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59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 | import OAI.Geometry.IsometricImmersion.Energy.DirectedDefectCost
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology
namespace SmoothLocal.Weighted
open SmoothLocal.Geometry
def directedFluxCostConstant (D M lambda S MI X : ℝ) : ℝ :=
6*X*D*Real.exp (lambda*S+MI)*(M+1)
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
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
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
end
end OAI
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