| import OAI.Geometry.IsometricImmersion.Caps.CapJacobian |
| import OAI.Geometry.IsometricImmersion.Energy.RegionSobolev |
| import OAI.Geometry.IsometricImmersion.Caps.EllipticGradient |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Function MeasureTheory |
| open scoped ContDiff Topology Interval |
|
|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Sobolev |
|
|
| theorem capRectangle_isCompact (tl tr sb st : ℝ) : IsCompact (closedRectangle tl tr sb st) := by |
| have he : closedRectangle tl tr sb st = |
| (fun q : ℝ × ℝ => boxPoint q.1 q.2) '' (Icc tl tr ×ˢ Icc sb st) := by |
| ext p |
| constructor |
| · intro hp |
| exact ⟨(p 0,p 1),⟨hp.1,hp.2⟩,boxPoint_eta p⟩ |
| · rintro ⟨q,hq,rfl⟩ |
| exact boxPoint_mem hq.1 hq.2 |
| rw [he] |
| exact (isCompact_Icc.prod isCompact_Icc).image (by unfold boxPoint; fun_prop) |
|
|
| theorem rectangleIntegral_eq_coordinate_closed_area |
| {tl tr sb st : ℝ} {F : Coord → ℝ} (ht : tl ≤ tr) (hs : sb ≤ st) |
| (hF : ContinuousOn F (closedRectangle tl tr sb st)) : |
| rectangleIntegral tl tr sb st F = ∫ p in closedRectangle tl tr sb st, F p := by |
| rw [rectangleIntegral_eq_closed_area ht hs hF] |
| have hh := (volume_preserving_finTwoArrow ℝ).setIntegral_preimage_emb |
| (MeasurableEquiv.finTwoArrow (α := ℝ)).measurableEmbedding |
| (fun q : ℝ × ℝ => F (boxPoint q.1 q.2)) (Icc tl tr ×ˢ Icc sb st) |
| change (∫ p in closedRectangle tl tr sb st, F (boxPoint (p 0) (p 1))) = |
| ∫ q in Icc tl tr ×ˢ Icc sb st, F (boxPoint q.1 q.2) at hh |
| simpa only [boxPoint_eta] using hh.symm |
|
|
| theorem cap_image_integral_formula |
| {Y : ℝ → ℝ → ℝ} {e : OpenPartialHomeomorph (ℝ × ℝ) (ℝ × ℝ)} |
| (hYs : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2)) |
| (hvar : ∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2, 0 < deriv (fun r => Y r t) s) |
| (hsource : e.source = pairRectangle 2 (-2) 2) |
| (heq : (e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y) |
| {tl tr sb st : ℝ} (ht : tl ≤ tr) (hs : sb ≤ st) |
| (hbox : closedRectangle tl tr sb st ⊆ capChartDomain) |
| {f : Coord → ℝ} (hf : ContinuousOn f (capChart Y '' closedRectangle tl tr sb st)) : |
| (∫ p in capChart Y '' closedRectangle tl tr sb st, f p) = |
| rectangleIntegral tl tr sb st (fun p => coordPartial 1 (capFlowHeight Y) p*capPullback Y f p) := by |
| have hrect : MeasurableSet (closedRectangle tl tr sb st) := (capRectangle_isCompact _ _ _ _).isClosed.measurableSet |
| have hchart := capChart_contDiffOn hYs |
| have hd (p : Coord) (hp : p ∈ closedRectangle tl tr sb st) : |
| HasFDerivWithinAt (capChart Y) (fderiv ℝ (capChart Y) p) (closedRectangle tl tr sb st) p := |
| ((hchart.contDiffAt (capChartDomain_isOpen.mem_nhds (hbox hp))).differentiableAt (by simp)).hasFDerivAt.hasFDerivWithinAt |
| have hinj := (capChart_injOn_of_pairChart hsource heq).mono hbox |
| have hcomp : ContinuousOn (capPullback Y f) (closedRectangle tl tr sb st) := |
| hf.comp (hchart.continuousOn.mono hbox) (fun p hp => ⟨p,hp,rfl⟩) |
| have hJ : ContinuousOn (coordPartial 1 (capFlowHeight Y)) (closedRectangle tl tr sb st) := |
| (partial_contDiffOn (capFlowHeight_contDiffOn hYs) capChartDomain_isOpen 1).continuousOn.mono hbox |
| calc |
| _ = ∫ p in closedRectangle tl tr sb st, |(fderiv ℝ (capChart Y) p).det| • f (capChart Y p) := |
| integral_image_eq_integral_abs_det_fderiv_smul volume hrect hd hinj f |
| _ = ∫ p in closedRectangle tl tr sb st, coordPartial 1 (capFlowHeight Y) p*capPullback Y f p := by |
| apply setIntegral_congr_fun hrect |
| intro p hp |
| change |(fderiv ℝ (capChart Y) p).det| * f (capChart Y p) = _ |
| rw [capChart_abs_det_fderiv hYs hvar (hbox hp)] |
| rfl |
| _ = _ := (rectangleIntegral_eq_coordinate_closed_area ht hs (hJ.mul hcomp)).symm |
|
|
| theorem cap_rectangle_square_integral_bounds |
| {Y : ℝ → ℝ → ℝ} {e : OpenPartialHomeomorph (ℝ × ℝ) (ℝ × ℝ)} |
| (hYs : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2)) |
| (hvar : ∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2, 0 < deriv (fun r => Y r t) s) |
| (hsource : e.source = pairRectangle 2 (-2) 2) |
| (heq : (e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y) |
| {tl tr sb st M : ℝ} (ht : tl ≤ tr) (hs : sb ≤ st) |
| (hbox : closedRectangle tl tr sb st ⊆ capChartDomain) |
| (hJ : ∀ p ∈ closedRectangle tl tr sb st, |
| Real.exp (-2*M) ≤ coordPartial 1 (capFlowHeight Y) p ∧ |
| coordPartial 1 (capFlowHeight Y) p ≤ Real.exp (2*M)) |
| {f : Coord → ℝ} (hf : ContinuousOn f (capChart Y '' closedRectangle tl tr sb st)) : |
| Real.exp (-2*M)*rectangleIntegral tl tr sb st (fun p => (capPullback Y f p)^2) ≤ |
| ∫ p in capChart Y '' closedRectangle tl tr sb st, (f p)^2 ∧ |
| (∫ p in capChart Y '' closedRectangle tl tr sb st, (f p)^2) ≤ |
| Real.exp (2*M)*rectangleIntegral tl tr sb st (fun p => (capPullback Y f p)^2) := by |
| have hchart := capChart_contDiffOn hYs |
| have hfc : ContinuousOn (fun p => (capPullback Y f p)^2) (closedRectangle tl tr sb st) := |
| (hf.comp (hchart.continuousOn.mono hbox) (fun p hp => ⟨p,hp,rfl⟩)).pow 2 |
| have hJc : ContinuousOn (coordPartial 1 (capFlowHeight Y)) (closedRectangle tl tr sb st) := |
| (partial_contDiffOn (capFlowHeight_contDiffOn hYs) capChartDomain_isOpen 1).continuousOn.mono hbox |
| have hformula := cap_image_integral_formula hYs hvar hsource heq ht hs hbox (hf.pow 2) |
| change (∫ p in capChart Y '' closedRectangle tl tr sb st, (f p)^2) = |
| rectangleIntegral tl tr sb st (fun p => coordPartial 1 (capFlowHeight Y) p*(capPullback Y f p)^2) at hformula |
| rw [hformula] |
| constructor |
| · have hh := rectangleIntegral_mono ht hs (continuousOn_const.mul hfc) (hJc.mul hfc) |
| (fun p hp => mul_le_mul_of_nonneg_right (hJ p hp).1 (sq_nonneg _)) |
| simpa only [Pi.mul_def, rectangleIntegral_const_mul] using hh |
| · have hh := rectangleIntegral_mono ht hs (hJc.mul hfc) (continuousOn_const.mul hfc) |
| (fun p hp => mul_le_mul_of_nonneg_right (hJ p hp).2 (sq_nonneg _)) |
| simpa only [Pi.mul_def, rectangleIntegral_const_mul] using hh |
|
|
| theorem cap_pullback_square_integral_le_image |
| {Y : ℝ → ℝ → ℝ} {e : OpenPartialHomeomorph (ℝ × ℝ) (ℝ × ℝ)} |
| (hYs : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2)) |
| (hvar : ∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2, 0 < deriv (fun r => Y r t) s) |
| (hsource : e.source = pairRectangle 2 (-2) 2) |
| (heq : (e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y) |
| {tl tr sb st M : ℝ} (ht : tl ≤ tr) (hs : sb ≤ st) |
| (hbox : closedRectangle tl tr sb st ⊆ capChartDomain) |
| (hJ : ∀ p ∈ closedRectangle tl tr sb st, |
| Real.exp (-2*M) ≤ coordPartial 1 (capFlowHeight Y) p ∧ |
| coordPartial 1 (capFlowHeight Y) p ≤ Real.exp (2*M)) |
| {f : Coord → ℝ} (hf : ContinuousOn f (capChart Y '' closedRectangle tl tr sb st)) : |
| rectangleIntegral tl tr sb st (fun p => (capPullback Y f p)^2) ≤ |
| Real.exp (2*M)*(∫ p in capChart Y '' closedRectangle tl tr sb st, (f p)^2) := by |
| have hh := (cap_rectangle_square_integral_bounds hYs hvar hsource heq ht hs hbox hJ hf).1 |
| have hscale : Real.exp (2*M)*Real.exp (-2*M) = 1 := by rw [←Real.exp_add]; simp |
| calc |
| _ = Real.exp (2*M)*(Real.exp (-2*M)*rectangleIntegral tl tr sb st (fun p => (capPullback Y f p)^2)) := by |
| rw [←mul_assoc,hscale,one_mul] |
| _ ≤ _ := mul_le_mul_of_nonneg_left hh (Real.exp_pos _).le |
|
|
| end SmoothLocal.Flow |
|
|
| end |
|
|
| end OAI |
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|
