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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 | import OAI.Geometry.IsometricImmersion.Caps.EllipticC1
import OAI.Geometry.IsometricImmersion.Caps.EllipticRatio
import OAI.Geometry.IsometricImmersion.Caps.EllipticCutoffEdge
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped ContDiff Topology Interval
namespace SmoothLocal.Weighted
open SmoothLocal.Geometry
def threeEdgeDerivativeConstant (l0 l1 r1 r0 a0 a1 : ℝ) : ℝ :=
transitionDerivativeBound / (l1 - l0) + transitionDerivativeBound / (r0 - r1) +
transitionDerivativeBound / (a1 - a0)
theorem threeEdgeDerivativeConstant_nonneg {l0 l1 r1 r0 a0 a1 : ℝ}
(hL : l0 < l1) (hR : r1 < r0) (hS : a0 < a1) :
0 ≤ threeEdgeDerivativeConstant l0 l1 r1 r0 a0 a1 := by
have ht := transitionDerivativeBound_pos.le
have hl := sub_pos.mpr hL
have hr := sub_pos.mpr hR
have hs := sub_pos.mpr hS
unfold threeEdgeDerivativeConstant
positivity
theorem threeEdgeCutoff_partial_common {l0 l1 r1 r0 a0 a1 : ℝ}
(hL : l0 < l1) (hR : r1 < r0) (hS : a0 < a1) (p : Coord) (i : Fin 2) :
|coordPartial i (threeEdgeCutoff l0 l1 r1 r0 a0 a1) p| ≤
threeEdgeDerivativeConstant l0 l1 r1 r0 a0 a1 := by
have hb := threeEdgeCutoff_partial_bounds hL hR hS p
have ht := transitionDerivativeBound_pos.le
have hl : 0 ≤ transitionDerivativeBound / (l1 - l0) := div_nonneg ht (sub_pos.mpr hL).le
have hr : 0 ≤ transitionDerivativeBound / (r0 - r1) := div_nonneg ht (sub_pos.mpr hR).le
have hs : 0 ≤ transitionDerivativeBound / (a1 - a0) := div_nonneg ht (sub_pos.mpr hS).le
unfold threeEdgeDerivativeConstant
fin_cases i <;> simp only [Fin.mk_zero, Fin.mk_one] <;> linarith [hb.1, hb.2]
theorem ellipticWeightC1_errors_eq {b c : ℝ} (A B C chi K : Coord → ℝ)
{p : Coord} (hp : p 1 < b) :
ellipticErrorT B (ellipticWeightC1 b c chi K) p =
ellipticErrorT B (ellipticWeight b c chi K) p ∧
ellipticErrorS A C (ellipticWeightC1 b c chi K) p =
ellipticErrorS A C (ellipticWeight b c chi K) p := by
have he : (fun q => A q * ellipticWeightC1 b c chi K q) =ᶠ[𝓝 p]
(fun q => A q * ellipticWeight b c chi K q) := by
filter_upwards [(isOpen_lt (continuous_apply 1) continuous_const).mem_nhds hp] with q hq
rw [ellipticWeightC1_eq hq.le]
have hderiv : coordPartial 1 (fun q => A q * ellipticWeightC1 b c chi K q) p =
coordPartial 1 (fun q => A q * ellipticWeight b c chi K q) p := by
unfold coordPartial
rw [he.fderiv_eq]
constructor
· rw [ellipticErrorT, ellipticErrorT, ellipticWeightC1_partial_eq hp 0,
ellipticWeightC1_eq hp.le]
· rw [ellipticErrorS, ellipticErrorS, hderiv, ellipticWeightC1_eq hp.le]
theorem threeEdge_ellipticWeight_boundary {l0 l1 r1 r0 a0 a1 b c : ℝ}
(hL : l0 < l1) (hR : r1 < r0) (hS : a0 < a1) (K : Coord → ℝ) :
zeroRectangleBoundary l0 r0 a0 b
(ellipticWeightC1 b c (threeEdgeCutoff l0 l1 r1 r0 a0 a1) K) := by
constructor
· intro s hs
have hl := threeEdgeCutoff_zero hL hR hS (boxPoint l0 s)
(Or.inl (by simp [boxPoint]))
have hr := threeEdgeCutoff_zero hL hR hS (boxPoint r0 s)
(Or.inr (Or.inl (by simp [boxPoint])))
constructor <;> simp [ellipticWeightC1, hl, hr]
· intro t ht
have hb := threeEdgeCutoff_zero hL hR hS (boxPoint t a0)
(Or.inr (Or.inr (by simp [boxPoint])))
constructor
· simp [ellipticWeightC1, hb]
· exact ellipticWeightC1_zero (by simp [boxPoint])
variable {U : Set Coord} {K G1 B C u f : Coord → ℝ}
{l0 l1 r1 r0 a0 a1 b c MK g0 M : ℝ}
theorem threeEdge_elliptic_energy_bound
(hL : l0 < l1) (hR : r1 < r0) (hS : a0 < a1)
(ht : l0 ≤ r0) (hs : a0 ≤ b) (hc : 0 < c)
(hMK : 0 ≤ MK) (hg0 : 0 < g0) (hM : 0 ≤ M)
(hU : IsOpen U) (hK : ContDiffOn ℝ ∞ K U) (hG1 : ContDiffOn ℝ ∞ G1 U)
(hB : ContDiffOn ℝ ∞ B U) (hC : ContDiffOn ℝ ∞ C U) (hu : ContDiffOn ℝ ∞ u U)
(hbox : closedRectangle l0 r0 a0 b ⊆ U)
(hEq : ∀ p ∈ U, multiplierOperator (fun q => G1 q * K q) B C u p = f p)
(hKi : ∀ p ∈ closedRectangle l0 r0 a0 b, ∀ i : Fin 2, |coordPartial i K p| ≤ MK)
(hGlow : ∀ p ∈ closedRectangle l0 r0 a0 b, g0 ≤ G1 p)
(hAhi : ∀ p ∈ closedRectangle l0 r0 a0 b, |G1 p * K p| ≤ M)
(hAs : ∀ p ∈ closedRectangle l0 r0 a0 b,
|coordPartial 1 (fun q => G1 q * K q) p| ≤ M)
(hBb : ∀ p ∈ closedRectangle l0 r0 a0 b, |B p| ≤ M)
(hCb : ∀ p ∈ closedRectangle l0 r0 a0 b, |C p| ≤ M) :
rectangleIntegral l0 r0 a0 b
(ellipticEnergy (fun q => G1 q * K q)
(ellipticWeight b c (threeEdgeCutoff l0 l1 r1 r0 a0 a1) K) u) ≤
(b - a0) ^ 6 * rectangleIntegral l0 r0 a0 b (fun p => (f p) ^ 2) +
((b - a0) ^ 6 + ellipticRatioConstant (b - a0) (g0 * c / 2)
(ellipticCutoffDerivativeConstant c MK (threeEdgeDerivativeConstant l0 l1 r1 r0 a0 a1)) M) *
rectangleIntegral l0 r0 a0 b (fun p => (u p) ^ 2) := by
let chi := threeEdgeCutoff l0 l1 r1 r0 a0 a1
let v := ellipticWeightC1 b c chi K
let X := threeEdgeDerivativeConstant l0 l1 r1 r0 a0 a1
let H := ellipticRatioConstant (b - a0) (g0 * c / 2) (ellipticCutoffDerivativeConstant c MK X) M
have hX : 0 ≤ X := threeEdgeDerivativeConstant_nonneg hL hR hS
have hchi : ContDiffOn ℝ ∞ chi U := (threeEdgeCutoff_contDiff _ _ _ _ _ _).contDiffOn
have hv : ContDiffOn ℝ 1 v U := ellipticWeightC1_contDiffOn hc
(hchi.of_le (by simp)) (hK.of_le (by simp))
have hA : ContDiffOn ℝ ∞ (fun q => G1 q * K q) U := hG1.mul hK
have hH : 0 ≤ H := ellipticRatioConstant_nonneg (sub_nonneg.mpr hs) (by positivity) hM
have hposD (p : Coord) (hpv : 0 < v p) : p 1 < b := by
by_contra hn
have hz : v p = 0 := ellipticWeightC1_zero (le_of_not_gt hn)
linarith
have hweightEq (p : Coord) (hp : p ∈ closedRectangle l0 r0 a0 b) :
v p = ellipticWeight b c chi K p := ellipticWeightC1_eq hp.2.2
have hposA (p : Coord) (hp : p ∈ closedRectangle l0 r0 a0 b) (hpv : 0 < v p) :
0 < G1 p * K p := by
have hd := hposD p hpv
have hcurv := ellipticWeight_pos_implies_curvature hc hd
(show 0 < ellipticWeight b c chi K p by simpa only [← hweightEq p hp] using hpv)
have hKpos : 0 < K p := (mul_pos (by positivity : 0 < c / 2) (sub_pos.mpr hd)).trans hcurv
exact mul_pos (hg0.trans_le (hGlow p hp)) hKpos
have hratio (p : Coord) (hp : p ∈ closedRectangle l0 r0 a0 b) (hpv : 0 < v p) :
(ellipticErrorT B v p) ^ 2 / v p +
(ellipticErrorS (fun q => G1 q * K q) C v p) ^ 2 / (G1 p * K p * v p) ≤ H := by
have hd := hposD p hpv
have herrors := ellipticWeightC1_errors_eq (c := c) (fun q => G1 q * K q) B C chi K hd
change (ellipticErrorT B (ellipticWeightC1 b c chi K) p) ^ 2 / v p +
(ellipticErrorS (fun q => G1 q * K q) C (ellipticWeightC1 b c chi K) p) ^ 2 /
(G1 p * K p * v p) ≤ H
rw [herrors.1, herrors.2, hweightEq p hp]
exact ellipticCutoff_actual_ratio hc (sub_pos.mpr hd)
(by unfold edgeDistance; linarith [hp.2.1]) hMK hX hg0 hM
((hchi.contDiffAt (hU.mem_nhds (hbox hp))).differentiableAt (by simp))
((hK.contDiffAt (hU.mem_nhds (hbox hp))).differentiableAt (by simp))
((hG1.contDiffAt (hU.mem_nhds (hbox hp))).differentiableAt (by simp))
(threeEdgeCutoff_range _ _ _ _ _ _ p)
(threeEdgeCutoff_partial_common hL hR hS p) (hKi p hp)
(show 0 < ellipticWeight b c chi K p by simpa only [← hweightEq p hp] using hpv)
(hGlow p hp) (hAhi p hp) (hAs p hp) (hBb p hp) (hCb p hp)
have hestimate := integrated_elliptic_energy_bound_C1 ht hs hU hA hB hC hv hu hbox
(threeEdge_ellipticWeight_boundary hL hR hS K) hEq ((b - a0) ^ 6) H
(by positivity) hH (fun p _ => ellipticWeightC1_nonneg _ _ _ _ p)
(fun p hp => (ellipticWeightC1_le_distance_six _ _ _ _ p
(threeEdgeCutoff_range _ _ _ _ _ _ p)).trans
(pow_le_pow_left₀ (sub_nonneg.mpr hp.2.2)
(by linarith [hp.2.1]) 6))
hposA hratio
have henergyEq : rectangleIntegral l0 r0 a0 b
(ellipticEnergy (fun q => G1 q * K q) v u) =
rectangleIntegral l0 r0 a0 b
(ellipticEnergy (fun q => G1 q * K q) (ellipticWeight b c chi K) u) := by
apply rectangleIntegral_congr ht hs
intro p hp
change v p * _ = ellipticWeight b c chi K p * _
rw [hweightEq p hp]
rw [henergyEq] at hestimate
exact hestimate
end SmoothLocal.Weighted
end
end OAI
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