℝ b y‖)] |
| have hsource : (∫ y,b y^2*‖B (y+a)-B y- |
| (cmcJacobian (p (y+a))-cmcJacobian (p y)) (gradient g y)‖^2)≤ |
| (D*‖a‖)^2*(∫ y,b y^2) := by |
| rw [← integral_const_mul] |
| apply integral_mono_of_nonneg (Eventually.of_forall fun _ ↦ by positivity) (hb2.const_mul _) |
| exact Eventually.of_forall fun y ↦ by |
| have hh := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (norm_nonneg _) |
| (linear_cmc_difference_source_bound hg hA hB y a) 2) (sq_nonneg (b y)) |
| dsimp only [D] |
| nlinarith |
| calc |
| (∫ y in Metric.ball 0 s,‖gradient g (y+a)-gradient g y‖^2) = |
| ∫ y in Metric.ball 0 s,b y^2*‖gradient g (y+a)-gradient g y‖^2 := by |
| apply setIntegral_congr_fun measurableSet_ball |
| intro y hy |
| dsimp only |
| rw [b.one_of_mem_closedBall (Metric.ball_subset_closedBall hy)] |
| simp |
| _ ≤ ∫ y,b y^2*‖gradient g (y+a)-gradient g y‖^2 := |
| setIntegral_le_integral hAi (ae_of_all _ fun _ ↦ by positivity) |
| _ ≤ 4096*(∫ y,(g (y+a)-g y)^2*‖fderiv ℝ b y‖^2)+256*(∫ y,b y^2*‖B (y+a)-B y- |
| (cmcJacobian (p (y+a))-cmcJacobian (p y)) (gradient g y)‖^2) := |
| hw.caccioppoli hg hp hpb hB.continuous hBb hgc (by linarith) ha b.contDiff b.hasCompactSupport hbs |
| _ ≤ 4096*(((K:ℝ)*‖a‖)^2*(∫ y,‖fderiv ℝ b y‖^2))+256*((D*‖a‖)^2*(∫ y,b y^2)) := by linarith |
| _ = C*‖a‖^2 := by dsimp only [C]; ring |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Filter Set |
| open scoped Topology RealInnerProductSpace |
| namespace CubicTorus |
| universe uHilbert |
|
|
| theorem hilbert_bounded_weak_subsequence {E : Type uHilbert} |
| [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] |
| [TopologicalSpace.SeparableSpace E] (f : ℕ → E) {M : ℝ} |
| (hb : ∀ i, ‖f i‖≤M) : |
| ∃ (u : E) (φ : ℕ → ℕ), ‖u‖≤M ∧ StrictMono φ ∧ |
| ∀ v : E, Tendsto (fun i ↦ ⟪f (φ i),v⟫) atTop (𝓝 ⟪u,v⟫) := by |
| let F : ℕ → WeakDual ℝ E := fun i ↦ StrongDual.toWeakDual ((InnerProductSpace.toDual ℝ E) (f i)) |
| have hm (i : ℕ) : F i∈WeakDual.toStrongDual ⁻¹' Metric.closedBall (0 : StrongDual ℝ E) M := by |
| simp only [mem_preimage,Metric.mem_closedBall,dist_zero_right] |
| change ‖(InnerProductSpace.toDual ℝ E) (f i)‖≤M |
| rw [(InnerProductSpace.toDual ℝ E).norm_map] |
| exact hb i |
| obtain ⟨L,hL,φ,hφ,hlim⟩ := (WeakDual.isSeqCompact_closedBall ℝ E (0 : StrongDual ℝ E) M) hm |
| let u : E := (InnerProductSpace.toDual ℝ E).symm (WeakDual.toStrongDual L) |
| refine ⟨u,φ,?_,hφ,fun v ↦ ?_⟩ |
| · change ‖(InnerProductSpace.toDual ℝ E).symm _‖≤M |
| rw [(InnerProductSpace.toDual ℝ E).symm.norm_map] |
| simpa only [mem_preimage,Metric.mem_closedBall,dist_zero_right] using hL |
| · have hh := ((WeakDual.eval_continuous v).tendsto L).comp hlim |
| simpa only [F,Function.comp_def,StrongDual.toWeakDual_apply,InnerProductSpace.toDual_apply_apply, |
| u,InnerProductSpace.toDual_symm_apply,WeakDual.toStrongDual_apply] using hh |
| |
| end CubicTorus |
| |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
| |
| lemma continuous_difference_quotient_pairing {f ψ : Plane → ℝ} |
| (hf : Continuous f) (hψ : Continuous ψ) (hc : HasCompactSupport ψ) |
| (t : ℝ) (v : Plane) : |
| (∫ y, (t⁻¹*(f (y+t • v)-f y))*ψ y)= |
| ∫ y, (t⁻¹*(ψ (y+t • (-v))-ψ y))*f y := by |
| have h₁ : Integrable (fun y ↦ f (y+t • v)*ψ y) := |
| ((hf.comp (continuous_id.add continuous_const)).mul hψ).integrable_of_hasCompactSupport hc.mul_left |
| have h₂ : Integrable (fun y ↦ f y*ψ y) := |
| (hf.mul hψ).integrable_of_hasCompactSupport hc.mul_left |
| have h₃ : Integrable (fun y ↦ f y*ψ (y+t • (-v))) := |
| (hf.mul (hψ.comp (continuous_id.add continuous_const))).integrable_of_hasCompactSupport |
| (hc.comp_homeomorph (Homeomorph.addRight (t • (-v)))).mul_left |
| have he : (∫ y, f (y+t • v)*ψ y)=(∫ y, f y*ψ (y+t • (-v))) := by |
| rw [← integral_add_right_eq_self _ (t • (-v))] |
| simp |
| simp_rw [mul_assoc,← mul_comm (f _),sub_mul,mul_sub] |
| rw [integral_sub (h₁.const_mul _) (h₂.const_mul _), |
| integral_sub (h₃.const_mul _) (h₂.const_mul _),integral_const_mul,integral_const_mul, |
| integral_const_mul,he] |
| |
| theorem continuous_difference_quotient_test_limit {f ψ : Plane → ℝ} |
| (hf : Continuous f) (hψ : ContDiff ℝ 1 ψ) (hc : HasCompactSupport ψ) (v : Plane) : |
| Tendsto (fun t : ℝ ↦ ∫ y, (t⁻¹*(f (y+t • v)-f y))*ψ y) (𝓝[>] 0) |
| (𝓝 (-(∫ y, f y*fderiv ℝ ψ y v))) := by |
| obtain ⟨L,hL⟩ := hψ.lipschitzWith_of_hasCompactSupport hc (by simp) |
| have hh := LipschitzWith.integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul' |
| (μ := (volume : Measure Plane)) hL hc hf (-v) |
| have he : (∫ y, lineDeriv ℝ ψ y (-v)*f y)= -(∫ y, f y*fderiv ℝ ψ y v) := by |
| simp_rw [(hψ.differentiable (by simp)).differentiableAt.lineDeriv_eq_fderiv, |
| map_neg,neg_mul,integral_neg,mul_comm _ (f _)] |
| rw [he] at hh |
| convert hh using 1 |
| funext t |
| exact continuous_difference_quotient_pairing hf hψ.continuous hc t v |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem continuous_L2_weak_directional_derivative {f : Plane → ℝ} |
| (hf : Continuous f) {M : ℝ} (hb : ∀ y, ‖f y‖≤M) |
| {r δ C : ℝ} (hδ : 0<δ) (hC : 0≤C) (v : Plane) |
| (he : ∀ t∈Ioc 0 δ, (∫ y in Metric.ball 0 r, (f (y+t • v)-f y)^2)≤C*t^2) : |
| ∃ u : Lp ℝ 2 (volume.restrict (Metric.ball (0 : Plane) r)), ‖u‖≤Real.sqrt C ∧ |
| ∀ ψ : Plane → ℝ, ContDiff ℝ 1 ψ → HasCompactSupport ψ → tsupport ψ⊆Metric.ball 0 r → |
| (∫ y in Metric.ball 0 r, u y*ψ y)= -(∫ y, f y*fderiv ℝ ψ y v) := by |
| let μ : Measure Plane := volume.restrict (Metric.ball 0 r) |
| let : IsFiniteMeasure μ := isFiniteMeasure_restrict.mpr measure_ball_lt_top.ne |
| let : Fact ((2 : ℝ≥0∞)≠⊤) := ⟨by norm_num⟩ |
| let t : ℕ → ℝ := fun i ↦ δ*(1/((i:ℝ)+1)) |
| have ht (i : ℕ) : t i∈Ioc 0 δ := by |
| have hn : (1:ℝ)≤(i:ℝ)+1 := by linarith [Nat.cast_nonneg (α := ℝ) i] |
| constructor |
| · dsimp only [t]; positivity |
| · dsimp only [t] |
| exact mul_le_of_le_one_right hδ.le (by rw [div_le_one (by linarith : (0:ℝ)<(i:ℝ)+1)]; exact hn) |
| have htlim : Tendsto t atTop (𝓝[>] 0) := by |
| apply tendsto_nhdsWithin_iff.mpr |
| refine ⟨?_,Eventually.of_forall (fun i ↦ (ht i).1)⟩ |
| simpa only [mul_zero] using tendsto_one_div_add_atTop_nhds_zero_nat.const_mul δ |
| let q : ℕ → Plane → ℝ := fun i y ↦ (t i)⁻¹*(f (y+t i • v)-f y) |
| have hqm (i : ℕ) : MemLp (q i) 2 μ := by |
| apply MemLp.of_bound (C := |(t i)⁻¹| *(2*M)) (by fun_prop) |
| filter_upwards [] with y |
| change ‖(t i)⁻¹*(f (y+t i • v)-f y)‖≤_ |
| rw [norm_mul,Real.norm_eq_abs] |
| exact mul_le_mul_of_nonneg_left ((norm_sub_le _ _).trans (by linarith [hb (y+t i • v),hb y])) |
| (abs_nonneg _) |
| let Q : ℕ → Lp ℝ 2 μ := fun i ↦ (hqm i).toLp (q i) |
| have hQ (i : ℕ) : ‖Q i‖≤Real.sqrt C := by |
| have heq : ‖Q i‖^2=(t i)⁻¹^2*(∫ y in Metric.ball 0 r, (f (y+t i • v)-f y)^2) := by |
| rw [L2_norm_sq_integral] |
| calc |
| _ = ∫ y, (q i y)^2 ∂μ := integral_congr_ae ((hqm i).coeFn_toLp.mono (fun y hy ↦ by rw [hy])) |
| _ = _ := by simp only [q,mul_pow,integral_const_mul,μ] |
| have hh := mul_le_mul_of_nonneg_left (he (t i) (ht i)) (sq_nonneg (t i)⁻¹) |
| have htne := (ht i).1.ne' |
| have hsq := Real.sq_sqrt hC |
| have hre : (t i)⁻¹^2*(C*(t i)^2)=C := by field_simp |
| rw [hre,← heq] at hh |
| nlinarith [norm_nonneg (Q i),Real.sqrt_nonneg C] |
| obtain ⟨u,φ,hu,hφ,hlim⟩ := hilbert_bounded_weak_subsequence Q hQ |
| refine ⟨u,hu,fun ψ hψ hc hs ↦ ?_⟩ |
| have hψm : MemLp ψ 2 μ := hψ.continuous.memLp_of_hasCompactSupport hc |
| let V : Lp ℝ 2 μ := hψm.toLp ψ |
| have hpair (i : ℕ) : ⟪Q i,V⟫=(∫ y, q i y*ψ y) := by |
| rw [L2.inner_def,integral_mul_eq_restrict_of_tsupport measurableSet_ball (q i) ψ hs] |
| apply integral_congr_ae |
| filter_upwards [(hqm i).coeFn_toLp,hψm.coeFn_toLp] with y hy hψy |
| simp only [RCLike.inner_apply,RCLike.conj_to_real] |
| rw [hy,hψy] |
| ring |
| have hupair : ⟪u,V⟫=(∫ y in Metric.ball 0 r, u y*ψ y) := by |
| rw [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [hψm.coeFn_toLp] with y hy |
| simp only [RCLike.inner_apply,RCLike.conj_to_real] |
| rw [hy] |
| ring |
| have hlim₁ := (hlim V) |
| simp only [hpair,hupair] at hlim₁ |
| have hlim₂ := (continuous_difference_quotient_test_limit hf hψ hc v).comp |
| (htlim.comp hφ.tendsto_atTop) |
| exact tendsto_nhds_unique hlim₁ hlim₂ |
| |
| end CubicTorus |
| |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
| |
| lemma continuous_local_cutoff {f η : Plane → ℝ} {U : Set Plane} |
| (hU : IsOpen U) (hf : ContinuousOn f U) (hη : Continuous η) |
| (hs : tsupport η⊆U) : Continuous (fun y ↦ η y*f y) := by |
| rw [continuous_iff_continuousAt] |
| intro y |
| by_cases hy : y∈tsupport η |
| · exact hη.continuousAt.mul ((hf y (hs hy)).continuousAt (hU.mem_nhds (hs hy))) |
| · apply (continuousAt_const (y := (0:ℝ))).congr_of_eventuallyEq |
| filter_upwards [notMem_tsupport_iff_eventuallyEq.mp hy] with z hz |
| simp [hz] |
| |
| lemma C1_gradient_continuousOn {g : Plane → ℝ} {U : Set Plane} |
| (hg : ContDiffOn ℝ 1 g U) (hU : IsOpen U) : ContinuousOn (gradient g) U := |
| (InnerProductSpace.toDual ℝ Plane).symm.continuous.comp_continuousOn |
| (hg.continuousOn_fderiv_of_isOpen hU le_rfl) |
| |
| theorem WeakCMCGraph.weak_hessian {g : Plane → ℝ} {K : ℝ≥0} |
| (hg : LipschitzWith K g) (hK : (K:ℝ)≤1) {H R : ℝ} (hR : 0<R) |
| (hgc : ContDiffOn ℝ 1 g (Metric.ball 0 R)) |
| (hw : WeakCMCGraph g H (Metric.ball 0 R)) (v w : Plane) (hv : ‖v‖≤1) (hw' : ‖w‖≤1) : |
| ∃ u : Lp ℝ 2 (volume.restrict (Metric.ball (0 : Plane) (R/4))), |
| ∀ ψ : Plane → ℝ, ContDiff ℝ 1 ψ → HasCompactSupport ψ → |
| tsupport ψ⊆Metric.ball 0 (R/4) → |
| (∫ y in Metric.ball 0 (R/4), u y*ψ y)= |
| -(∫ y, ⟪gradient g y,w⟫*fderiv ℝ ψ y v) := by |
| let b : ContDiffBump (0 : Plane) := ⟨R/2,3*R/4,by linarith,by linarith⟩ |
| have hbs : tsupport b⊆Metric.ball 0 R := by |
| rw [b.tsupport_eq] |
| exact Metric.closedBall_subset_ball (by dsimp [b]; linarith) |
| let f : Plane → ℝ := fun y ↦ b y*⟪gradient g y,w⟫ |
| have hfc : Continuous f := continuous_local_cutoff Metric.isOpen_ball |
| ((C1_gradient_continuousOn hgc Metric.isOpen_ball).inner continuousOn_const) b.continuous hbs |
| have hgrad (y : Plane) : ‖gradient g y‖≤(K:ℝ) := by |
| rw [plane_gradient_norm]; exact norm_fderiv_le_of_lipschitz ℝ hg |
| have hfb (y : Plane) : ‖f y‖≤(K:ℝ) := by |
| dsimp only [f] |
| rw [norm_mul,Real.norm_eq_abs,abs_of_nonneg (b.nonneg (x := y))] |
| have hi : ‖⟪gradient g y,w⟫‖≤(K:ℝ) := (norm_inner_le_norm _ _).trans |
| (by nlinarith [hgrad y,norm_nonneg (gradient g y),norm_nonneg w,K.coe_nonneg]) |
| exact (mul_le_mul_of_nonneg_left hi (b.nonneg (x := y))).trans |
| (mul_le_of_le_one_left K.coe_nonneg (b.le_one (x := y))) |
| have hfeq {y : Plane} (hy : ‖y‖≤R/2) : f y=⟪gradient g y,w⟫ := by |
| dsimp only [f] |
| rw [b.one_of_mem_closedBall (by simpa only [Metric.mem_closedBall,dist_zero_right] using hy),one_mul] |
| obtain ⟨C,hC,henergy⟩ := hw.local_gradient_energy hg hK hgc |
| (show 0<R/4 by linarith) (show R/4<R/2 by linarith) |
| have he : ∀ t∈Ioc 0 (R/4), (∫ y in Metric.ball 0 (R/4), (f (y+t • v)-f y)^2)≤C*t^2 := by |
| intro t ht |
| have ha : ‖t • v‖≤t := by |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos ht.1] |
| exact mul_le_of_le_one_right ht.1.le hv |
| have haR : R/2+‖t • v‖≤R := by linarith [ht.2] |
| have hfc' : Continuous (fun y ↦ (f (y+t • v)-f y)^2) := |
| ((hfc.comp (continuous_id.add continuous_const)).sub hfc).pow 2 |
| have hfi : IntegrableOn (fun y ↦ (f (y+t • v)-f y)^2) (Metric.ball 0 (R/4)) := hfc'.continuousOn.integrableOn_compact (isCompact_closedBall (0 : Plane) (R/4)) |
| |>.mono_set Metric.ball_subset_closedBall |
| have hgm := measurable_plane_gradient g |
| have hgi : IntegrableOn (fun y ↦ ‖gradient g (y+t • v)-gradient g y‖^2) (Metric.ball 0 (R/4)) := by |
| apply (integrableOn_const (C := (2*(K:ℝ))^2) measure_ball_lt_top.ne).mono' |
| (((hgm.comp (measurable_id.add_const _)).sub hgm).norm.pow_const 2).aestronglyMeasurable |
| filter_upwards [] with y |
| rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)] |
| exact pow_le_pow_left₀ (norm_nonneg _) (gradient_difference_norm_le hg y (t • v)) 2 |
| calc |
| _ ≤ ∫ y in Metric.ball 0 (R/4), ‖gradient g (y+t • v)-gradient g y‖^2 := by |
| apply integral_mono_ae hfi hgi |
| filter_upwards [ae_restrict_mem measurableSet_ball] with y hy |
| have hy' : ‖y‖<R/4 := by simpa only [Metric.mem_ball,dist_zero_right] using hy |
| rw [hfeq ((norm_add_le y (t • v)).trans (by linarith [ht.2])),hfeq (by linarith),← inner_sub_left] |
| have hi := (norm_inner_le_norm (𝕜 := ℝ) (gradient g (y+t • v)-gradient g y) w).trans |
| (mul_le_of_le_one_right (norm_nonneg _) hw') |
| simpa only [Real.norm_eq_abs,sq_abs] using pow_le_pow_left₀ (norm_nonneg _) hi 2 |
| _ ≤ C*‖t • v‖^2 := henergy (t • v) haR |
| _ ≤ C*t^2 := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (norm_nonneg _) ha 2) hC |
| obtain ⟨u,_hu,hud⟩ := continuous_L2_weak_directional_derivative hfc hfb |
| (show 0<R/4 by linarith) hC v he |
| refine ⟨u,fun ψ hψ hc hs ↦ ?_⟩ |
| rw [hud ψ hψ hc hs] |
| congr 1 |
| apply integral_congr_ae |
| filter_upwards [] with y |
| by_cases hy : y∈tsupport ψ |
| · rw [hfeq (by |
| have hh : ‖y‖<R/4 := by simpa only [Metric.mem_ball,dist_zero_right] using hs hy |
| linarith)] |
| · rw [fderiv_of_notMem_tsupport ℝ hy,zero_apply,mul_zero,mul_zero] |
| |
| end CubicTorus |
| |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
| |
| def differenceStep (δ : ℝ) (i : ℕ) : ℝ := δ*(1/((i:ℝ)+1)) |
| def vectorDifference (f : Plane → Plane) (v : Plane) (t : ℝ) (y : Plane) : Plane := |
| t⁻¹ • (f (y+t • v)-f y) |
| |
| lemma differenceStep_mem {δ : ℝ} (hδ : 0<δ) (i : ℕ) : differenceStep δ i∈Ioc 0 δ := by |
| have hn : (1:ℝ)≤(i:ℝ)+1 := by linarith [Nat.cast_nonneg (α := ℝ) i] |
| constructor |
| · dsimp only [differenceStep]; positivity |
| · dsimp only [differenceStep] |
| exact mul_le_of_le_one_right hδ.le (by rw [div_le_one (by linarith : (0:ℝ)<(i:ℝ)+1)]; exact hn) |
| |
| lemma differenceStep_tendsto {δ : ℝ} (hδ : 0<δ) : Tendsto (differenceStep δ) atTop (𝓝[>] 0) := by |
| apply tendsto_nhdsWithin_iff.mpr |
| refine ⟨?_,Eventually.of_forall (fun i ↦ (differenceStep_mem hδ i).1)⟩ |
| change Tendsto (fun i : ℕ ↦ δ*(1/((i:ℝ)+1))) atTop (𝓝 0) |
| simpa only [mul_zero] using tendsto_one_div_add_atTop_nhds_zero_nat.const_mul δ |
| |
| lemma L2_vector_norm_sq_integral {X : Type*} [MeasurableSpace X] {μ : Measure X} |
| (u : Lp Plane 2 μ) : ‖u‖^2=∫ x, ‖u x‖^2 ∂μ := by |
| rw [← real_inner_self_eq_norm_sq,L2.inner_def] |
| simp only [real_inner_self_eq_norm_sq] |
| |
| lemma vectorDifference_memLp {f : Plane → Plane} (hf : Continuous f) |
| {M : ℝ} (hb : ∀ y, ‖f y‖≤M) (r t : ℝ) (v : Plane) : |
| MemLp (vectorDifference f v t) 2 (volume.restrict (Metric.ball 0 r)) := by |
| let : IsFiniteMeasure (volume.restrict (Metric.ball (0 : Plane) r)) := |
| isFiniteMeasure_restrict.mpr measure_ball_lt_top.ne |
| apply MemLp.of_bound (C := |t⁻¹| *(2*M)) (by unfold vectorDifference; fun_prop) |
| filter_upwards [] with y |
| rw [vectorDifference,norm_smul,Real.norm_eq_abs] |
| exact mul_le_mul_of_nonneg_left ((norm_sub_le _ _).trans (by linarith [hb (y+t • v),hb y])) |
| (abs_nonneg _) |
| |
| theorem vector_L2_difference_subsequence {f : Plane → Plane} |
| (hf : Continuous f) {M : ℝ} (hb : ∀ y, ‖f y‖≤M) |
| {r δ C : ℝ} (hδ : 0<δ) (hC : 0≤C) (v : Plane) |
| (he : ∀ t∈Ioc 0 δ, (∫ y in Metric.ball 0 r, ‖f (y+t • v)-f y‖^2)≤C*t^2) : |
| ∃ (u : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) r))) (φ : ℕ → ℕ), |
| ‖u‖≤Real.sqrt C ∧ StrictMono φ ∧ |
| ∀ V : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) r)), |
| Tendsto (fun i ↦ ⟪(vectorDifference_memLp hf hb r (differenceStep δ (φ i)) v).toLp |
| (vectorDifference f v (differenceStep δ (φ i))),V⟫) atTop (𝓝 ⟪u,V⟫) := by |
| let μ : Measure Plane := volume.restrict (Metric.ball 0 r) |
| let : IsFiniteMeasure μ := isFiniteMeasure_restrict.mpr measure_ball_lt_top.ne |
| let : Fact ((2 : ℝ≥0∞)≠⊤) := ⟨by norm_num⟩ |
| let Q : ℕ → Lp Plane 2 μ := fun i ↦ (vectorDifference_memLp hf hb r (differenceStep δ i) v).toLp |
| (vectorDifference f v (differenceStep δ i)) |
| have hQ (i : ℕ) : ‖Q i‖≤Real.sqrt C := by |
| have heq : ‖Q i‖^2=(differenceStep δ i)⁻¹^2* |
| (∫ y in Metric.ball 0 r, ‖f (y+differenceStep δ i • v)-f y‖^2) := by |
| rw [L2_vector_norm_sq_integral] |
| calc |
| _ = ∫ y, ‖vectorDifference f v (differenceStep δ i) y‖^2 ∂μ := integral_congr_ae |
| ((vectorDifference_memLp hf hb r (differenceStep δ i) v).coeFn_toLp.mono (fun y hy ↦ by rw [hy])) |
| _ = _ := by simp only [vectorDifference,norm_smul,Real.norm_eq_abs,mul_pow,sq_abs,integral_const_mul,μ] |
| have hh := mul_le_mul_of_nonneg_left (he _ (differenceStep_mem hδ i)) (sq_nonneg (differenceStep δ i)⁻¹) |
| have htne := (differenceStep_mem hδ i).1.ne' |
| have hsq := Real.sq_sqrt hC |
| have hre : (differenceStep δ i)⁻¹^2*(C*(differenceStep δ i)^2)=C := by field_simp |
| rw [hre,← heq] at hh |
| nlinarith [norm_nonneg (Q i),Real.sqrt_nonneg C] |
| exact hilbert_bounded_weak_subsequence Q hQ |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma continuous_local_vector_cutoff {f : Plane → Plane} {η : Plane → ℝ} {U : Set Plane} |
| (hU : IsOpen U) (hf : ContinuousOn f U) (hη : Continuous η) |
| (hs : tsupport η⊆U) : Continuous (fun y ↦ η y • f y) := by |
| rw [continuous_iff_continuousAt] |
| intro y |
| by_cases hy : y∈tsupport η |
| · exact hη.continuousAt.smul ((hf y (hs hy)).continuousAt (hU.mem_nhds (hs hy))) |
| · apply (continuousAt_const (y := (0:Plane))).congr_of_eventuallyEq |
| filter_upwards [notMem_tsupport_iff_eventuallyEq.mp hy] with z hz |
| simp [hz] |
|
|
| theorem WeakCMCGraph.gradient_difference_data {g : Plane → ℝ} {K : ℝ≥0} |
| (hg : LipschitzWith K g) (hK : (K:ℝ)≤1) {H R : ℝ} (hR : 0<R) |
| (hgc : ContDiffOn ℝ 1 g (Metric.ball 0 R)) |
| (hw : WeakCMCGraph g H (Metric.ball 0 R)) : |
| ∃ (G : Plane → Plane) (C : ℝ), Continuous G ∧ (∀ y, ‖G y‖≤1) ∧ 0≤C ∧ |
| (∀ y, ‖y‖≤R/2 → G y=gradient g y) ∧ |
| ∀ v : Plane, ‖v‖≤1 → ∀ t∈Ioc 0 (R/4), |
| (∫ y in Metric.ball 0 (R/4), ‖G (y+t • v)-G y‖^2)≤C*t^2 := by |
| let b : ContDiffBump (0 : Plane) := ⟨R/2,3*R/4,by linarith,by linarith⟩ |
| have hbs : tsupport b⊆Metric.ball 0 R := by |
| rw [b.tsupport_eq] |
| exact Metric.closedBall_subset_ball (by dsimp [b]; linarith) |
| let G : Plane → Plane := fun y ↦ b y • gradient g y |
| have hGc : Continuous G := continuous_local_vector_cutoff Metric.isOpen_ball |
| (C1_gradient_continuousOn hgc Metric.isOpen_ball) b.continuous hbs |
| have hgrad (y : Plane) : ‖gradient g y‖≤(K:ℝ) := by |
| rw [plane_gradient_norm]; exact norm_fderiv_le_of_lipschitz ℝ hg |
| have hGb (y : Plane) : ‖G y‖≤1 := by |
| dsimp only [G] |
| rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg b.nonneg] |
| exact ((mul_le_mul_of_nonneg_left (hgrad y) b.nonneg).trans |
| (mul_le_of_le_one_left K.coe_nonneg b.le_one)).trans hK |
| have hGeq {y : Plane} (hy : ‖y‖≤R/2) : G y=gradient g y := by |
| dsimp only [G] |
| rw [b.one_of_mem_closedBall (by simpa only [Metric.mem_closedBall,dist_zero_right] using hy),one_smul] |
| obtain ⟨C,hC,henergy⟩ := hw.local_gradient_energy hg hK hgc |
| (show 0<R/4 by linarith) (show R/4<R/2 by linarith) |
| refine ⟨G,C,hGc,hGb,hC,fun _ hy ↦ hGeq hy,fun v hv t ht ↦ ?_⟩ |
| have ha : ‖t • v‖≤t := by |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos ht.1] |
| exact mul_le_of_le_one_right ht.1.le hv |
| calc |
| _ = ∫ y in Metric.ball 0 (R/4), ‖gradient g (y+t • v)-gradient g y‖^2 := by |
| apply setIntegral_congr_fun measurableSet_ball |
| intro y hy |
| have hy' : ‖y‖<R/4 := by simpa only [Metric.mem_ball,dist_zero_right] using hy |
| dsimp only |
| rw [hGeq ((norm_add_le y (t • v)).trans (by linarith [ht.2])),hGeq (by linarith)] |
| _ ≤ C*‖t • v‖^2 := henergy (t • v) (by linarith [ht.2]) |
| _ ≤ C*t^2 := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (norm_nonneg _) ha 2) hC |
| |
| end CubicTorus |
| |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
| |
| theorem LinearCMCEquation.gradient_difference_data {g : Plane → ℝ} {K : ℝ≥0} |
| (hg : LipschitzWith K g) {p B : Plane → Plane} (hp : Continuous p) (hpb : ∀ y,‖p y‖≤1) |
| {LA LB : ℝ≥0} (hA : LipschitzWith LA (fun y ↦ cmcJacobian (p y))) (hB : LipschitzWith LB B) |
| {M R : ℝ} (hBb : ∀ y,‖B y‖≤M) (hR : 0<R) |
| (hgc : ContDiffOn ℝ 1 g (Metric.ball 0 R)) |
| (hw : LinearCMCEquation g p B (Metric.ball 0 R)) : |
| ∃ (G : Plane → Plane) (C : ℝ), Continuous G ∧ (∀ y, ‖G y‖≤(K:ℝ)) ∧ 0≤C ∧ |
| (∀ y, ‖y‖≤R/2 → G y=gradient g y) ∧ |
| ∀ v : Plane, ‖v‖≤1 → ∀ t∈Ioc 0 (R/4), |
| (∫ y in Metric.ball 0 (R/4), ‖G (y+t • v)-G y‖^2)≤C*t^2 := by |
| let b : ContDiffBump (0 : Plane) := ⟨R/2,3*R/4,by linarith,by linarith⟩ |
| have hbs : tsupport b⊆Metric.ball 0 R := by |
| rw [b.tsupport_eq] |
| exact Metric.closedBall_subset_ball (by dsimp [b]; linarith) |
| let G : Plane → Plane := fun y ↦ b y • gradient g y |
| have hGc : Continuous G := continuous_local_vector_cutoff Metric.isOpen_ball |
| (C1_gradient_continuousOn hgc Metric.isOpen_ball) b.continuous hbs |
| have hgrad (y : Plane) : ‖gradient g y‖≤(K:ℝ) := by |
| rw [plane_gradient_norm]; exact norm_fderiv_le_of_lipschitz ℝ hg |
| have hGb (y : Plane) : ‖G y‖≤(K:ℝ) := by |
| dsimp only [G] |
| rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg b.nonneg] |
| exact (mul_le_mul_of_nonneg_left (hgrad y) b.nonneg).trans |
| (mul_le_of_le_one_left K.coe_nonneg b.le_one) |
| have hGeq {y : Plane} (hy : ‖y‖≤R/2) : G y=gradient g y := by |
| dsimp only [G] |
| rw [b.one_of_mem_closedBall (by simpa only [Metric.mem_closedBall,dist_zero_right] using hy),one_smul] |
| obtain ⟨C,hC,henergy⟩ := hw.local_gradient_energy hg hp hpb hA hB hBb hgc |
| (show 0<R/4 by linarith) (show R/4<R/2 by linarith) |
| refine ⟨G,C,hGc,hGb,hC,fun _ hy ↦ hGeq hy,fun v hv t ht ↦ ?_⟩ |
| have ha : ‖t • v‖≤t := by |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos ht.1] |
| exact mul_le_of_le_one_right ht.1.le hv |
| calc |
| _ = ∫ y in Metric.ball 0 (R/4), ‖gradient g (y+t • v)-gradient g y‖^2 := by |
| apply setIntegral_congr_fun measurableSet_ball |
| intro y hy |
| have hy' : ‖y‖<R/4 := by simpa only [Metric.mem_ball,dist_zero_right] using hy |
| dsimp only |
| rw [hGeq ((norm_add_le y (t • v)).trans (by linarith [ht.2])),hGeq (by linarith)] |
| _ ≤ C*‖t • v‖^2 := henergy (t • v) (by linarith [ht.2]) |
| _ ≤ C*t^2 := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (norm_nonneg _) ha 2) hC |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma lipschitz_vector_quotient_bound {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {f : Plane → E} {K : ℝ≥0} (hf : LipschitzWith K f) {t : ℝ} (ht : t≠0) (y v : Plane) : |
| ‖t⁻¹ • (f (y+t • v)-f y)‖≤(K:ℝ)*‖v‖ := by |
| have hh : ‖f (y+t • v)-f y‖≤(K:ℝ)*(|t| *‖v‖) := by |
| simpa only [dist_eq_norm,add_sub_cancel_left,norm_smul,Real.norm_eq_abs] using hf.dist_le_mul (y+t • v) y |
| rw [norm_smul,Real.norm_eq_abs,abs_inv] |
| calc |
| _ ≤ |t|⁻¹*((K:ℝ)*(|t| *‖v‖)) := mul_le_mul_of_nonneg_left hh (inv_nonneg.mpr (abs_nonneg _)) |
| _ = _ := by field_simp |
|
|
| lemma differentiable_vector_quotient_limit {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {f : Plane → E} {y : Plane} (hf : DifferentiableAt ℝ f y) (v : Plane) |
| {t : ℕ → ℝ} (ht : Tendsto t atTop (𝓝[>] 0)) : |
| Tendsto (fun i ↦ (t i)⁻¹ • (f (y+t i • v)-f y)) atTop (𝓝 (fderiv ℝ f y v)) := |
| (hf.hasFDerivAt.hasLineDerivAt v).tendsto_slope_zero_right.comp ht |
|
|
| lemma linear_source_difference_limit {f : Plane → Plane} (hf : Continuous f) |
| {M : ℝ} (hb : ∀ y,‖f y‖≤M) {A : Plane → Plane →L[ℝ] Plane} {B : Plane → Plane} |
| {LA LB : ℝ≥0} (hA : LipschitzWith LA A) (hB : LipschitzWith LB B) |
| (hAd : Differentiable ℝ A) (hBd : Differentiable ℝ B) |
| {r : ℝ} {t : ℕ → ℝ} (ht : ∀ i,0<t i) (htlim : Tendsto t atTop (𝓝[>] 0)) |
| (v : Plane) {ψ : Plane → ℝ} (hψ : ContDiff ℝ 1 ψ) (hc : HasCompactSupport ψ) : |
| Tendsto (fun i ↦ ∫ y in Metric.ball 0 r,fderiv ℝ ψ y |
| (((t i)⁻¹ • (A (y+t i • v)-A y)) (f (y+t i • v))-(t i)⁻¹ • (B (y+t i • v)-B y))) atTop |
| (𝓝 (∫ y in Metric.ball 0 r,fderiv ℝ ψ y (fderiv ℝ A y v (f y)-fderiv ℝ B y v))) := by |
| let μ : Measure Plane := volume.restrict (Metric.ball 0 r) |
| let : IsFiniteMeasure μ := isFiniteMeasure_restrict.mpr measure_ball_lt_top.ne |
| obtain ⟨L,hL⟩ := hψ.lipschitzWith_of_hasCompactSupport hc (by simp) |
| apply tendsto_integral_of_dominated_convergence (fun _ ↦ (L:ℝ)*((LA:ℝ)*‖v‖*M+(LB:ℝ)*‖v‖)) |
| · intro i |
| apply Continuous.aestronglyMeasurable |
| have hcA : Continuous (fun y : Plane ↦ (t i)⁻¹ • (A (y+t i • v)-A y)) := |
| ((hA.continuous.comp (continuous_id.add continuous_const)).sub hA.continuous).const_smul ((t i)⁻¹ : ℝ) |
| have hcB : Continuous (fun y : Plane ↦ (t i)⁻¹ • (B (y+t i • v)-B y)) := |
| ((hB.continuous.comp (continuous_id.add continuous_const)).sub hB.continuous).const_smul ((t i)⁻¹ : ℝ) |
| exact (hψ.continuous_fderiv (by simp)).clm_apply |
| ((hcA.clm_apply (hf.comp (continuous_id.add continuous_const))).sub hcB) |
| · exact integrable_const _ |
| · intro i |
| exact Eventually.of_forall fun y ↦ by |
| have h₁ := (ContinuousLinearMap.le_opNorm ((t i)⁻¹ • (A (y+t i • v)-A y)) (f (y+t i • v))).trans |
| (mul_le_mul (lipschitz_vector_quotient_bound hA (ht i).ne' y v) (hb _) (norm_nonneg _) (by positivity)) |
| have h₂ := lipschitz_vector_quotient_bound hB (ht i).ne' y v |
| exact (ContinuousLinearMap.le_opNorm _ _).trans (mul_le_mul |
| (norm_fderiv_le_of_lipschitz ℝ hL) ((norm_sub_le _ _).trans (add_le_add h₁ h₂)) (norm_nonneg _) L.coe_nonneg) |
| · exact Eventually.of_forall fun y ↦ by |
| have hy : Tendsto (fun i ↦ y+t i • v) atTop (𝓝 y) := by |
| simpa only [zero_smul,add_zero] using |
| (tendsto_const_nhds (x := y)).add ((tendsto_nhds_of_tendsto_nhdsWithin htlim).smul (tendsto_const_nhds (x := v))) |
| apply (fderiv ℝ ψ y).continuous.tendsto _ |>.comp |
| have heval : Continuous (fun z : (Plane →L[ℝ] Plane) × Plane ↦ z.1 z.2) := |
| continuous_fst.clm_apply continuous_snd |
| exact ((heval.tendsto _).comp ((differentiable_vector_quotient_limit (hAd y) v htlim).prodMk_nhds |
| (hf.continuousAt.tendsto.comp hy))).sub (differentiable_vector_quotient_limit (hBd y) v htlim) |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem vector_weak_quotient_derivative {f : Plane → Plane} (hf : Continuous f) |
| {r : ℝ} {t : ℕ → ℝ} (htlim : Tendsto t atTop (𝓝[>] 0)) (v w : Plane) |
| {Q : ℕ → Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) r))} |
| {u : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) r))} |
| (hQ : ∀ i, (Q i : Plane → Plane)=ᵐ[volume.restrict (Metric.ball (0 : Plane) r)] vectorDifference f v (t i)) |
| (hw : ∀ V : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) r)), |
| Tendsto (fun i ↦ ⟪Q i,V⟫) atTop (𝓝 ⟪u,V⟫)) |
| {ψ : Plane → ℝ} (hψ : ContDiff ℝ 1 ψ) (hc : HasCompactSupport ψ) |
| (hs : tsupport ψ⊆Metric.ball 0 r) : |
| (∫ y in Metric.ball 0 r, ⟪u y,w⟫*ψ y)= -(∫ y, ⟪f y,w⟫*fderiv ℝ ψ y v) := by |
| let μ : Measure Plane := volume.restrict (Metric.ball 0 r) |
| let W : Plane → Plane := fun y ↦ ψ y • w |
| have hWc : Continuous W := hψ.continuous.smul continuous_const |
| have hWs : HasCompactSupport W := hc.smul_right |
| have hWm : MemLp W 2 μ := hWc.memLp_of_hasCompactSupport hWs |
| let V : Lp Plane 2 μ := hWm.toLp W |
| have hp (i : ℕ) : ⟪Q i,V⟫=(∫ y, ((t i)⁻¹*(⟪f (y+t i • v),w⟫-⟪f y,w⟫))*ψ y) := by |
| rw [L2.inner_def,integral_mul_eq_restrict_of_tsupport measurableSet_ball _ ψ hs] |
| apply integral_congr_ae |
| filter_upwards [hQ i,hWm.coeFn_toLp] with y hy hy' |
| rw [hy,hy'] |
| simp only [W,vectorDifference,inner_smul_right,inner_smul_left,inner_sub_left,RCLike.conj_to_real] |
| ring |
| have hup : ⟪u,V⟫=(∫ y in Metric.ball 0 r, ⟪u y,w⟫*ψ y) := by |
| rw [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [hWm.coeFn_toLp] with y hy |
| rw [hy] |
| simp only [W,inner_smul_right] |
| ring |
| have hlim₁ := hw V |
| simp_rw [hp,hup] at hlim₁ |
| have hlim₂ := (continuous_difference_quotient_test_limit (hf.inner (continuous_const (y := w))) hψ hc v).comp htlim |
| exact tendsto_nhds_unique hlim₁ hlim₂ |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma weak_linear_cmc_pairing {f : Plane → Plane} {p : Plane → Plane} |
| (hp : Continuous p) (hpb : ∀ y,‖p y‖≤1) {r : ℝ} {t : ℕ → ℝ} (v : Plane) |
| {Q : ℕ → Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) r))} |
| {u : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) r))} |
| (hQ : ∀ i,(Q i : Plane → Plane)=ᵐ[volume.restrict (Metric.ball (0 : Plane) r)] vectorDifference f v (t i)) |
| (hw : ∀ V : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) r)), |
| Tendsto (fun i ↦ ⟪Q i,V⟫) atTop (𝓝 ⟪u,V⟫)) |
| {ψ : Plane → ℝ} (hψ : ContDiff ℝ 1 ψ) (hc : HasCompactSupport ψ) : |
| Tendsto (fun i ↦ ∫ y in Metric.ball 0 r,fderiv ℝ ψ y (cmcJacobian (p y) (vectorDifference f v (t i) y))) atTop |
| (𝓝 (∫ y in Metric.ball 0 r,fderiv ℝ ψ y (cmcJacobian (p y) (u y)))) := by |
| let μ : Measure Plane := volume.restrict (Metric.ball 0 r) |
| let : IsFiniteMeasure μ := isFiniteMeasure_restrict.mpr measure_ball_lt_top.ne |
| obtain ⟨L,hL⟩ := hψ.lipschitzWith_of_hasCompactSupport hc (by simp) |
| have hgrad (y : Plane) : ‖gradient ψ y‖≤(L:ℝ) := by |
| rw [plane_gradient_norm]; exact norm_fderiv_le_of_lipschitz ℝ hL |
| let W : Plane → Plane := fun y ↦ cmcJacobian (p y) (gradient ψ y) |
| have hWcont : Continuous W := (cmcJacobian_contDiff.continuous.comp hp).clm_apply |
| ((InnerProductSpace.toDual ℝ Plane).symm.continuous.comp (hψ.continuous_fderiv (by simp))) |
| have hWb (y : Plane) : ‖W y‖≤2*(L:ℝ) := |
| (cmcJacobian_norm_le (hpb y) _).trans (by gcongr; exact hgrad y) |
| have hW : MemLp W 2 μ := MemLp.of_bound hWcont.aestronglyMeasurable (C := 2*(L:ℝ)) (Eventually.of_forall hWb) |
| let V : Lp Plane 2 μ := hW.toLp W |
| have hid (y z : Plane) : ⟪z,W y⟫=fderiv ℝ ψ y (cmcJacobian (p y) z) := by |
| dsimp only [W] |
| rw [cmcJacobian_symmetric,real_inner_comm,inner_gradient_left] |
| have hpair (i : ℕ) : ⟪Q i,V⟫=(∫ y,fderiv ℝ ψ y (cmcJacobian (p y) (vectorDifference f v (t i) y)) ∂μ) := by |
| rw [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [hQ i,hW.coeFn_toLp] with y hy hz |
| rw [hy,hz,hid] |
| have hup : ⟪u,V⟫=(∫ y,fderiv ℝ ψ y (cmcJacobian (p y) (u y)) ∂μ) := by |
| rw [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [hW.coeFn_toLp] with y hy |
| rw [hy,hid] |
| have hh := hw V |
| simpa only [hpair,hup,μ] using hh |
|
|
| lemma Lp.cmc_differential_integrable {p : Plane → Plane} |
| (hp : Continuous p) (hpb : ∀ y,‖p y‖≤1) (r : ℝ) |
| (u : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) r))) |
| {ψ : Plane → ℝ} (hψ : ContDiff ℝ 1 ψ) (hc : HasCompactSupport ψ) : |
| Integrable (fun y ↦ fderiv ℝ ψ y (cmcJacobian (p y) (u y))) |
| (volume.restrict (Metric.ball (0 : Plane) r)) := by |
| let μ : Measure Plane := volume.restrict (Metric.ball 0 r) |
| let : IsFiniteMeasure μ := isFiniteMeasure_restrict.mpr measure_ball_lt_top.ne |
| obtain ⟨L,hL⟩ := hψ.lipschitzWith_of_hasCompactSupport hc (by simp) |
| let W : Plane → Plane := fun y ↦ cmcJacobian (p y) (gradient ψ y) |
| have hWc : Continuous W := (cmcJacobian_contDiff.continuous.comp hp).clm_apply |
| ((InnerProductSpace.toDual ℝ Plane).symm.continuous.comp (hψ.continuous_fderiv (by simp))) |
| have hWb (y : Plane) : ‖W y‖≤2*(L:ℝ) := by |
| have hg : ‖gradient ψ y‖≤(L:ℝ) := by |
| rw [plane_gradient_norm]; exact norm_fderiv_le_of_lipschitz ℝ hL |
| exact (cmcJacobian_norm_le (hpb y) _).trans (by linarith) |
| have hW : MemLp W 2 μ := MemLp.of_bound hWc.aestronglyMeasurable (C := 2*(L:ℝ)) (Eventually.of_forall hWb) |
| apply (L2.integrable_inner (𝕜 := ℝ) u (hW.toLp W)).congr |
| filter_upwards [hW.coeFn_toLp] with y hy |
| rw [hy] |
| dsimp only [W] |
| rw [cmcJacobian_symmetric,real_inner_comm,inner_gradient_left] |
|
|
| theorem weak_L2_linear_cmc_chain {f : Plane → Plane} (hf : Continuous f) |
| {M : ℝ} (hb : ∀ y,‖f y‖≤M) {p B : Plane → Plane} (hp : Continuous p) (hpb : ∀ y,‖p y‖≤1) |
| {LA LB : ℝ≥0} (hA : LipschitzWith LA (fun y ↦ cmcJacobian (p y))) (hB : LipschitzWith LB B) |
| (hAd : Differentiable ℝ (fun y ↦ cmcJacobian (p y))) (hBd : Differentiable ℝ B) |
| {r : ℝ} {t : ℕ → ℝ} (ht : ∀ i,0<t i) (htlim : Tendsto t atTop (𝓝[>] 0)) |
| (v : Plane) {Q : ℕ → Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) r))} |
| {u : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) r))} |
| (hQ : ∀ i,(Q i : Plane → Plane)=ᵐ[volume.restrict (Metric.ball (0 : Plane) r)] vectorDifference f v (t i)) |
| (hw : ∀ V : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) r)), |
| Tendsto (fun i ↦ ⟪Q i,V⟫) atTop (𝓝 ⟪u,V⟫)) |
| {ψ : Plane → ℝ} (hψ : ContDiff ℝ 1 ψ) (hc : HasCompactSupport ψ) : |
| Tendsto (fun i ↦ ∫ y in Metric.ball 0 r,fderiv ℝ ψ y ((t i)⁻¹ • |
| ((cmcJacobian (p (y+t i • v)) (f (y+t i • v))-B (y+t i • v))- |
| (cmcJacobian (p y) (f y)-B y)))) atTop |
| (𝓝 ((∫ y in Metric.ball 0 r,fderiv ℝ ψ y (cmcJacobian (p y) (u y)))+ |
| (∫ y in Metric.ball 0 r,fderiv ℝ ψ y |
| (fderiv ℝ (fun z ↦ cmcJacobian (p z)) y v (f y)-fderiv ℝ B y v)))) := by |
| have h₁ := weak_linear_cmc_pairing hp hpb v hQ hw hψ hc |
| have h₂ := linear_source_difference_limit hf hb hA hB hAd hBd ht htlim v hψ hc (r := r) |
| convert h₁.add h₂ using 1 |
| funext i |
| let A : Plane → Plane →L[ℝ] Plane := fun y ↦ cmcJacobian (p y) |
| have hD : Continuous (fderiv ℝ ψ) := hψ.continuous_fderiv (by simp) |
| have hAi : Continuous (fun y : Plane ↦ A y (vectorDifference f v (t i) y)) := |
| hA.continuous.clm_apply (((hf.comp (continuous_id.add continuous_const)).sub hf).const_smul ((t i)⁻¹ : ℝ)) |
| have hSi : Continuous (fun y : Plane ↦ ((t i)⁻¹ • (A (y+t i • v)-A y)) (f (y+t i • v))- |
| (t i)⁻¹ • (B (y+t i • v)-B y)) := by |
| have hcA : Continuous (fun y : Plane ↦ (t i)⁻¹ • (A (y+t i • v)-A y)) := |
| ((hA.continuous.comp (continuous_id.add continuous_const)).sub hA.continuous).const_smul ((t i)⁻¹ : ℝ) |
| have hcB : Continuous (fun y : Plane ↦ (t i)⁻¹ • (B (y+t i • v)-B y)) := |
| ((hB.continuous.comp (continuous_id.add continuous_const)).sub hB.continuous).const_smul ((t i)⁻¹ : ℝ) |
| exact (hcA.clm_apply (hf.comp (continuous_id.add continuous_const))).sub hcB |
| have hi₁ := (hD.clm_apply hAi).continuousOn.integrableOn_compact (μ := (volume : Measure Plane)) (isCompact_closedBall (0 : Plane) r) |
| |>.mono_set Metric.ball_subset_closedBall |
| have hi₂ := (hD.clm_apply hSi).continuousOn.integrableOn_compact (μ := (volume : Measure Plane)) (isCompact_closedBall (0 : Plane) r) |
| |>.mono_set Metric.ball_subset_closedBall |
| rw [← integral_add hi₁ hi₂] |
| apply setIntegral_congr_fun measurableSet_ball |
| intro y _hy |
| dsimp only [A,vectorDifference] |
| simp only [map_sub,map_smul,smul_eq_mul,sub_apply,smul_apply] |
| ring |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem weak_L2_cmc_chain {f : Plane → Plane} (hf : Continuous f) |
| (hb : ∀ y, ‖f y‖≤1) {r C : ℝ} (hC : 0≤C) {t : ℕ → ℝ} |
| (ht : ∀ i, 0<t i) (htlim : Tendsto t atTop (𝓝 0)) |
| (v : Plane) |
| (henergy : ∀ i, (∫ y in Metric.ball 0 r, ‖f (y+t i • v)-f y‖^2)≤C*(t i)^2) |
| {Q : ℕ → Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) r))} |
| {u : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) r))} |
| (hQ : ∀ i, (Q i : Plane → Plane)=ᵐ[volume.restrict (Metric.ball (0 : Plane) r)] vectorDifference f v (t i)) |
| (hw : ∀ V : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) r)), |
| Tendsto (fun i ↦ ⟪Q i,V⟫) atTop (𝓝 ⟪u,V⟫)) |
| {ψ : Plane → ℝ} (hψ : ContDiff ℝ 1 ψ) (hc : HasCompactSupport ψ) : |
| Tendsto (fun i ↦ ∫ y in Metric.ball 0 r, |
| fderiv ℝ ψ y ((t i)⁻¹ • (cmcFlux (f (y+t i • v))-cmcFlux (f y)))) atTop |
| (𝓝 (∫ y in Metric.ball 0 r, fderiv ℝ ψ y (cmcJacobian (f y) (u y)))) := by |
| let μ : Measure Plane := volume.restrict (Metric.ball 0 r) |
| let : IsFiniteMeasure μ := isFiniteMeasure_restrict.mpr measure_ball_lt_top.ne |
| obtain ⟨L,hL⟩ := hψ.lipschitzWith_of_hasCompactSupport hc (by simp) |
| obtain ⟨B,hB,hrem⟩ := cmcFlux_quadratic_remainder |
| have hgrad (y : Plane) : ‖gradient ψ y‖≤(L:ℝ) := by |
| rw [plane_gradient_norm]; exact norm_fderiv_le_of_lipschitz ℝ hL |
| let W : Plane → Plane := fun y ↦ cmcJacobian (f y) (gradient ψ y) |
| have hWcont : Continuous W := (cmcJacobian_contDiff.continuous.comp hf).clm_apply |
| ((InnerProductSpace.toDual ℝ Plane).symm.continuous.comp (hψ.continuous_fderiv (by simp))) |
| have hWb (y : Plane) : ‖W y‖≤2*(L:ℝ) := |
| (cmcJacobian_norm_le (hb y) _).trans (by gcongr; exact hgrad y) |
| have hW : MemLp W 2 μ := MemLp.of_bound hWcont.aestronglyMeasurable (C := 2*(L:ℝ)) (Eventually.of_forall hWb) |
| let V : Lp Plane 2 μ := hW.toLp W |
| have hid (y z : Plane) : ⟪z,W y⟫=fderiv ℝ ψ y (cmcJacobian (f y) z) := by |
| dsimp only [W] |
| rw [cmcJacobian_symmetric,real_inner_comm,inner_gradient_left] |
| have hp (i : ℕ) : ⟪Q i,V⟫=(∫ y, fderiv ℝ ψ y (cmcJacobian (f y) (vectorDifference f v (t i) y)) ∂μ) := by |
| rw [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [hQ i,hW.coeFn_toLp] with y hy hy' |
| rw [hy,hy',hid] |
| have hup : ⟪u,V⟫=(∫ y, fderiv ℝ ψ y (cmcJacobian (f y) (u y)) ∂μ) := by |
| rw [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [hW.coeFn_toLp] with y hy |
| rw [hy,hid] |
| have hlim := hw V |
| rw [hup] at hlim |
| simp_rw [hp] at hlim |
| let Err : ℕ → Plane → ℝ := fun i y ↦ fderiv ℝ ψ y ((t i)⁻¹ • |
| (cmcFlux (f (y+t i • v))-cmcFlux (f y)-cmcJacobian (f y) (f (y+t i • v)-f y))) |
| have herr (i : ℕ) : ‖∫ y, Err i y ∂μ‖≤((L:ℝ)*B*C)*t i := by |
| have hti : 0<(t i)⁻¹ := inv_pos.mpr (ht i) |
| have hei : Integrable (fun y ↦ ‖f (y+t i • v)-f y‖^2) μ := by |
| apply (integrable_const (4:ℝ)).mono' (by fun_prop) |
| filter_upwards [] with y |
| rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)] |
| have hh : ‖f (y+t i • v)-f y‖≤2 := (norm_sub_le _ _).trans (by linarith [hb (y+t i • v),hb y]) |
| nlinarith [norm_nonneg (f (y+t i • v)-f y)] |
| calc |
| _ ≤ ∫ y, ((L:ℝ)*B*(t i)⁻¹)*‖f (y+t i • v)-f y‖^2 ∂μ := by |
| apply norm_integral_le_of_norm_le (hei.const_mul _) |
| filter_upwards [] with y |
| dsimp only [Err] |
| calc |
| _ ≤ (L:ℝ)*‖(t i)⁻¹ • (cmcFlux (f (y+t i • v))-cmcFlux (f y)-cmcJacobian (f y) (f (y+t i • v)-f y))‖ := |
| (ContinuousLinearMap.le_opNorm _ _).trans (mul_le_mul_of_nonneg_right |
| (norm_fderiv_le_of_lipschitz ℝ hL) (norm_nonneg _)) |
| _ = (L:ℝ)*((t i)⁻¹*‖cmcFlux (f (y+t i • v))-cmcFlux (f y)-cmcJacobian (f y) (f (y+t i • v)-f y)‖) := by |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr (ht i))] |
| _ ≤ (L:ℝ)*((t i)⁻¹*(B*‖f (y+t i • v)-f y‖^2)) := by |
| exact mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left |
| (hrem _ _ (hb y) (hb (y+t i • v))) hti.le) L.coe_nonneg |
| _ = _ := by ring |
| _ = ((L:ℝ)*B*(t i)⁻¹)*(∫ y, ‖f (y+t i • v)-f y‖^2 ∂μ) := integral_const_mul _ _ |
| _ ≤ ((L:ℝ)*B*(t i)⁻¹)*(C*(t i)^2) := by |
| simpa only [mul_comm] using mul_le_mul (henergy i) |
| (le_refl ((L:ℝ)*B*(t i)⁻¹)) (by positivity) (mul_nonneg hC (sq_nonneg (t i))) |
| _ = _ := by field_simp |
| have herrlim : Tendsto (fun i ↦ ∫ y, Err i y ∂μ) atTop (𝓝 0) := |
| squeeze_zero_norm herr (by simpa using htlim.const_mul ((L:ℝ)*B*C)) |
| have hsum := hlim.add herrlim |
| simp only [add_zero] at hsum |
| convert hsum using 1 |
| funext i |
| have hEi : Integrable (Err i) μ := by |
| apply (integrable_const ((L:ℝ)*(t i)⁻¹*8)).mono' ?_ ?_ |
| · apply Continuous.aestronglyMeasurable |
| dsimp only [Err] |
| have hD := hψ.continuous_fderiv (by simp) |
| have hF := cmcFlux_contDiff.continuous |
| have hJ := cmcJacobian_contDiff.continuous |
| fun_prop |
| · filter_upwards [] with y |
| dsimp only [Err] |
| have hrem8 : ‖cmcFlux (f (y+t i • v))-cmcFlux (f y)-cmcJacobian (f y) (f (y+t i • v)-f y)‖≤8 := by |
| have hd : ‖f (y+t i • v)-f y‖≤2 := (norm_sub_le _ _).trans (by linarith [hb (y+t i • v),hb y]) |
| have h₁ := cmcFlux_difference_bound (hb y) (hb (y+t i • v)) |
| have h₂ := cmcJacobian_norm_le (hb y) (f (y+t i • v)-f y) |
| exact (norm_sub_le _ _).trans (by linarith) |
| calc |
| _ ≤ (L:ℝ)*‖(t i)⁻¹ • (cmcFlux (f (y+t i • v))-cmcFlux (f y)-cmcJacobian (f y) (f (y+t i • v)-f y))‖ := |
| (ContinuousLinearMap.le_opNorm _ _).trans (mul_le_mul_of_nonneg_right |
| (norm_fderiv_le_of_lipschitz ℝ hL) (norm_nonneg _)) |
| _ = (L:ℝ)*((t i)⁻¹*‖cmcFlux (f (y+t i • v))-cmcFlux (f y)-cmcJacobian (f y) (f (y+t i • v)-f y)‖) := by |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr (ht i))] |
| _ ≤ (L:ℝ)*((t i)⁻¹*8) := mul_le_mul_of_nonneg_left |
| (mul_le_mul_of_nonneg_left hrem8 (inv_pos.mpr (ht i)).le) L.coe_nonneg |
| _ = _ := by ring |
| have hJi : Integrable (fun y ↦ fderiv ℝ ψ y (cmcJacobian (f y) (vectorDifference f v (t i) y))) μ := by |
| have hi := L2.integrable_inner (𝕜 := ℝ) (Q i) V |
| apply hi.congr |
| filter_upwards [hQ i,hW.coeFn_toLp] with y hy hy' |
| rw [hy,hy',hid] |
| rw [← integral_add hJi hEi] |
| apply integral_congr_ae |
| filter_upwards [] with y |
| dsimp only [Err,vectorDifference] |
| simp only [map_sub,map_smul,smul_eq_mul] |
| ring |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| end CubicTorus |
| end |
| end |
| end |
|
|
| end OAI |
|
|
