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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 | import OAI.Geometry.IsometricImmersion.Caps.EllipticEnergy
import Mathlib.Analysis.Calculus.LocalExtr.Basic
import Mathlib.Tactic.NormNum
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped ContDiff Topology Interval
namespace SmoothLocal.Weighted
open SmoothLocal.Geometry
theorem weighted_young (w x y : ℝ) (hw : 0 < w) :
|x * y| ≤ (w * x ^ 2 + y ^ 2 / w) / 2 := by
have hminus : 0 ≤ w * x ^ 2 - 2 * x * y + y ^ 2 / w := by
have hn := div_nonneg (sq_nonneg (w * x - y)) hw.le
have he : (w * x - y) ^ 2 / w = w * x ^ 2 - 2 * x * y + y ^ 2 / w := by
field_simp [ne_of_gt hw]
ring
rw [he] at hn
exact hn
have hplus : 0 ≤ w * x ^ 2 + 2 * x * y + y ^ 2 / w := by
have hn := div_nonneg (sq_nonneg (w * x + y)) hw.le
have he : (w * x + y) ^ 2 / w = w * x ^ 2 + 2 * x * y + y ^ 2 / w := by
field_simp [ne_of_gt hw]
ring
rw [he] at hn
exact hn
apply abs_le.mpr
constructor <;> nlinarith
theorem elliptic_error_scalar_bound
(A v u x y eT eS H : ℝ) (hv : 0 < v) (hA : 0 < A)
(hratio : eT ^ 2 / v + eS ^ 2 / (A * v) ≤ H) :
|u * (eT * x + eS * y)| ≤ v * (x ^ 2 + A * y ^ 2) / 2 + H * u ^ 2 / 2 := by
have hx := weighted_young v x (u * eT) hv
have hy := weighted_young (A * v) y (u * eS) (mul_pos hA hv)
calc
|u * (eT * x + eS * y)| = |x * (u * eT) + y * (u * eS)| := by congr 1; ring
_ ≤ |x * (u * eT)| + |y * (u * eS)| := abs_add_le _ _
_ ≤ (v * x ^ 2 + (u * eT) ^ 2 / v) / 2 +
(A * v * y ^ 2 + (u * eS) ^ 2 / (A * v)) / 2 := add_le_add hx hy
_ = v * (x ^ 2 + A * y ^ 2) / 2 +
(eT ^ 2 / v + eS ^ 2 / (A * v)) * u ^ 2 / 2 := by
field_simp [ne_of_gt hv, ne_of_gt hA]
ring
_ ≤ v * (x ^ 2 + A * y ^ 2) / 2 + H * u ^ 2 / 2 := by
nlinarith [mul_le_mul_of_nonneg_right hratio (sq_nonneg u)]
variable {A B C v u f : Coord → ℝ} {U : Set Coord} {p : Coord}
theorem ellipticErrors_zero_of_nonneg
(hU : IsOpen U) (hp : p ∈ U) (hA : DifferentiableAt ℝ A p)
(hv : DifferentiableAt ℝ v p) (hvn : ∀ q ∈ U, 0 ≤ v q) (hzero : v p = 0) :
ellipticErrorT B v p = 0 ∧ ellipticErrorS A C v p = 0 := by
have hmin : IsLocalMin v p := by
change ∀ᶠ q in 𝓝 p, v p ≤ v q
filter_upwards [hU.mem_nhds hp] with q hq
simpa only [hzero] using hvn q hq
have hdv := hmin.fderiv_eq_zero
have hpv (i : Fin 2) : coordPartial i v p = 0 := by simp [coordPartial, hdv]
constructor
· simp [ellipticErrorT, hpv 0, hzero]
· rw [ellipticErrorS, HessianCalculus.coordPartial_mul_at hA hv 1]
simp [hpv 1, hzero]
theorem ellipticError_abs_bound
(hU : IsOpen U) (hp : p ∈ U) (hA : DifferentiableAt ℝ A p)
(hv : DifferentiableAt ℝ v p) (hvn : ∀ q ∈ U, 0 ≤ v q)
(H : ℝ) (hH : 0 ≤ H)
(hpos : 0 < v p → 0 < A p)
(hratio : 0 < v p → (ellipticErrorT B v p) ^ 2 / v p +
(ellipticErrorS A C v p) ^ 2 / (A p * v p) ≤ H) :
|ellipticError A B C v u p| ≤ ellipticEnergy A v u p / 2 + H * (u p) ^ 2 / 2 := by
by_cases hzero : v p = 0
· have hz := ellipticErrors_zero_of_nonneg (B := B) (C := C) hU hp hA hv hvn hzero
simp only [ellipticError, ellipticEnergy, hz.1, hz.2, hzero,
zero_mul, add_zero, mul_zero, abs_zero, zero_div, zero_add]
exact div_nonneg (mul_nonneg hH (sq_nonneg _)) (by norm_num)
· have hvpos : 0 < v p := lt_of_le_of_ne (hvn p hp) (Ne.symm hzero)
exact elliptic_error_scalar_bound (A p) (v p) (u p)
(coordPartial 0 u p) (coordPartial 1 u p)
(ellipticErrorT B v p) (ellipticErrorS A C v p) H hvpos (hpos hvpos) (hratio hvpos)
theorem ellipticEnergy_nonneg (A v u : Coord → ℝ) (p : Coord)
(hv : 0 ≤ v p) (hA : 0 < v p → 0 < A p) : 0 ≤ ellipticEnergy A v u p := by
by_cases hz : v p = 0
· simp [ellipticEnergy, hz]
· have hvpos : 0 < v p := lt_of_le_of_ne hv (Ne.symm hz)
exact mul_nonneg hv (add_nonneg (sq_nonneg _) (mul_nonneg (hA hvpos).le (sq_nonneg _)))
variable {tl tr sb st : ℝ}
theorem rectangleIntegral_sub {F G : Coord → ℝ} (ht : tl ≤ tr) (hs : sb ≤ st)
(hF : ContinuousOn F (closedRectangle tl tr sb st))
(hG : ContinuousOn G (closedRectangle tl tr sb st)) :
rectangleIntegral tl tr sb st (fun p => F p - G p) =
rectangleIntegral tl tr sb st F - rectangleIntegral tl tr sb st G := by
simp only [sub_eq_add_neg]
have hneg : ContinuousOn (fun p => -G p) (closedRectangle tl tr sb st) := hG.neg
rw [rectangleIntegral_add ht hs hF hneg, rectangleIntegral_neg]
theorem rectangleIntegral_mono {F G : Coord → ℝ} (ht : tl ≤ tr) (hs : sb ≤ st)
(hF : ContinuousOn F (closedRectangle tl tr sb st))
(hG : ContinuousOn G (closedRectangle tl tr sb st))
(hle : ∀ p ∈ closedRectangle tl tr sb st, F p ≤ G p) :
rectangleIntegral tl tr sb st F ≤ rectangleIntegral tl tr sb st G := by
rw [rectangleIntegral_eq_area ht hs hF, rectangleIntegral_eq_area ht hs hG]
apply setIntegral_mono_on (rectangle_area_integrable hF) (rectangle_area_integrable hG)
(measurableSet_Ioc.prod measurableSet_Ioc)
intro p hp
exact hle (boxPoint p.1 p.2) (boxPoint_mem
(Ioc_subset_Icc_self hp.1) (Ioc_subset_Icc_self hp.2))
theorem integrated_elliptic_energy_bound
(ht : tl ≤ tr) (hs : sb ≤ st) (hU : IsOpen U)
(hA : ContDiffOn ℝ ∞ A U) (hB : ContDiffOn ℝ ∞ B U)
(hC : ContDiffOn ℝ ∞ C U) (hv : ContDiffOn ℝ ∞ v U) (hu : ContDiffOn ℝ ∞ u U)
(hbox : closedRectangle tl tr sb st ⊆ U) (hboundary : zeroRectangleBoundary tl tr sb st v)
(hEq : ∀ p ∈ U, multiplierOperator A B C u p = f p)
(V H : ℝ) (hV : 0 ≤ V) (hH : 0 ≤ H)
(hvn : ∀ p ∈ U, 0 ≤ v p)
(hvV : ∀ p ∈ closedRectangle tl tr sb st, v p ≤ V)
(hpos : ∀ p ∈ closedRectangle tl tr sb st, 0 < v p → 0 < A p)
(hratio : ∀ p ∈ closedRectangle tl tr sb st, 0 < v p →
(ellipticErrorT B v p) ^ 2 / v p + (ellipticErrorS A C v p) ^ 2 / (A p * v p) ≤ H) :
rectangleIntegral tl tr sb st (ellipticEnergy A v u) ≤
V * rectangleIntegral tl tr sb st (fun p => (f p) ^ 2) +
(V + H) * rectangleIntegral tl tr sb st (fun p => (u p) ^ 2) := by
have hf : ContDiffOn ℝ ∞ f U :=
(ellipticOperator_contDiffOn hU hA hB hC hu).congr (fun p hp => (hEq p hp).symm)
obtain ⟨_, _, hE, hErr⟩ := elliptic_fields_contDiffOn hU hA hB hC hv hu
have hSource := (hv.mul hu).mul hf
have hpoint (p : Coord) (hp : p ∈ closedRectangle tl tr sb st) :
-(v p * u p * f p) - ellipticError A B C v u p ≤
(1 / 2 : ℝ) * ellipticEnergy A v u p +
((V + H) / 2) * (u p) ^ 2 + (V / 2) * (f p) ^ 2 := by
have hpU := hbox hp
have hErrBound := ellipticError_abs_bound (u := u) hU hpU
((hA.contDiffAt (hU.mem_nhds hpU)).differentiableAt (by simp))
((hv.contDiffAt (hU.mem_nhds hpU)).differentiableAt (by simp))
hvn H hH (hpos p hp) (hratio p hp)
have hyoung : -(u p * f p) ≤ ((u p) ^ 2 + (f p) ^ 2) / 2 := by
nlinarith [sq_nonneg (u p + f p)]
have hm := mul_le_mul_of_nonneg_left hyoung (hvn p hpU)
have hcost := mul_le_mul (hvV p hp) (le_refl (((u p) ^ 2 + (f p) ^ 2) / 2))
(div_nonneg (add_nonneg (sq_nonneg (u p)) (sq_nonneg (f p)))
(by norm_num : (0 : ℝ) ≤ 2)) hV
have hsource : -(v p * u p * f p) ≤ V * ((u p) ^ 2 + (f p) ^ 2) / 2 := by
nlinarith [hm, hcost]
nlinarith [hsource, neg_le_abs (ellipticError A B C v u p)]
have hEhalf : ContinuousOn (fun p => (1 / 2 : ℝ) * ellipticEnergy A v u p)
(closedRectangle tl tr sb st) := (contDiffOn_const.mul hE).continuousOn.mono hbox
have hUhalf : ContinuousOn (fun p => ((V + H) / 2) * (u p) ^ 2)
(closedRectangle tl tr sb st) := (contDiffOn_const.mul (hu.pow 2)).continuousOn.mono hbox
have hFhalf : ContinuousOn (fun p => (V / 2) * (f p) ^ 2)
(closedRectangle tl tr sb st) := (contDiffOn_const.mul (hf.pow 2)).continuousOn.mono hbox
have hi := rectangleIntegral_mono ht hs
((hSource.neg.sub hErr).continuousOn.mono hbox) ((hEhalf.add hUhalf).add hFhalf) hpoint
simp only [Pi.add_def] at hi
have hsum : ContinuousOn (fun p => (1 / 2 : ℝ) * ellipticEnergy A v u p +
((V + H) / 2) * (u p) ^ 2) (closedRectangle tl tr sb st) := hEhalf.add hUhalf
have heq := integrated_elliptic_equation ht hs hU hA hB hC hv hu hbox hboundary hEq
have hleft : rectangleIntegral tl tr sb st
(fun p => -(v p * u p * f p) - ellipticError A B C v u p) =
rectangleIntegral tl tr sb st (ellipticEnergy A v u) := by
rw [rectangleIntegral_sub ht hs (hSource.neg.continuousOn.mono hbox)
(hErr.continuousOn.mono hbox), rectangleIntegral_neg]
exact heq.symm
rw [hleft,
rectangleIntegral_add ht hs hsum hFhalf,
rectangleIntegral_add ht hs hEhalf hUhalf,
rectangleIntegral_const_mul tl tr sb st (1 / 2 : ℝ) (ellipticEnergy A v u),
rectangleIntegral_const_mul tl tr sb st ((V + H) / 2) (fun p => (u p) ^ 2),
rectangleIntegral_const_mul tl tr sb st (V / 2) (fun p => (f p) ^ 2)] at hi
nlinarith
end SmoothLocal.Weighted
end
end OAI
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