| import Mathlib |
| import OAI.Geometry.CubicTorus.Estimates.DifferenceStep |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma differential_integral_restrict {ψ : Plane → ℝ} {S : Set Plane} |
| (hs : tsupport ψ⊆S) (F : Plane → Plane) : |
| (∫ y in S, fderiv ℝ ψ y (F y))=(∫ y, fderiv ℝ ψ y (F y)) := by |
| apply setIntegral_eq_integral_of_forall_compl_eq_zero |
| intro y hy |
| rw [fderiv_of_notMem_tsupport ℝ (fun hy' ↦ hy (hs hy')),zero_apply] |
|
|
| theorem WeakCMCGraph.linearized_equation {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)) |
| (hcmc : WeakCMCGraph g H (Metric.ball 0 R)) (v : Plane) (hv : ‖v‖≤1) : |
| ∃ u : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) (R/4))), |
| (∀ (w : Plane) (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,w⟫*ψ y)= |
| -(∫ y, ⟪gradient g y,w⟫*fderiv ℝ ψ y v)) ∧ |
| (∀ ψ : Plane → ℝ, ContDiff ℝ 1 ψ → HasCompactSupport ψ → |
| tsupport ψ⊆Metric.ball 0 (R/4) → |
| (∫ y in Metric.ball 0 (R/4), fderiv ℝ ψ y (cmcJacobian (gradient g y) (u y)))=0) := by |
| obtain ⟨G,C,hGc,hGb,hC,hGeq,henergy⟩ := hcmc.gradient_difference_data hg hK hR hgc |
| have hδ : 0<R/4 := by linarith |
| obtain ⟨u,φ,_hu,hφ,hw⟩ := vector_L2_difference_subsequence hGc hGb hδ hC v (henergy v hv) |
| let t : ℕ → ℝ := fun i ↦ differenceStep (R/4) (φ i) |
| let Q : ℕ → Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) (R/4))) := fun i ↦ |
| (vectorDifference_memLp hGc hGb (R/4) (t i) v).toLp (vectorDifference G v (t i)) |
| have ht (i : ℕ) : t i∈Ioc 0 (R/4) := differenceStep_mem hδ (φ i) |
| have htlim : Tendsto t atTop (𝓝[>] 0) := (differenceStep_tendsto hδ).comp hφ.tendsto_atTop |
| have hQ (i : ℕ) : (Q i : Plane → Plane)=ᵐ[volume.restrict (Metric.ball (0 : Plane) (R/4))] |
| vectorDifference G v (t i) := (vectorDifference_memLp hGc hGb (R/4) (t i) v).coeFn_toLp |
| have hweak : ∀ V : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) (R/4))), |
| Tendsto (fun i ↦ ⟪Q i,V⟫) atTop (𝓝 ⟪u,V⟫) := hw |
| have hty (i : ℕ) : ‖t i • v‖≤t i := by |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos (ht i).1] |
| exact mul_le_of_le_one_right (ht i).1.le hv |
| have hGy (y : Plane) (hy : y∈Metric.ball (0 : Plane) (R/4)) : G y=gradient g y := by |
| apply hGeq |
| have hh : ‖y‖<R/4 := by simpa only [Metric.mem_ball,dist_zero_right] using hy |
| linarith |
| have hGyt (i : ℕ) (y : Plane) (hy : y∈Metric.ball (0 : Plane) (R/4)) : |
| G (y+t i • v)=gradient g (y+t i • v) := by |
| apply hGeq |
| have hh : ‖y‖<R/4 := by simpa only [Metric.mem_ball,dist_zero_right] using hy |
| exact (norm_add_le _ _).trans (by linarith [hty i,(ht i).2]) |
| refine ⟨u,?_,?_⟩ |
| · intro w ψ hψ hc hs |
| rw [vector_weak_quotient_derivative hGc htlim v w hQ hweak hψ hc hs] |
| congr 1 |
| apply integral_congr_ae |
| filter_upwards [] with y |
| by_cases hy : y∈tsupport ψ |
| · rw [hGy y (hs hy)] |
| · rw [fderiv_of_notMem_tsupport ℝ hy,zero_apply,mul_zero,mul_zero] |
| · intro ψ hψ hc hs |
| have hlim := weak_L2_cmc_chain hGc hGb hC (fun i ↦ (ht i).1) |
| (tendsto_nhds_of_tendsto_nhdsWithin htlim) v (fun i ↦ henergy v hv (t i) (ht i)) hQ hweak hψ hc |
| have hzero (i : ℕ) : (∫ y in Metric.ball 0 (R/4), |
| fderiv ℝ ψ y ((t i)⁻¹ • (cmcFlux (G (y+t i • v))-cmcFlux (G y))))=0 := by |
| calc |
| _ = ∫ y in Metric.ball 0 (R/4), |
| fderiv ℝ ψ y ((t i)⁻¹ • (cmcFlux (gradient g (y+t i • v))-cmcFlux (gradient g y))) := by |
| apply setIntegral_congr_fun measurableSet_ball |
| intro y hy |
| dsimp only |
| rw [hGy y hy,hGyt i y hy] |
| _ = ∫ y, fderiv ℝ ψ y ((t i)⁻¹ • |
| (cmcFlux (gradient g (y+t i • v))-cmcFlux (gradient g y))) := differential_integral_restrict hs _ |
| _ = (t i)⁻¹*(∫ y, fderiv ℝ ψ y |
| (cmcFlux (gradient g (y+t i • v))-cmcFlux (gradient g y))) := by |
| simp only [map_smul,smul_eq_mul,integral_const_mul] |
| _ = 0 := by |
| rw [hcmc.difference_test hg hδ (by linarith [hty i,(ht i).2]) hψ hc hs,mul_zero] |
| have he : (∫ y in Metric.ball 0 (R/4), fderiv ℝ ψ y (cmcJacobian (G y) (u y)))=0 := |
| tendsto_nhds_unique hlim (by simp only [hzero]; exact tendsto_const_nhds) |
| rw [← he] |
| apply setIntegral_congr_fun measurableSet_ball |
| intro y hy |
| dsimp only |
| rw [hGy y hy] |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem LinearCMCEquation.weak_derivative {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) |
| (hAc : ContDiff ℝ 1 (fun y ↦ cmcJacobian (p y))) (hBc : ContDiff ℝ 1 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)) (v : Plane) (hv : ‖v‖≤1) : |
| ∃ u : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) (R/4))), |
| (∀ (w : Plane) (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,w⟫*ψ y)= |
| -(∫ y, ⟪gradient g y,w⟫*fderiv ℝ ψ y v)) ∧ |
| (∀ ψ : Plane → ℝ, ContDiff ℝ 1 ψ → HasCompactSupport ψ → |
| tsupport ψ⊆Metric.ball 0 (R/4) → |
| (∫ y in Metric.ball 0 (R/4), fderiv ℝ ψ y (cmcJacobian (p y) (u y)- |
| (fderiv ℝ B y v-fderiv ℝ (fun z ↦ cmcJacobian (p z)) y v (gradient g y))))=0) := by |
| obtain ⟨G,C,hGc,hGb,hC,hGeq,henergy⟩ := hw.gradient_difference_data hg hp hpb hA hB hBb hR hgc |
| have hδ : 0<R/4 := by linarith |
| obtain ⟨u,φ,_hu,hφ,hwk⟩ := vector_L2_difference_subsequence hGc hGb hδ hC v (henergy v hv) |
| let t : ℕ → ℝ := fun i ↦ differenceStep (R/4) (φ i) |
| let Q : ℕ → Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) (R/4))) := fun i ↦ |
| (vectorDifference_memLp hGc hGb (R/4) (t i) v).toLp (vectorDifference G v (t i)) |
| have ht (i : ℕ) : t i∈Ioc 0 (R/4) := differenceStep_mem hδ (φ i) |
| have htlim : Tendsto t atTop (𝓝[>] 0) := (differenceStep_tendsto hδ).comp hφ.tendsto_atTop |
| have hQ (i : ℕ) : (Q i : Plane → Plane)=ᵐ[volume.restrict (Metric.ball (0 : Plane) (R/4))] |
| vectorDifference G v (t i) := (vectorDifference_memLp hGc hGb (R/4) (t i) v).coeFn_toLp |
| have hweak : ∀ V : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) (R/4))), |
| Tendsto (fun i ↦ ⟪Q i,V⟫) atTop (𝓝 ⟪u,V⟫) := hwk |
| have hty (i : ℕ) : ‖t i • v‖≤t i := by |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos (ht i).1] |
| exact mul_le_of_le_one_right (ht i).1.le hv |
| have hGy (y : Plane) (hy : y∈Metric.ball (0 : Plane) (R/4)) : G y=gradient g y := by |
| apply hGeq |
| have hh : ‖y‖<R/4 := by simpa only [Metric.mem_ball,dist_zero_right] using hy |
| linarith |
| have hGyt (i : ℕ) (y : Plane) (hy : y∈Metric.ball (0 : Plane) (R/4)) : |
| G (y+t i • v)=gradient g (y+t i • v) := by |
| apply hGeq |
| have hh : ‖y‖<R/4 := by simpa only [Metric.mem_ball,dist_zero_right] using hy |
| exact (norm_add_le _ _).trans (by linarith [hty i,(ht i).2]) |
| refine ⟨u,?_,?_⟩ |
| · intro w ψ hψ hc hs |
| rw [vector_weak_quotient_derivative hGc htlim v w hQ hweak hψ hc hs] |
| congr 1 |
| apply integral_congr_ae |
| filter_upwards [] with y |
| by_cases hy : y∈tsupport ψ |
| · rw [hGy y (hs hy)] |
| · rw [fderiv_of_notMem_tsupport ℝ hy,zero_apply,mul_zero,mul_zero] |
| · intro ψ hψ hc hs |
| have hlim := weak_L2_linear_cmc_chain hGc hGb hp hpb hA hB |
| (hAc.differentiable (by simp)) (hBc.differentiable (by simp)) |
| (fun i ↦ (ht i).1) htlim v hQ hweak hψ hc |
| have hzero (i : ℕ) : (∫ y in Metric.ball 0 (R/4), |
| fderiv ℝ ψ y ((t i)⁻¹ • ((cmcJacobian (p (y+t i • v)) (G (y+t i • v))-B (y+t i • v))- |
| (cmcJacobian (p y) (G y)-B y))))=0 := by |
| calc |
| _ = ∫ y in Metric.ball 0 (R/4), fderiv ℝ ψ y ((t i)⁻¹ • |
| ((cmcJacobian (p (y+t i • v)) (gradient g (y+t i • v))-B (y+t i • v))- |
| (cmcJacobian (p y) (gradient g y)-B y))) := by |
| apply setIntegral_congr_fun measurableSet_ball |
| intro y hy |
| dsimp only |
| rw [hGy y hy,hGyt i y hy] |
| _ = ∫ y, fderiv ℝ ψ y ((t i)⁻¹ • |
| ((cmcJacobian (p (y+t i • v)) (gradient g (y+t i • v))-B (y+t i • v))- |
| (cmcJacobian (p y) (gradient g y)-B y))) := differential_integral_restrict hs _ |
| _ = (t i)⁻¹*(∫ y, fderiv ℝ ψ y |
| ((cmcJacobian (p (y+t i • v)) (gradient g (y+t i • v))-B (y+t i • v))- |
| (cmcJacobian (p y) (gradient g y)-B y))) := by |
| simp only [map_smul,smul_eq_mul,integral_const_mul] |
| _ = 0 := by |
| rw [hw.difference_test hg hp hpb hB.continuous hBb hδ |
| (by linarith [hty i,(ht i).2]) hψ hc hs,mul_zero] |
| have he : (∫ y in Metric.ball 0 (R/4), fderiv ℝ ψ y (cmcJacobian (p y) (u y)))+ |
| (∫ y in Metric.ball 0 (R/4),fderiv ℝ ψ y |
| (fderiv ℝ (fun z ↦ cmcJacobian (p z)) y v (G y)-fderiv ℝ B y v))=0 := |
| tendsto_nhds_unique hlim (by simp only [hzero]; exact tendsto_const_nhds) |
| have hAi := Lp.cmc_differential_integrable hp hpb (R/4) u hψ hc |
| have hSc : Continuous (fun y ↦ fderiv ℝ ψ y |
| (fderiv ℝ (fun z ↦ cmcJacobian (p z)) y v (G y)-fderiv ℝ B y v)) := |
| (hψ.continuous_fderiv (by simp)).clm_apply |
| ((((hAc.continuous_fderiv (by simp)).clm_apply continuous_const).clm_apply hGc).sub |
| ((hBc.continuous_fderiv (by simp)).clm_apply continuous_const)) |
| have hSi := hSc.continuousOn.integrableOn_compact (μ := (volume : Measure Plane)) |
| (isCompact_closedBall (0 : Plane) (R/4)) |>.mono_set Metric.ball_subset_closedBall |
| rw [← integral_add hAi hSi] at he |
| rw [← he] |
| apply setIntegral_congr_fun measurableSet_ball |
| intro y hy |
| dsimp only |
| rw [hGy y hy] |
| simp only [map_sub] |
| ring |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma C1_gradient_continuous {f : Plane → ℝ} (hf : ContDiff ℝ 1 f) : |
| Continuous (gradient f) := |
| (InnerProductSpace.toDual ℝ Plane).symm.continuous.comp (hf.continuous_fderiv (by simp)) |
|
|
| lemma plane_gradient_support {f : Plane → ℝ} : tsupport (gradient f)⊆tsupport f := by |
| exact (tsupport_comp_subset (g := (InnerProductSpace.toDual ℝ Plane).symm) (by simp) _).trans |
| (tsupport_fderiv_subset ℝ) |
|
|
| lemma plane_gradient_compact {f : Plane → ℝ} (hf : HasCompactSupport f) : |
| HasCompactSupport (gradient f) := |
| (hf.fderiv ℝ).comp_left (map_zero (InnerProductSpace.toDual ℝ Plane).symm) |
|
|
| lemma compact_C1_gradient_memLp {f : Plane → ℝ} (hf : ContDiff ℝ 1 f) |
| (hc : HasCompactSupport f) : MemLp (gradient f) 2 := |
| (C1_gradient_continuous hf).memLp_of_hasCompactSupport (plane_gradient_compact hc) |
|
|
| lemma compact_test_poincare (R : ℝ) : ∃ C : ℝ, 0≤C ∧ |
| ∀ f : Plane → ℝ, ∀ hf : ContDiff ℝ 1 f, ∀ hc : HasCompactSupport f, |
| tsupport f⊆Metric.ball 0 R → |
| ‖(hf.continuous.memLp_of_hasCompactSupport hc : MemLp f 2).toLp f‖≤ |
| C*‖(compact_C1_gradient_memLp hf hc).toLp (gradient f)‖ := by |
| let A : ℝ≥0 := eLpNormLESNormFDerivOfEqInnerConst (volume : Measure Plane) 1 |
| let B : ℝ≥0∞ := volume (Metric.ball (0 : Plane) R)^(1/2:ℝ) |
| have hB : B≠⊤ := ENNReal.rpow_ne_top_of_nonneg (by norm_num) measure_ball_lt_top.ne |
| refine ⟨(A:ℝ)*B.toReal,by positivity,?_⟩ |
| intro f hf hc hs |
| have hd := hf.continuous_fderiv (by simp : (1 : WithTop ℕ∞)≠0) |
| have hm : MemLp (fderiv ℝ f) 2 := hd.memLp_of_hasCompactSupport (hc.fderiv ℝ) |
| have hsub : Function.support (fderiv ℝ f)⊆Metric.ball 0 R := |
| (subset_tsupport (fderiv ℝ f)).trans ((tsupport_fderiv_subset ℝ).trans hs) |
| have hgns := eLpNorm_le_eLpNorm_fderiv_of_eq_inner (volume : Measure Plane) hf hc |
| (p := 1) (p' := 2) (by norm_num) (by simp [Plane]) (by norm_num [Plane]) |
| have hcomp := eLpNorm_le_eLpNorm_mul_rpow_measure_univ (μ := volume.restrict (Metric.ball 0 R)) |
| (f := fderiv ℝ f) (p := 1) (q := 2) (by norm_num) hd.aestronglyMeasurable.restrict |
| rw [eLpNorm_restrict_eq_of_support_subset (μ := volume) (p := 1) (f := fderiv ℝ f) hd.aestronglyMeasurable hsub, |
| eLpNorm_restrict_eq_of_support_subset (μ := volume) (p := 2) (f := fderiv ℝ f) hd.aestronglyMeasurable hsub,Measure.restrict_apply_univ] at hcomp |
| norm_num only [ENNReal.toReal_one,ENNReal.toReal_ofNat,div_one] at hcomp |
| have hcomp' : eLpNorm (fderiv ℝ f) 1 volume≤eLpNorm (fderiv ℝ f) 2 volume*B := by |
| exact hcomp |
| have he : eLpNorm f 2 volume ≤ (A:ℝ≥0∞)*(eLpNorm (fderiv ℝ f) 2 volume*B) := |
| hgns.trans (mul_le_mul_right hcomp' _) |
| have hgn : eLpNorm (gradient f) 2 volume=eLpNorm (fderiv ℝ f) 2 volume := |
| eLpNorm_congr_norm_ae (C1_gradient_continuous hf).aestronglyMeasurable |
| hd.aestronglyMeasurable (ae_of_all _ (plane_gradient_norm f)) |
| rw [Lp.norm_toLp,Lp.norm_toLp,hgn] |
| have hh := ENNReal.toReal_mono (ENNReal.mul_ne_top ENNReal.coe_ne_top |
| (ENNReal.mul_ne_top hm.ne hB)) he |
| simpa only [ENNReal.toReal_mul,ENNReal.coe_toReal,mul_assoc,mul_comm,mul_left_comm] using hh |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| def PlaneWeakGradient (S : Set Plane) (u : Plane → ℝ) (G : Plane → Plane) : Prop := |
| ∀ (v : Plane) (ψ : Plane → ℝ), ContDiff ℝ (⊤ : ℕ∞) ψ → HasCompactSupport ψ → |
| tsupport ψ⊆S → (∫ y, ⟪G y,v⟫*ψ y)= -(∫ y,u y*fderiv ℝ ψ y v) |
|
|
| lemma PlaneWeakGradient.congr {S : Set Plane} {u u' : Plane → ℝ} {G G' : Plane → Plane} |
| (h : PlaneWeakGradient S u G) (hu : u'=ᵐ[volume] u) (hG : G'=ᵐ[volume] G) : |
| PlaneWeakGradient S u' G' := by |
| intro v ψ hψ hc hs |
| have ha : (∫ y,⟪G' y,v⟫*ψ y)=(∫ y,⟪G y,v⟫*ψ y) := |
| integral_congr_ae (hG.mono fun _ he ↦ by dsimp only; rw [he]) |
| have hb : (∫ y,u' y*fderiv ℝ ψ y v)=(∫ y,u y*fderiv ℝ ψ y v) := |
| integral_congr_ae (hu.mono fun _ he ↦ by dsimp only; rw [he]) |
| rw [ha,hb] |
| exact h v ψ hψ hc hs |
|
|
| lemma compact_C1_weak_gradient {f : Plane → ℝ} (hf : ContDiff ℝ 1 f) |
| (_hc : HasCompactSupport f) (S : Set Plane) : PlaneWeakGradient S f (gradient f) := by |
| intro v ψ hψ hc _hs |
| have hi₁ : Integrable (fun y ↦ fderiv ℝ f y v*ψ y) := |
| (((hf.continuous_fderiv (by simp)).clm_apply continuous_const).mul hψ.continuous).integrable_of_hasCompactSupport |
| hc.mul_left |
| have hi₂ : Integrable (fun y ↦ f y*fderiv ℝ ψ y v) := |
| (hf.continuous.mul ((hψ.continuous_fderiv (by simp)).clm_apply continuous_const)).integrable_of_hasCompactSupport |
| (hc.fderiv_apply ℝ v).mul_left |
| have hi₃ : Integrable (fun y ↦ f y*ψ y) := |
| (hf.continuous.mul hψ.continuous).integrable_of_hasCompactSupport hc.mul_left |
| have hh := integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable hi₁ hi₂ hi₃ |
| (fun _ _ ↦ (hf.differentiable (by simp)).differentiableAt) |
| (fun _ _ ↦ (hψ.differentiable (by simp)).differentiableAt) |
| simpa only [inner_gradient_left,neg_neg] using congrArg Neg.neg hh.symm |
|
|
| lemma L2_pair_scalar_test {u : Lp ℝ 2 (volume : Measure Plane)} {φ : Plane → ℝ} |
| (hφ : MemLp φ 2 volume) : ⟪u,hφ.toLp φ⟫=(∫ y,u y*φ y) := by |
| rw [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [hφ.coeFn_toLp] with y hy |
| simp only [hy,RCLike.inner_apply,conj_trivial] |
| ring |
|
|
| lemma L2_pair_vector_test {G : Lp Plane 2 (volume : Measure Plane)} {φ : Plane → ℝ} |
| (hφ : Continuous φ) (hc : HasCompactSupport φ) (v : Plane) : |
| let hm : MemLp (fun y ↦ φ y • v) 2 volume := |
| (hφ.smul continuous_const).memLp_of_hasCompactSupport hc.smul_right |
| ⟪G,hm.toLp (fun y ↦ φ y • v)⟫=(∫ y,⟪G y,v⟫*φ y) := by |
| dsimp only |
| rw [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [((hφ.smul continuous_const).memLp_of_hasCompactSupport hc.smul_right : |
| MemLp (fun y ↦ φ y • v) 2 volume).coeFn_toLp] with y hy |
| simp only [Pi.smul_def'] at hy |
| rw [hy,inner_smul_right,mul_comm] |
|
|
| lemma PlaneWeakGradient.L2_limit {S : Set Plane} |
| {u : ℕ → Lp ℝ 2 (volume : Measure Plane)} {G : ℕ → Lp Plane 2 (volume : Measure Plane)} |
| {u₀ : Lp ℝ 2 (volume : Measure Plane)} {G₀ : Lp Plane 2 (volume : Measure Plane)} |
| (hu : Tendsto u atTop (𝓝 u₀)) (hG : Tendsto G atTop (𝓝 G₀)) |
| (hweak : ∀ i,PlaneWeakGradient S (u i) (G i)) : PlaneWeakGradient S u₀ G₀ := by |
| intro v ψ hψ hc hs |
| let a : Plane → Plane := fun y ↦ ψ y • v |
| have ha : MemLp a 2 volume := (hψ.continuous.smul continuous_const).memLp_of_hasCompactSupport hc.smul_right |
| let A := ha.toLp a |
| let b : Plane → ℝ := fun y ↦ fderiv ℝ ψ y v |
| have hb : MemLp b 2 volume := ((hψ.continuous_fderiv (by simp)).clm_apply continuous_const).memLp_of_hasCompactSupport |
| (hc.fderiv_apply ℝ v) |
| let B := hb.toLp b |
| have he (i : ℕ) : ⟪G i,A⟫= -⟪u i,B⟫ := by |
| rw [L2_pair_vector_test hψ.continuous hc v,L2_pair_scalar_test hb] |
| exact hweak i v ψ hψ hc hs |
| have hleft := hG.inner (𝕜 := ℝ) (tendsto_const_nhds (x := A)) |
| have hright := (hu.inner (𝕜 := ℝ) (tendsto_const_nhds (x := B))).neg |
| have hh := tendsto_nhds_unique (hleft.congr' (Eventually.of_forall he)) hright |
| rw [L2_pair_vector_test hψ.continuous hc v,L2_pair_scalar_test hb] at hh |
| exact hh |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma continuous_local_scalar_product {f η : Plane → ℝ} {S : Set Plane} |
| (hS : IsOpen S) (hf : ContinuousOn f S) (hη : Continuous η) |
| (hs : tsupport η⊆S) : Continuous (fun y ↦ f y*η y) := by |
| rw [continuous_iff_continuousAt] |
| intro y |
| by_cases hy : y∈tsupport η |
| · exact ((hf y (hs hy)).continuousAt (hS.mem_nhds (hs hy))).mul hη.continuousAt |
| · apply (continuousAt_const (y := (0:ℝ))).congr_of_eventuallyEq |
| filter_upwards [notMem_tsupport_iff_eventuallyEq.mp hy] with z hz |
| simp [hz] |
|
|
| lemma C1_local_integration_by_parts {g ψ : Plane → ℝ} {S : Set Plane} |
| (hS : IsOpen S) (hg : Continuous g) (hgc : ContDiffOn ℝ 1 g S) |
| (hψ : ContDiff ℝ 1 ψ) (hc : HasCompactSupport ψ) (hs : tsupport ψ⊆S) (v : Plane) : |
| (∫ y, fderiv ℝ g y v*ψ y)= -(∫ y,g y*fderiv ℝ ψ y v) := by |
| have hdf : ContinuousOn (fun y ↦ fderiv ℝ g y v) S := |
| (hgc.continuousOn_fderiv_of_isOpen hS (by simp)).clm_apply continuousOn_const |
| have hi₁ : Integrable (fun y ↦ fderiv ℝ g y v*ψ y) := |
| (continuous_local_scalar_product hS hdf hψ.continuous hs).integrable_of_hasCompactSupport hc.mul_left |
| have hi₂ : Integrable (fun y ↦ g y*fderiv ℝ ψ y v) := |
| (hg.mul ((hψ.continuous_fderiv (by simp)).clm_apply continuous_const)).integrable_of_hasCompactSupport |
| (hc.fderiv_apply ℝ v).mul_left |
| have hi₃ : Integrable (fun y ↦ g y*ψ y) := |
| (hg.mul hψ.continuous).integrable_of_hasCompactSupport hc.mul_left |
| have hh := integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable hi₁ hi₂ hi₃ |
| (fun y hy ↦ ((hgc y (hs hy)).contDiffAt (hS.mem_nhds (hs hy))).differentiableAt (by simp)) |
| (fun _ _ ↦ (hψ.differentiable (by simp)).differentiableAt) |
| simpa only [neg_neg] using congrArg Neg.neg hh.symm |
|
|
| lemma C1_mixed_test_identity {g ψ : Plane → ℝ} {S : Set Plane} |
| (hS : IsOpen S) (hg : Continuous g) (hgc : ContDiffOn ℝ 1 g S) |
| (hψ : ContDiff ℝ (⊤ : ℕ∞) ψ) (hc : HasCompactSupport ψ) (hs : tsupport ψ⊆S) |
| (v w : Plane) : |
| (∫ y, ⟪gradient g y,w⟫*fderiv ℝ ψ y v)= |
| (∫ y, ⟪gradient g y,v⟫*fderiv ℝ ψ y w) := by |
| have hv : ContDiff ℝ (⊤ : ℕ∞) (fun y ↦ fderiv ℝ ψ y v) := |
| (hψ.fderiv_right (by simp)).clm_apply contDiff_const |
| have hw : ContDiff ℝ (⊤ : ℕ∞) (fun y ↦ fderiv ℝ ψ y w) := |
| (hψ.fderiv_right (by simp)).clm_apply contDiff_const |
| simp only [inner_gradient_left] |
| rw [C1_local_integration_by_parts hS hg hgc (hv.of_le (by simp)) |
| (hc.fderiv_apply ℝ v) ((tsupport_fderiv_apply_subset (f := ψ) ℝ v).trans hs) w, |
| C1_local_integration_by_parts hS hg hgc (hw.of_le (by simp)) |
| (hc.fderiv_apply ℝ w) ((tsupport_fderiv_apply_subset (f := ψ) ℝ w).trans hs) v] |
| congr 1 |
| apply integral_congr_ae |
| filter_upwards [] with y |
| congr 1 |
| rw [fderiv_fderiv_apply hψ,fderiv_fderiv_apply hψ] |
| exact (hψ.contDiffAt.isSymmSndFDerivAt (by simp only [minSmoothness_of_isRCLikeNormedField]; exact WithTop.coe_le_coe.mpr le_top)) w v |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma indicator_inner_test_integral {S : Set Plane} (hS : MeasurableSet S) |
| (U : Plane → Plane) (v : Plane) (ψ : Plane → ℝ) : |
| (∫ y,⟪S.indicator U y,v⟫*ψ y)=(∫ y in S,⟪U y,v⟫*ψ y) := by |
| rw [← integral_indicator hS] |
| apply integral_congr_ae |
| filter_upwards [] with y |
| by_cases hy : y∈S <;> simp [hy] |
|
|
| lemma indicator_differential_integral {S : Set Plane} (hS : MeasurableSet S) |
| (U : Plane → Plane) (A : Plane → Plane →L[ℝ] Plane) (ψ : Plane → ℝ) : |
| (∫ y,fderiv ℝ ψ y (A y (S.indicator U y)))= |
| (∫ y in S,fderiv ℝ ψ y (A y (U y))) := by |
| rw [← integral_indicator hS] |
| apply integral_congr_ae |
| filter_upwards [] with y |
| by_cases hy : y∈S <;> simp [hy] |
|
|
| theorem WeakCMCGraph.scalar_linearized_equation {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)) |
| (hcmc : WeakCMCGraph g H (Metric.ball 0 R)) (v : Plane) (hv : ‖v‖≤1) : |
| ∃ (u : Lp ℝ 2 (volume : Measure Plane)) (U : Lp Plane 2 (volume : Measure Plane)), |
| (u : Plane → ℝ)=ᵐ[volume.restrict (Metric.ball 0 (R/4))] |
| (fun y ↦ fderiv ℝ g y v) ∧ |
| PlaneWeakGradient (Metric.ball 0 (R/4)) u U ∧ |
| (∀ ψ : Plane → ℝ, ContDiff ℝ (⊤ : ℕ∞) ψ → HasCompactSupport ψ → |
| tsupport ψ⊆Metric.ball 0 (R/4) → |
| (∫ y,fderiv ℝ ψ y (cmcJacobian (gradient g y) (U y)))=0) := by |
| let S : Set Plane := Metric.ball 0 (R/4) |
| obtain ⟨F,hF,hlin⟩ := hcmc.linearized_equation hg hK hR hgc v hv |
| have hFmem : MemLp (S.indicator (F : Plane → Plane)) 2 volume := |
| (memLp_indicator_iff_restrict measurableSet_ball).mpr (Lp.memLp F) |
| let U := hFmem.toLp (S.indicator (F : Plane → Plane)) |
| let b : ContDiffBump (0 : Plane) := ⟨R/4,R/2,by linarith,by linarith⟩ |
| let f : Plane → ℝ := fun y ↦ fderiv ℝ g y v*b y |
| have hbs : tsupport b⊆Metric.ball 0 R := by |
| rw [b.tsupport_eq] |
| exact Metric.closedBall_subset_ball (by dsimp [b]; linarith) |
| have hfc : Continuous f := continuous_local_scalar_product Metric.isOpen_ball |
| ((hgc.continuousOn_fderiv_of_isOpen Metric.isOpen_ball (by simp)).clm_apply continuousOn_const) |
| b.continuous hbs |
| have hfs : HasCompactSupport f := b.hasCompactSupport.mul_left |
| have hfmem : MemLp f 2 volume := hfc.memLp_of_hasCompactSupport hfs |
| let u := hfmem.toLp f |
| have hfe (y : Plane) (hy : y∈S) : f y=fderiv ℝ g y v := by |
| dsimp only [f] |
| rw [b.one_of_mem_closedBall (Metric.ball_subset_closedBall hy),mul_one] |
| refine ⟨u,U,?_,?_,?_⟩ |
| · filter_upwards [ae_restrict_of_ae hfmem.coeFn_toLp,ae_restrict_mem (measurableSet_ball : MeasurableSet S)] with y hy hs |
| exact hy.trans (hfe y hs) |
| · intro w ψ hψ hc hs |
| have hψs : tsupport ψ⊆Metric.ball 0 R := hs.trans (Metric.ball_subset_ball (by linarith)) |
| calc |
| _ = ∫ y,⟪S.indicator (F : Plane → Plane) y,w⟫*ψ y := by |
| apply integral_congr_ae |
| filter_upwards [hFmem.coeFn_toLp] with y hy |
| rw [hy] |
| _ = ∫ y in S,⟪F y,w⟫*ψ y := indicator_inner_test_integral measurableSet_ball _ _ _ |
| _ = -(∫ y,⟪gradient g y,w⟫*fderiv ℝ ψ y v) := hF w ψ (hψ.of_le (by simp)) hc hs |
| _ = -(∫ y,⟪gradient g y,v⟫*fderiv ℝ ψ y w) := by |
| rw [C1_mixed_test_identity Metric.isOpen_ball hg.continuous hgc hψ hc hψs v w] |
| _ = -(∫ y,u y*fderiv ℝ ψ y w) := by |
| congr 1 |
| apply integral_congr_ae |
| filter_upwards [hfmem.coeFn_toLp] with y hy |
| rw [hy,inner_gradient_left] |
| by_cases hyS : y∈S |
| · rw [hfe y hyS] |
| · rw [fderiv_of_notMem_tsupport ℝ (fun h ↦ hyS (hs h)),zero_apply,mul_zero,mul_zero] |
| · intro ψ hψ hc hs |
| calc |
| _ = ∫ y,fderiv ℝ ψ y (cmcJacobian (gradient g y) (S.indicator (F : Plane → Plane) y)) := by |
| apply integral_congr_ae |
| filter_upwards [hFmem.coeFn_toLp] with y hy |
| rw [hy] |
| _ = ∫ y in S,fderiv ℝ ψ y (cmcJacobian (gradient g y) (F y)) := |
| indicator_differential_integral measurableSet_ball _ _ _ |
| _ = 0 := hlin ψ (hψ.of_le (by simp)) hc hs |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem LinearCMCEquation.scalar_derivative {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) |
| (hAc : ContDiff ℝ 1 (fun y ↦ cmcJacobian (p y))) (hBc : ContDiff ℝ 1 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)) (v : Plane) (hv : ‖v‖≤1) : |
| ∃ (u : Lp ℝ 2 (volume : Measure Plane)) (U : Lp Plane 2 (volume : Measure Plane)), |
| (u : Plane → ℝ)=ᵐ[volume.restrict (Metric.ball 0 (R/4))] |
| (fun y ↦ fderiv ℝ g y v) ∧ |
| PlaneWeakGradient (Metric.ball 0 (R/4)) u U ∧ |
| (∀ ψ : Plane → ℝ, ContDiff ℝ (⊤ : ℕ∞) ψ → HasCompactSupport ψ → |
| tsupport ψ⊆Metric.ball 0 (R/4) → |
| (∫ y,fderiv ℝ ψ y (cmcJacobian (p y) (U y)- |
| (fderiv ℝ B y v-fderiv ℝ (fun z ↦ cmcJacobian (p z)) y v (gradient g y))))=0) := by |
| let S : Set Plane := Metric.ball 0 (R/4) |
| obtain ⟨F,hF,hlin⟩ := hw.weak_derivative hg hp hpb hA hB hAc hBc hBb hR hgc v hv |
| have hFmem : MemLp (S.indicator (F : Plane → Plane)) 2 volume := |
| (memLp_indicator_iff_restrict measurableSet_ball).mpr (Lp.memLp F) |
| let U := hFmem.toLp (S.indicator (F : Plane → Plane)) |
| let b : ContDiffBump (0 : Plane) := ⟨R/4,R/2,by linarith,by linarith⟩ |
| let f : Plane → ℝ := fun y ↦ fderiv ℝ g y v*b y |
| have hbs : tsupport b⊆Metric.ball 0 R := by |
| rw [b.tsupport_eq] |
| exact Metric.closedBall_subset_ball (by dsimp [b]; linarith) |
| have hfc : Continuous f := continuous_local_scalar_product Metric.isOpen_ball |
| ((hgc.continuousOn_fderiv_of_isOpen Metric.isOpen_ball (by simp)).clm_apply continuousOn_const) |
| b.continuous hbs |
| have hfs : HasCompactSupport f := b.hasCompactSupport.mul_left |
| have hfmem : MemLp f 2 volume := hfc.memLp_of_hasCompactSupport hfs |
| let u := hfmem.toLp f |
| have hfe (y : Plane) (hy : y∈S) : f y=fderiv ℝ g y v := by |
| dsimp only [f] |
| rw [b.one_of_mem_closedBall (Metric.ball_subset_closedBall hy),mul_one] |
| refine ⟨u,U,?_,?_,?_⟩ |
| · filter_upwards [ae_restrict_of_ae hfmem.coeFn_toLp,ae_restrict_mem (measurableSet_ball : MeasurableSet S)] with y hy hs |
| exact hy.trans (hfe y hs) |
| · intro w ψ hψ hc hs |
| have hψs : tsupport ψ⊆Metric.ball 0 R := hs.trans (Metric.ball_subset_ball (by linarith)) |
| calc |
| _ = ∫ y,⟪S.indicator (F : Plane → Plane) y,w⟫*ψ y := by |
| apply integral_congr_ae |
| filter_upwards [hFmem.coeFn_toLp] with y hy |
| rw [hy] |
| _ = ∫ y in S,⟪F y,w⟫*ψ y := indicator_inner_test_integral measurableSet_ball _ _ _ |
| _ = -(∫ y,⟪gradient g y,w⟫*fderiv ℝ ψ y v) := hF w ψ (hψ.of_le (by simp)) hc hs |
| _ = -(∫ y,⟪gradient g y,v⟫*fderiv ℝ ψ y w) := by |
| rw [C1_mixed_test_identity Metric.isOpen_ball hg.continuous hgc hψ hc hψs v w] |
| _ = -(∫ y,u y*fderiv ℝ ψ y w) := by |
| congr 1 |
| apply integral_congr_ae |
| filter_upwards [hfmem.coeFn_toLp] with y hy |
| rw [hy,inner_gradient_left] |
| by_cases hyS : y∈S |
| · rw [hfe y hyS] |
| · rw [fderiv_of_notMem_tsupport ℝ (fun h ↦ hyS (hs h)),zero_apply,mul_zero,mul_zero] |
| · intro ψ hψ hc hs |
| calc |
| _ = ∫ y in S,fderiv ℝ ψ y (cmcJacobian (p y) (U y)- |
| (fderiv ℝ B y v-fderiv ℝ (fun z ↦ cmcJacobian (p z)) y v (gradient g y))) := |
| (differential_integral_restrict hs _).symm |
| _ = ∫ y in S,fderiv ℝ ψ y (cmcJacobian (p y) (F y)- |
| (fderiv ℝ B y v-fderiv ℝ (fun z ↦ cmcJacobian (p z)) y v (gradient g y))) := by |
| apply integral_congr_ae |
| filter_upwards [ae_restrict_of_ae hFmem.coeFn_toLp, |
| ae_restrict_mem (measurableSet_ball : MeasurableSet S)] with y hy hyS |
| rw [hy,Set.indicator_of_mem hyS] |
| _ = 0 := hlin ψ (hψ.of_le (by simp)) hc hs |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma C1_gradient_add {f g : Plane → ℝ} (hf : ContDiff ℝ 1 f) (hg : ContDiff ℝ 1 g) : |
| gradient (f+g)=gradient f+gradient g := by |
| funext x |
| simp only [gradient,fderiv_add ((hf.differentiable (by simp)).differentiableAt) |
| ((hg.differentiable (by simp)).differentiableAt),map_add,Pi.add_apply] |
|
|
| lemma C1_gradient_smul {f : Plane → ℝ} (hf : ContDiff ℝ 1 f) (a : ℝ) : |
| gradient (a • f)=a • gradient f := by |
| funext x |
| simp only [gradient,fderiv_const_smul ((hf.differentiable (by simp)).differentiableAt), |
| map_smul,Pi.smul_apply] |
|
|
| lemma C1_gradient_sub {f g : Plane → ℝ} (hf : ContDiff ℝ 1 f) (hg : ContDiff ℝ 1 g) : |
| gradient (f-g)=gradient f-gradient g := by |
| funext x |
| simp only [gradient,fderiv_sub ((hf.differentiable (by simp)).differentiableAt) |
| ((hg.differentiable (by simp)).differentiableAt),map_sub,Pi.sub_apply] |
|
|
| def smoothGradientSpace (S : Set Plane) : Submodule ℝ (Lp Plane 2 (volume : Measure Plane)) where |
| carrier := {G | ∃ f : Plane → ℝ, ∃ hf : ContDiff ℝ (⊤ : ℕ∞) f, |
| ∃ hc : HasCompactSupport f, tsupport f⊆S ∧ |
| G=(compact_C1_gradient_memLp (hf.of_le (by simp)) hc).toLp (gradient f)} |
| zero_mem' := by |
| refine ⟨0,contDiff_const,HasCompactSupport.zero,by simp,?_⟩ |
| apply Lp.ext |
| filter_upwards [Lp.coeFn_zero (E := Plane) (p := 2) (μ := (volume : Measure Plane)), |
| (compact_C1_gradient_memLp (f := 0) contDiff_const HasCompactSupport.zero).coeFn_toLp] |
| with x hz hx |
| rw [hz,hx] |
| simp [gradient] |
| add_mem' := by |
| rintro G K ⟨f,hf,hc,hs,rfl⟩ ⟨g,hg,hgc,hgs,rfl⟩ |
| refine ⟨f+g,hf.add hg,hc.add hgc,(tsupport_add f g).trans (union_subset hs hgs),?_⟩ |
| have he := C1_gradient_add (hf.of_le (by simp)) (hg.of_le (by simp)) |
| apply Lp.ext |
| filter_upwards [Lp.coeFn_add |
| ((compact_C1_gradient_memLp (hf.of_le (by simp)) hc).toLp (gradient f)) |
| ((compact_C1_gradient_memLp (hg.of_le (by simp)) hgc).toLp (gradient g)), |
| (compact_C1_gradient_memLp (hf.of_le (by simp)) hc).coeFn_toLp, |
| (compact_C1_gradient_memLp (hg.of_le (by simp)) hgc).coeFn_toLp, |
| (compact_C1_gradient_memLp ((hf.add hg).of_le (by simp)) (hc.add hgc)).coeFn_toLp] |
| with x hx hfx hgx hfgx |
| rw [hx] |
| simp only [Pi.add_def] at hfgx ⊢ |
| rw [hfgx] |
| change _ + _ = gradient (f+g) x |
| rw [hfx,hgx,he] |
| rfl |
| smul_mem' := by |
| rintro a G ⟨f,hf,hc,hs,rfl⟩ |
| have hcf : HasCompactSupport (a • f) := hc.smul_left |
| refine ⟨a • f,contDiff_const.smul hf,hcf, |
| (tsupport_smul_subset_right (fun _ ↦ a) f).trans hs,?_⟩ |
| apply Lp.ext |
| filter_upwards [Lp.coeFn_smul a |
| ((compact_C1_gradient_memLp (hf.of_le (by simp)) hc).toLp (gradient f)), |
| (compact_C1_gradient_memLp (hf.of_le (by simp)) hc).coeFn_toLp, |
| (compact_C1_gradient_memLp ((contDiff_const.smul hf).of_le (by simp)) hcf).coeFn_toLp] |
| with x hx hfx hax |
| rw [hx] |
| simp only [Pi.smul_def,Pi.smul_def'] at hax ⊢ |
| rw [hax] |
| change a • _ = gradient (a • f) x |
| rw [hfx,C1_gradient_smul (hf.of_le (by simp)) a] |
| rfl |
|
|
| def zeroBoundaryGradientSpace (S : Set Plane) : Submodule ℝ (Lp Plane 2 (volume : Measure Plane)) := |
| (smoothGradientSpace S).topologicalClosure |
|
|
| lemma zeroBoundaryGradientSpace_isClosed (S : Set Plane) : |
| IsClosed (zeroBoundaryGradientSpace S : Set (Lp Plane 2 (volume : Measure Plane))) := |
| (smoothGradientSpace S).isClosed_topologicalClosure |
|
|
| lemma zeroBoundaryGradientSpace_approximation {S : Set Plane} |
| {G : Lp Plane 2 (volume : Measure Plane)} (hG : G∈zeroBoundaryGradientSpace S) : |
| ∃ f : ℕ → Plane → ℝ, ∃ hf : ∀ i, ContDiff ℝ (⊤ : ℕ∞) (f i), |
| ∃ hc : ∀ i, HasCompactSupport (f i), |
| (∀ i,tsupport (f i)⊆S) ∧ |
| Tendsto (fun i ↦ (compact_C1_gradient_memLp ((hf i).of_le (by simp)) (hc i)).toLp |
| (gradient (f i))) atTop (𝓝 G) := by |
| obtain ⟨F,hF,hlim⟩ := mem_closure_iff_seq_limit.mp hG |
| choose f hf hc hs he using hF |
| refine ⟨f,hf,hc,hs,?_⟩ |
| convert hlim using 1 |
| funext i |
| exact (he i).symm |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma compact_test_poincare_difference {R C : ℝ} |
| (hP : ∀ f : Plane → ℝ, ∀ hf : ContDiff ℝ 1 f, ∀ hc : HasCompactSupport f, |
| tsupport f⊆Metric.ball 0 R → |
| ‖(hf.continuous.memLp_of_hasCompactSupport hc : MemLp f 2).toLp f‖≤ |
| C*‖(compact_C1_gradient_memLp hf hc).toLp (gradient f)‖) |
| {f g : Plane → ℝ} (hf : ContDiff ℝ 1 f) (hg : ContDiff ℝ 1 g) |
| (hc : HasCompactSupport f) (hgc : HasCompactSupport g) |
| (hs : tsupport f⊆Metric.ball 0 R) (hgs : tsupport g⊆Metric.ball 0 R) : |
| dist ((hf.continuous.memLp_of_hasCompactSupport hc : MemLp f 2).toLp f) |
| ((hg.continuous.memLp_of_hasCompactSupport hgc : MemLp g 2).toLp g)≤ |
| C*dist ((compact_C1_gradient_memLp hf hc).toLp (gradient f)) |
| ((compact_C1_gradient_memLp hg hgc).toLp (gradient g)) := by |
| have hp := hP (f-g) (hf.sub hg) (hc.sub hgc) |
| ((tsupport_sub f g).trans (union_subset hs hgs)) |
| rw [Lp.norm_toLp,Lp.norm_toLp,C1_gradient_sub hf hg] at hp |
| simpa only [Lp.dist_edist,Lp.edist_toLp_toLp] using hp |
|
|
| lemma cauchySeq_of_distance_comparison {X Y : Type*} [PseudoMetricSpace X] [PseudoMetricSpace Y] |
| {f : ℕ → X} {g : ℕ → Y} {C : ℝ} (hC : 0≤C) |
| (hg : CauchySeq g) (hbound : ∀ i j,dist (f i) (f j)≤C*dist (g i) (g j)) : |
| CauchySeq f := by |
| rw [Metric.cauchySeq_iff] at hg ⊢ |
| intro ε hε |
| have hCp : 0<C+1 := by linarith |
| obtain ⟨N,hN⟩ := hg (ε/(C+1)) (div_pos hε hCp) |
| refine ⟨N,fun i hi j hj ↦ ?_⟩ |
| calc |
| dist (f i) (f j) ≤ C*dist (g i) (g j) := hbound i j |
| _ ≤ (C+1)*dist (g i) (g j) := by nlinarith [(dist_nonneg : 0≤dist (g i) (g j))] |
| _ < ε := (lt_div_iff₀' hCp).mp (hN i hi j hj) |
|
|
| theorem zeroBoundaryGradientSpace_primitive {R : ℝ} |
| {G : Lp Plane 2 (volume : Measure Plane)} |
| (hG : G∈zeroBoundaryGradientSpace (Metric.ball 0 R)) : |
| ∃ u : Lp ℝ 2 (volume : Measure Plane), PlaneWeakGradient univ u G ∧ |
| ∃ f : ℕ → Plane → ℝ, ∃ hf : ∀ i,ContDiff ℝ (⊤ : ℕ∞) (f i), |
| ∃ hc : ∀ i,HasCompactSupport (f i), (∀ i,tsupport (f i)⊆Metric.ball 0 R) ∧ |
| Tendsto (fun i ↦ ((hf i).continuous.memLp_of_hasCompactSupport (hc i) : |
| MemLp (f i) 2 volume).toLp (f i)) atTop (𝓝 u) ∧ |
| Tendsto (fun i ↦ (compact_C1_gradient_memLp ((hf i).of_le (by simp)) (hc i)).toLp |
| (gradient (f i))) atTop (𝓝 G) := by |
| obtain ⟨C,hC,hP⟩ := compact_test_poincare R |
| obtain ⟨f,hf,hc,hs,hlim⟩ := zeroBoundaryGradientSpace_approximation hG |
| let U (i : ℕ) : Lp ℝ 2 (volume : Measure Plane) := |
| ((hf i).continuous.memLp_of_hasCompactSupport (hc i) : MemLp (f i) 2 volume).toLp (f i) |
| let F (i : ℕ) : Lp Plane 2 (volume : Measure Plane) := |
| (compact_C1_gradient_memLp ((hf i).of_le (by simp)) (hc i)).toLp (gradient (f i)) |
| have hbound (i j : ℕ) : dist (U i) (U j)≤C*dist (F i) (F j) := |
| compact_test_poincare_difference hP ((hf i).of_le (by simp)) ((hf j).of_le (by simp)) |
| (hc i) (hc j) (hs i) (hs j) |
| obtain ⟨u,hu⟩ := cauchySeq_tendsto_of_complete |
| (cauchySeq_of_distance_comparison hC hlim.cauchySeq hbound) |
| refine ⟨u,?_,f,hf,hc,hs,hu,hlim⟩ |
| apply PlaneWeakGradient.L2_limit hu hlim |
| intro i |
| apply (compact_C1_weak_gradient ((hf i).of_le (by simp)) (hc i) univ).congr |
| · exact ((hf i).continuous.memLp_of_hasCompactSupport (hc i) : MemLp (f i) 2 volume).coeFn_toLp |
| · exact (compact_C1_gradient_memLp ((hf i).of_le (by simp)) (hc i)).coeFn_toLp |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem plane_harmonic_replacement (R : ℝ) |
| (G : Lp Plane 2 (volume : Measure Plane)) : |
| ∃ W : Lp Plane 2 (volume : Measure Plane), |
| W∈zeroBoundaryGradientSpace (Metric.ball 0 R) ∧ |
| ‖W‖≤‖G‖ ∧ |
| (∀ F∈zeroBoundaryGradientSpace (Metric.ball 0 R), ⟪G-W,F⟫=0) ∧ |
| ∃ w : Lp ℝ 2 (volume : Measure Plane), PlaneWeakGradient univ w W := by |
| let K := zeroBoundaryGradientSpace (Metric.ball (0 : Plane) R) |
| let : IsClosed (K : Set (Lp Plane 2 (volume : Measure Plane))) := |
| zeroBoundaryGradientSpace_isClosed _ |
| let W := K.starProjection G |
| have hW : W∈K := K.starProjection_apply_mem G |
| obtain ⟨w,hw,_⟩ := zeroBoundaryGradientSpace_primitive hW |
| exact ⟨W,hW,K.norm_starProjection_apply_le G, |
| fun F hF ↦ K.starProjection_inner_eq_zero G F hF,w,hw⟩ |
|
|
| lemma L2_pair_gradient_test {G : Lp Plane 2 (volume : Measure Plane)} |
| {ψ : Plane → ℝ} (hψ : ContDiff ℝ 1 ψ) (hc : HasCompactSupport ψ) : |
| ⟪G,(compact_C1_gradient_memLp hψ hc).toLp (gradient ψ)⟫= |
| ∫ y, fderiv ℝ ψ y (G y) := by |
| rw [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [(compact_C1_gradient_memLp hψ hc).coeFn_toLp] with y hy |
| simp only [hy,inner_gradient_right,conj_trivial] |
|
|
| lemma harmonic_replacement_test {R : ℝ} {G W : Lp Plane 2 (volume : Measure Plane)} |
| (ho : ∀ F∈zeroBoundaryGradientSpace (Metric.ball 0 R), ⟪G-W,F⟫=0) |
| {ψ : Plane → ℝ} (hψ : ContDiff ℝ (⊤ : ℕ∞) ψ) (hc : HasCompactSupport ψ) |
| (hs : tsupport ψ⊆Metric.ball 0 R) : |
| (∫ y, fderiv ℝ ψ y ((G-W) y))=0 := by |
| rw [← L2_pair_gradient_test (hψ.of_le (by simp)) hc] |
| apply ho |
| exact (smoothGradientSpace (Metric.ball 0 R)).le_topologicalClosure |
| ⟨ψ,hψ,hc,hs,rfl⟩ |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| open Laplacian |
| namespace CubicTorus |
|
|
| lemma L2_scalar_test_integrable (u : Lp ℝ 2 (volume : Measure Plane)) |
| {ψ : Plane → ℝ} (hψ : MemLp ψ 2 volume) : Integrable (fun y ↦ u y*ψ y) := by |
| apply (L2.integrable_inner (𝕜 := ℝ) u (hψ.toLp ψ)).congr |
| filter_upwards [hψ.coeFn_toLp] with y hy |
| simp only [hy,RCLike.inner_apply,conj_trivial] |
| ring |
|
|
| lemma L2_vector_test_integrable (G : Lp Plane 2 (volume : Measure Plane)) |
| {F : Plane → Plane} (hF : MemLp F 2 volume) : Integrable (fun y ↦ ⟪G y,F y⟫) := by |
| apply (L2.integrable_inner (𝕜 := ℝ) G (hF.toLp F)).congr |
| filter_upwards [hF.coeFn_toLp] with y hy |
| exact congrArg (inner ℝ (G y)) hy |
|
|
| lemma PlaneWeakGradient.Lp_sub {S : Set Plane} |
| {u w : Lp ℝ 2 (volume : Measure Plane)} {G W : Lp Plane 2 (volume : Measure Plane)} |
| (hu : PlaneWeakGradient S u G) (hw : PlaneWeakGradient S w W) : |
| PlaneWeakGradient S ((u-w : Lp ℝ 2 volume) : Plane → ℝ) |
| ((G-W : Lp Plane 2 volume) : Plane → Plane) := by |
| intro v ψ hψ hc hs |
| let a : Plane → Plane := fun y ↦ ψ y • v |
| have ha : MemLp a 2 volume := (hψ.continuous.smul continuous_const).memLp_of_hasCompactSupport hc.smul_right |
| let A := ha.toLp a |
| let b : Plane → ℝ := fun y ↦ fderiv ℝ ψ y v |
| have hb : MemLp b 2 volume := ((hψ.continuous_fderiv (by simp)).clm_apply continuous_const).memLp_of_hasCompactSupport |
| (hc.fderiv_apply ℝ v) |
| let B := hb.toLp b |
| have h₁ : ⟪G,A⟫= -⟪u,B⟫ := by |
| rw [L2_pair_vector_test hψ.continuous hc v,L2_pair_scalar_test hb] |
| exact hu v ψ hψ hc hs |
| have h₂ : ⟪W,A⟫= -⟪w,B⟫ := by |
| rw [L2_pair_vector_test hψ.continuous hc v,L2_pair_scalar_test hb] |
| exact hw v ψ hψ hc hs |
| rw [← L2_pair_vector_test hψ.continuous hc v,← L2_pair_scalar_test hb] |
| change ⟪G-W,A⟫= -⟪u-w,B⟫ |
| rw [inner_sub_left,inner_sub_left,h₁,h₂] |
| ring |
|
|
| end CubicTorus |
| end |
| end |
| end |
|
|
| end OAI |
|
|
