| import OAI.Geometry.IsometricImmersion.Energy.DirectedDefectCost |
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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 directedFluxCostConstant (D M lambda S MI X : ℝ) : ℝ := |
| 6*X*D*Real.exp (lambda*S+MI)*(M+1) |
|
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| theorem ellipticWeightC1_energy_nonneg |
| {b c glow : ℝ} {chi K G1 u : Coord → ℝ} {p : Coord} |
| (hc : 0 < c) (hg : 0 < glow) (hG1 : glow ≤ G1 p) (hp : p 1 ≤ b) : |
| 0 ≤ ellipticEnergy (fun q => G1 q*K q) (ellipticWeightC1 b c chi K) u p := by |
| apply ellipticEnergy_nonneg _ _ _ p (ellipticWeightC1_nonneg _ _ _ _ _) |
| intro hv |
| have hplt : p 1 < b := by |
| by_contra h |
| have heq : p 1 = b := le_antisymm hp (le_of_not_gt h) |
| rw [ellipticWeightC1_zero heq.ge] at hv |
| exact (lt_irrefl (0 : ℝ)) hv |
| have hvraw : 0 < ellipticWeight b c chi K p := by |
| simpa only [ellipticWeightC1_eq hp] using hv |
| have hk := ellipticWeight_pos_implies_curvature hc hplt hvraw |
| have hKpos : 0 < K p := (mul_pos (by positivity : 0 < c/2) (sub_pos.mpr hplt)).trans hk |
| exact mul_pos (hg.trans_le hG1) hKpos |
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| theorem directedCostConstants_nonneg {D M lambda S MI X c2 : ℝ} |
| (hD : 0 ≤ D) (hM : 0 ≤ M) (hlambda : 0 ≤ lambda) (hX : 0 ≤ X) (hc2 : 0 ≤ c2) : |
| 0 ≤ directedFluxCostConstant D M lambda S MI X ∧ |
| 0 ≤ directedDefectCostConstant D M lambda S MI c2 := by |
| have hN := (directedWeightJetBudget_bounds hD hM hlambda).1 |
| constructor |
| · unfold directedFluxCostConstant |
| positivity |
| · unfold directedDefectCostConstant directedQuadraticCostConstant |
| positivity |
|
|
| end SmoothLocal.Weighted |
|
|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.Model |
|
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| theorem actual_cap_directed_costs_le_elliptic |
| {g0 eta : MetricField} {U : Set Coord} {Y : ℝ → ℝ → ℝ} |
| {L R A0 b kappa c1 c2 D glow lambda epsilon S MI M : ℝ} |
| {G1 B C I u : Coord → ℝ} {p : Coord} |
| (hL : -2 < L) (hR : R < 2) (hA0 : -2 < A0) |
| (hk : 0 < kappa) (hc1 : 0 < c1) (hc2 : 0 ≤ c2) (hg : 0 < glow) |
| (hD : 0 ≤ D) (hM : 0 ≤ M) (hlambda : 0 ≤ lambda) |
| (heps : 0 ≤ epsilon) (heps1 : epsilon ≤ 1) |
| (hbackground : ∀ q ∈ U, gaussianCurvature g0 q = modelCurvature kappa q) |
| (hsupport : tsupport eta ⊆ patchBox) (hpU : capChart Y p ∈ U) |
| (hdisp : |capFlowHeight Y p-p 1| ≤ (1 : ℝ)/50) |
| (hbelow : p 1 ≤ b) (hdD : edgeDistance b p ≤ D) |
| (ht : |p 0| ≤ 2) (hs : |p 1| ≤ S) (hIv : |I p| ≤ MI) |
| (hId : DifferentiableAt ℝ I p) (hIi : ∀ i : Fin 2, |coordPartial i I p| ≤ M) |
| (hG1 : glow ≤ G1 p) |
| (hAd : DifferentiableAt ℝ (fun q => G1 q*capPullback Y (gaussianCurvature (g0+eta)) q) p) |
| (hAv : |G1 p*capPullback Y (gaussianCurvature (g0+eta)) p| ≤ M) |
| (hAi : ∀ i : Fin 2, |
| |coordPartial i (fun q => G1 q*capPullback Y (gaussianCurvature (g0+eta)) q) p| ≤ M) |
| (hBv : |B p| ≤ M) (hCv : |C p| ≤ M) |
| (hcoercive : 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 A := fun q => G1 q*capPullback Y (gaussianCurvature (g0+eta)) q |
| let chi := capSpatialCutoff L R A0 |
| let c := capOverlapThreshold kappa c1 D |
| let X := capSpatialDerivativeConstant L R A0 |
| |directedCutoffError A chi I u b lambda epsilon p| + |
| directedCoercivityDefect A B C chi I u b lambda epsilon c2 p ≤ |
| (directedFluxCostConstant D M lambda S MI X + directedDefectCostConstant D M lambda S MI c2)* |
| (D+1/(glow*c))* |
| ellipticEnergy A (ellipticWeightC1 b c (capOuterSpatialCutoff L R A0) |
| (capPullback Y (gaussianCurvature (g0+eta)))) u p := by |
| 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 c := capOverlapThreshold kappa c1 D |
| let X := capSpatialDerivativeConstant L R A0 |
| let E := ellipticEnergy A (ellipticWeightC1 b c (capOuterSpatialCutoff L R A0) K) u p |
| let CE := D+1/(glow*c) |
| let CF := directedFluxCostConstant D M lambda S MI X |
| let CD := directedDefectCostConstant D M lambda S MI c2 |
| change |directedCutoffError A chi I u b lambda epsilon p| + |
| directedCoercivityDefect A B C chi I u b lambda epsilon c2 p ≤ (CF+CD)*CE*E |
| have hc : 0 < c := capOverlapThreshold_pos hk hc1 hD |
| have hX : 0 ≤ X := (abs_nonneg _).trans (capSpatialCutoff_partial_uniform hL hR hA0 p 0) |
| have hEnonneg : 0 ≤ E := ellipticWeightC1_energy_nonneg hc hg hG1 hbelow |
| have hCE : 0 ≤ CE := by dsimp only [CE]; positivity |
| have hconst := directedCostConstants_nonneg (S := S) (MI := MI) hD hM hlambda hX hc2 |
| have hCF : 0 ≤ CF := hconst.1 |
| have hCD : 0 ≤ CD := hconst.2 |
| have hd : 0 ≤ edgeDistance b p := sub_nonneg.mpr hbelow |
| have hfluxcost : |directedCutoffError A chi I u b lambda epsilon p| ≤ CF*weightedGradient b u p := |
| directedCutoffError_abs_le_seven hlambda heps heps1 ht hd hdD hs hIv hAv |
| (capSpatialCutoff_partial_uniform hL hR hA0 p) |
| have hdefcost : directedCoercivityDefect A B C chi I u b lambda epsilon c2 p ≤ CD*weightedGradient b u p := |
| directedCoercivityDefect_le_seven hAd hId hlambda hM hc2 heps heps1 ht hd hdD hs hIv hIi |
| hAv hBv hCv hAi (capSpatialCutoff_range L R A0 p) |
| by_cases hedge : p 1 < b |
| · have hdpos : 0 < edgeDistance b p := sub_pos.mpr hedge |
| have hflux : |directedCutoffError A chi I u b lambda epsilon p| ≤ CF*CE*E := by |
| by_cases hz : directedCutoffError A chi I u b lambda epsilon p = 0 |
| · rw [hz, abs_zero] |
| positivity |
| · have hder : ∃ i : Fin 2, coordPartial i chi p ≠ 0 := by |
| by_contra h |
| have hh : ∀ i : Fin 2, coordPartial i chi p = 0 := by |
| intro i |
| by_contra hi |
| exact h ⟨i,hi⟩ |
| exact hz (multiplierCutoffError_zero A chi (directedM b lambda I) |
| (directedN b lambda epsilon I) u p (hh 0) (hh 1)) |
| obtain ⟨i,hi⟩ := hder |
| have hgrad : weightedGradient b u p ≤ CE*E := |
| capOuterElliptic_gradient_on_actual_cutoff_support hL hR hA0 hk hc1 hD hg |
| hbackground hsupport hpU hdisp hdpos hdD hG1 hi |
| exact hfluxcost.trans (by |
| have hh := mul_le_mul_of_nonneg_left hgrad hCF |
| convert hh using 1 |
| ring) |
| have hdef : directedCoercivityDefect A B C chi I u b lambda epsilon c2 p ≤ CD*CE*E := by |
| by_cases hz : directedCoercivityDefect A B C chi I u b lambda epsilon c2 p = 0 |
| · rw [hz] |
| positivity |
| · have hgrad : weightedGradient b u p ≤ CE*E := |
| capOuterElliptic_gradient_on_defect hL hR hA0 hk hc1 hD hg hdpos hdD hG1 hcoercive hz |
| exact hdefcost.trans (by |
| have hh := mul_le_mul_of_nonneg_left hgrad hCD |
| convert hh using 1 |
| ring) |
| have hh := add_le_add hflux hdef |
| convert hh using 1 |
| ring |
| · have heq : p 1 = b := le_antisymm hbelow (le_of_not_gt hedge) |
| rw [directedCutoffError_edge_zero A chi I u b lambda epsilon p heq, |
| directedCoercivityDefect_edge_zero A B C chi I u b lambda epsilon c2 p hAd hId heq, |
| abs_zero, zero_add] |
| positivity |
|
|
| end SmoothLocal.Flow |
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| end |
|
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| end OAI |
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