| import OAI.Geometry.IsometricImmersion.Caps.CapCutoffRectangle |
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| namespace OAI |
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| noncomputable section |
| open Set Filter |
| open scoped ContDiff Topology |
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| namespace SmoothLocal.Weighted |
| open SmoothLocal.Geometry |
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|
| def capDistanceBound (A b : ℝ) : ℝ := b-capOuterBottom A |
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| def capOverlapEnergyCost (L R A b kappa c1 c2 glow M lambda MI : ℝ) : ℝ := |
| let D := capDistanceBound A b |
| let c := capOverlapThreshold kappa c1 D |
| (directedFluxCostConstant D M lambda 2 MI (capSpatialDerivativeConstant L R A) + |
| directedDefectCostConstant D M lambda 2 MI c2)*(D+1/(glow*c)) |
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| def capEllipticRatioBudget (L R A b kappa c1 glow MK M : ℝ) : ℝ := |
| let D := capDistanceBound A b |
| let c := capOverlapThreshold kappa c1 D |
| ellipticRatioConstant D (glow*c/2) |
| (ellipticCutoffDerivativeConstant c MK |
| (threeEdgeDerivativeConstant (capOuterLeft L) (capOuterLeftOne L) (capOuterRightOne R) |
| (capOuterRight R) (capOuterBottom A) (capOuterBottomOne A))) M |
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| theorem directedForcingEnergy_bound |
| {chi I f : Coord → ℝ} {p : Coord} {b lambda D S MI : ℝ} |
| (hlambda : 0 ≤ lambda) (hd : 0 ≤ edgeDistance b p) (hdD : edgeDistance b p ≤ D) |
| (hs : |p 1| ≤ S) (hI : |I p| ≤ MI) (hchi : 0 ≤ chi p ∧ chi p ≤ 1) : |
| directedForcingEnergy chi I f b lambda p ≤ D^8*Real.exp (lambda*S+MI)*(f p)^2 := by |
| have hD : 0 ≤ D := hd.trans hdD |
| have hW0 : 0 ≤ directedWeight b lambda I p := by unfold directedWeight; positivity |
| have hW7 := directedWeight_le_seven hlambda hd hdD hs hI |
| have hdpow := pow_le_pow_left₀ hd hdD 7 |
| have hW : directedWeight b lambda I p ≤ D^8*Real.exp (lambda*S+MI) := by |
| calc |
| _ ≤ D*Real.exp (lambda*S+MI)*edgeDistance b p^7 := hW7 |
| _ ≤ D*Real.exp (lambda*S+MI)*D^7 := |
| mul_le_mul_of_nonneg_left hdpow (by positivity) |
| _ = _ := by ring |
| have hchiW := mul_le_mul hchi.2 hW hW0 (by norm_num : (0 : ℝ) ≤ 1) |
| simpa only [one_mul, directedForcingEnergy] using mul_le_mul_of_nonneg_right hchiW (sq_nonneg (f p)) |
|
|
| end SmoothLocal.Weighted |
|
|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.Model |
|
|
| theorem actual_cap_integrated_overlap_estimate |
| {g0 eta : MetricField} {U : Set Coord} {Y : ℝ → ℝ → ℝ} |
| {L R A0 b kappa c1 c2 glow lambda epsilon MI M MK : ℝ} |
| {G1 B C I u f : Coord → ℝ} |
| (hL : -2 < L) (hR : R < 2) (hA0 : -2 < A0) (hAb : A0 ≤ b) (hb : b ≤ 0) |
| (hk : 0 < kappa) (hc1 : 0 < c1) (hc2 : 0 < c2) (hg : 0 < glow) |
| (hM : 0 ≤ M) (hMK : 0 ≤ MK) (hlambda : 0 ≤ lambda) |
| (heps : 0 ≤ epsilon) (heps1 : epsilon ≤ 1) |
| (hK : ContDiffOn ℝ ∞ (capPullback Y (gaussianCurvature (g0+eta))) capChartDomain) |
| (hG1 : ContDiffOn ℝ ∞ G1 capChartDomain) (hB : ContDiffOn ℝ ∞ B capChartDomain) |
| (hC : ContDiffOn ℝ ∞ C capChartDomain) (hI : ContDiffOn ℝ ∞ I capChartDomain) |
| (hu : ContDiffOn ℝ ∞ u capChartDomain) |
| (hEq : ∀ p ∈ capChartDomain, |
| multiplierOperator (fun q => G1 q*capPullback Y (gaussianCurvature (g0+eta)) q) B C u p = f p) |
| (hbackground : ∀ q ∈ U, gaussianCurvature g0 q = modelCurvature kappa q) |
| (hsupport : tsupport eta ⊆ patchBox) |
| (himage : ∀ p ∈ closedRectangle (capOuterLeft L) (capOuterRight R) (capOuterBottom A0) b, |
| capChart Y p ∈ U) |
| (hdisp : ∀ p ∈ closedRectangle (capOuterLeft L) (capOuterRight R) (capOuterBottom A0) b, |
| |capFlowHeight Y p-p 1| ≤ (1 : ℝ)/50) |
| (hIv : ∀ p ∈ closedRectangle (capOuterLeft L) (capOuterRight R) (capOuterBottom A0) b, |I p| ≤ MI) |
| (hIi : ∀ p ∈ closedRectangle (capOuterLeft L) (capOuterRight R) (capOuterBottom A0) b, |
| ∀ i : Fin 2, |coordPartial i I p| ≤ M) |
| (hGlow : ∀ p ∈ closedRectangle (capOuterLeft L) (capOuterRight R) (capOuterBottom A0) b, glow ≤ G1 p) |
| (hAv : ∀ p ∈ closedRectangle (capOuterLeft L) (capOuterRight R) (capOuterBottom A0) b, |
| |G1 p*capPullback Y (gaussianCurvature (g0+eta)) p| ≤ M) |
| (hAi : ∀ p ∈ closedRectangle (capOuterLeft L) (capOuterRight R) (capOuterBottom A0) b, |
| ∀ i : Fin 2, |coordPartial i (fun q => G1 q*capPullback Y (gaussianCurvature (g0+eta)) q) p| ≤ M) |
| (hBv : ∀ p ∈ closedRectangle (capOuterLeft L) (capOuterRight R) (capOuterBottom A0) b, |B p| ≤ M) |
| (hCv : ∀ p ∈ closedRectangle (capOuterLeft L) (capOuterRight R) (capOuterBottom A0) b, |C p| ≤ M) |
| (hKi : ∀ p ∈ closedRectangle (capOuterLeft L) (capOuterRight R) (capOuterBottom A0) b, |
| ∀ i : Fin 2, |coordPartial i (capPullback Y (gaussianCurvature (g0+eta))) p| ≤ MK) |
| (hcoercive : ∀ p ∈ closedRectangle (capOuterLeft L) (capOuterRight R) (capOuterBottom A0) b, |
| capPullback Y (gaussianCurvature (g0+eta)) p ≤ c1*edgeDistance b p → |
| c2*directedWeight b lambda I p*((coordPartial 0 u p)^2+(coordPartial 1 u p)^2) ≤ |
| multiplierQuadratic (fun q => G1 q*capPullback Y (gaussianCurvature (g0+eta)) q) B C |
| (directedM b lambda I) (directedN b lambda epsilon I) u p) : |
| let D := capDistanceBound A0 b |
| let COverlap := capOverlapEnergyCost L R A0 b kappa c1 c2 glow M lambda MI |
| let HElliptic := capEllipticRatioBudget L R A0 b kappa c1 glow MK M |
| (c2/2)*rectangleIntegral (capOuterLeft L) (capOuterRight R) (capOuterBottom A0) b |
| (directedGradientEnergy (capSpatialCutoff L R A0) I u b lambda) ≤ |
| ((5/(2*c2))*(D^8*Real.exp (lambda*2+MI))+COverlap*D^6)* |
| rectangleIntegral (capOuterLeft L) (capOuterRight R) (capOuterBottom A0) b (fun p => (f p)^2) + |
| (COverlap*(D^6+HElliptic))* |
| rectangleIntegral (capOuterLeft L) (capOuterRight R) (capOuterBottom A0) b (fun p => (u p)^2) := by |
| let tl := capOuterLeft L |
| let tr := capOuterRight R |
| let sb := capOuterBottom A0 |
| let D := capDistanceBound A0 b |
| let c := capOverlapThreshold kappa c1 D |
| let A := fun q => G1 q*capPullback Y (gaussianCurvature (g0+eta)) q |
| let K := capPullback Y (gaussianCurvature (g0+eta)) |
| let chi := capSpatialCutoff L R A0 |
| let chi2 := capOuterSpatialCutoff L R A0 |
| let v := ellipticWeightC1 b c chi2 K |
| let E := ellipticEnergy A v u |
| let Err := directedCutoffError A chi I u b lambda epsilon |
| let Def := directedCoercivityDefect A B C chi I u b lambda epsilon c2 |
| let COverlap := capOverlapEnergyCost L R A0 b kappa c1 c2 glow M lambda MI |
| let HElliptic := capEllipticRatioBudget L R A0 b kappa c1 glow MK M |
| obtain ⟨htlo,htwidth,htone,hrone,hrwidth,htrhi,hsblo,hswidth,hsone,hlt,hrt⟩ := |
| capOuterRectangle_edges hL hR hA0 |
| have ht : tl ≤ tr := by dsimp only [tl,tr]; linarith |
| have hs : sb ≤ b := by dsimp only [sb]; linarith |
| have hD : 0 ≤ D := sub_nonneg.mpr hs |
| have hc : 0 < c := capOverlapThreshold_pos hk hc1 hD |
| have hbox : closedRectangle tl tr sb b ⊆ capChartDomain := |
| capOuterRectangle_subset_domain hL hR hA0 (by linarith) |
| have hA : ContDiffOn ℝ ∞ A capChartDomain := hG1.mul hK |
| have hchi : ContDiffOn ℝ ∞ chi capChartDomain := (capSpatialCutoff_contDiff _ _ _).contDiffOn |
| have hchi2 : ContDiffOn ℝ ∞ chi2 capChartDomain := (capOuterSpatialCutoff_contDiff _ _ _).contDiffOn |
| have hv : ContDiffOn ℝ 1 v capChartDomain := ellipticWeightC1_contDiffOn hc |
| (hchi2.of_le (by simp)) (hK.of_le (by simp)) |
| have hW := directedWeight_contDiffOn b lambda hI |
| have hm := hW.neg |
| have hn : ContDiffOn ℝ ∞ (directedN b lambda epsilon I) capChartDomain := |
| (contDiffOn_const.mul (contDiffOn_apply ℝ ℝ 0 _)).mul hW |
| have hut := partial_contDiffOn hu capChartDomain_isOpen 0 |
| have hus := partial_contDiffOn hu capChartDomain_isOpen 1 |
| have hgrad := (hut.pow 2).add (hus.pow 2) |
| have hQ := multiplierQuadratic_contDiffOn capChartDomain_isOpen hA hB hC hm hn hu |
| have hErr : ContinuousOn Err (closedRectangle tl tr sb b) := |
| (multiplierCutoffError_contDiffOn capChartDomain_isOpen hA hchi hm hn hu).continuousOn.mono hbox |
| have hDef : ContinuousOn Def (closedRectangle tl tr sb b) := |
| (hchi.continuousOn.mul ((continuousOn_const (c := (0 : ℝ))).sup |
| ((((contDiffOn_const (c := c2)).mul hW).mul hgrad).sub hQ).continuousOn)).mono hbox |
| obtain ⟨_,_,hEnergy,_⟩ := elliptic_fields_C1 capChartDomain_isOpen hA hB hC hv hu |
| have hE : ContinuousOn E (closedRectangle tl tr sb b) := hEnergy.mono hbox |
| have hX : 0 ≤ capSpatialDerivativeConstant L R A0 := |
| (abs_nonneg _).trans (capSpatialCutoff_partial_uniform hL hR hA0 (boxPoint 0 0) 0) |
| have hconst := directedCostConstants_nonneg (S := 2) (MI := MI) hD hM hlambda hX hc2.le |
| have hOverlap : 0 ≤ COverlap := by |
| dsimp only [COverlap,capOverlapEnergyCost] |
| exact mul_nonneg (add_nonneg hconst.1 hconst.2) (by positivity) |
| have hpoint (p : Coord) (hp : p ∈ closedRectangle tl tr sb b) : |Err p|+Def p ≤ COverlap*E p := by |
| have hpD := hbox hp |
| exact actual_cap_directed_costs_le_elliptic hL hR hA0 hk hc1 hc2.le hg hD hM hlambda heps heps1 |
| hbackground hsupport (himage p hp) (hdisp p hp) hp.2.2 |
| (by dsimp only [edgeDistance,D,capDistanceBound]; linarith [hp.2.1]) |
| (abs_lt.mpr hpD.1).le (abs_lt.mpr hpD.2).le (hIv p hp) |
| ((hI.contDiffAt (capChartDomain_isOpen.mem_nhds hpD)).differentiableAt (by simp)) (hIi p hp) |
| (hGlow p hp) ((hA.contDiffAt (capChartDomain_isOpen.mem_nhds hpD)).differentiableAt (by simp)) |
| (hAv p hp) (hAi p hp) (hBv p hp) (hCv p hp) (hcoercive p hp) |
| have hcostIntegral := rectangleIntegral_mono ht hs (hErr.abs.add hDef) (continuousOn_const.mul hE) hpoint |
| change rectangleIntegral tl tr sb b (fun p => |Err p| + Def p) ≤ |
| rectangleIntegral tl tr sb b (fun p => COverlap * E p) at hcostIntegral |
| rw [rectangleIntegral_add ht hs hErr.abs hDef, rectangleIntegral_const_mul] at hcostIntegral |
| have hEllRaw := threeEdge_elliptic_energy_bound htwidth hrwidth hswidth ht hs hc hMK hg hM |
| capChartDomain_isOpen hK hG1 hB hC hu hbox hEq hKi hGlow hAv (fun p hp => hAi p hp 1) hBv hCv |
| have hEll : rectangleIntegral tl tr sb b E ≤ |
| D^6*rectangleIntegral tl tr sb b (fun p => (f p)^2) + |
| (D^6+HElliptic)*rectangleIntegral tl tr sb b (fun p => (u p)^2) := by |
| calc |
| _ = rectangleIntegral tl tr sb b |
| (ellipticEnergy A (ellipticWeight b c |
| (threeEdgeCutoff (capOuterLeft L) (capOuterLeftOne L) (capOuterRightOne R) |
| (capOuterRight R) (capOuterBottom A0) (capOuterBottomOne A0)) K) u) := by |
| apply rectangleIntegral_congr ht hs |
| intro p hp |
| change ellipticWeightC1 b c chi2 K p * _ = _ |
| rw [ellipticWeightC1_eq hp.2.2] |
| change ellipticWeight b c (capOuterSpatialCutoff L R A0) K p * _ = _ |
| rw [capOuterSpatialCutoff_eq_threeEdge] |
| rfl |
| _ ≤ _ := hEllRaw |
| have hf : ContDiffOn ℝ ∞ f capChartDomain := |
| (ellipticOperator_contDiffOn capChartDomain_isOpen hA hB hC hu).congr (fun p hp => (hEq p hp).symm) |
| have hF : ContinuousOn (directedForcingEnergy chi I f b lambda) (closedRectangle tl tr sb b) := |
| (((hchi.mul hW).mul (hf.pow 2)).continuousOn.mono hbox) |
| have hf2 : ContinuousOn (fun p => (f p)^2) (closedRectangle tl tr sb b) := (hf.pow 2).continuousOn.mono hbox |
| have hforcing := rectangleIntegral_mono ht hs hF (continuousOn_const.mul hf2) |
| (fun p hp => directedForcingEnergy_bound hlambda (sub_nonneg.mpr hp.2.2) |
| (show edgeDistance b p ≤ D by dsimp only [edgeDistance,D,capDistanceBound]; linarith [hp.2.1]) |
| (abs_lt.mpr (hbox hp).2).le (hIv p hp) (capSpatialCutoff_range L R A0 p)) |
| change rectangleIntegral tl tr sb b (directedForcingEnergy chi I f b lambda) ≤ |
| rectangleIntegral tl tr sb b (fun p => (D^8*Real.exp (lambda*2+MI)) * (f p)^2) at hforcing |
| rw [rectangleIntegral_const_mul] at hforcing |
| obtain ⟨hsides,hbottom⟩ := capSpatialCutoff_outer_boundary (b := b) hL hR hA0 |
| have hdir := integrated_directed_energy_with_explicit_costs ht hs capChartDomain_isOpen hA hB hC |
| hchi hI hu hbox hsides hbottom hEq (fun p _ => (capSpatialCutoff_range L R A0 p).1) |
| lambda epsilon 2 c2 hc2 heps heps1 (fun p hp => (abs_lt.mpr (hbox hp).1).le) |
| norm_num only [show (1 : ℝ)+2^2=5 by norm_num] at hdir |
| have hsourceConstant : 0 ≤ (5 : ℝ)/(2*c2) := by positivity |
| have hcombined := add_le_add (mul_le_mul_of_nonneg_left hforcing hsourceConstant) |
| (hcostIntegral.trans (mul_le_mul_of_nonneg_left hEll hOverlap)) |
| change (c2/2)*rectangleIntegral tl tr sb b (directedGradientEnergy chi I u b lambda) ≤ _ |
| exact hdir.trans (by |
| convert hcombined using 1 <;> (first | rfl | ring)) |
|
|
| end SmoothLocal.Flow |
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|
| end |
|
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| end OAI |
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