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2fca478 | 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 | import Mathlib
import OAI.Geometry.PrescribedPotential.ChartTransport
import OAI.Geometry.PrescribedPotential.CoreBounds
namespace OAI
/-! Chart Transport Higher. -/
section
noncomputable section
open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory LineDeriv TemperedDistribution
open scoped ContDiff SchwartzMap Classical Laplacian
namespace GlobalElliptic
open SobolevChart
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E]
variable (e : OpenPartialHomeomorph E E) (he : ContDiffOn ℝ ∞ e e.source)
include he in
lemma transportCoefficient_smooth (v : E) (j : Fin (Module.finrank ℝ E)) :
ContDiffOn ℝ ∞ (fun x => ((stdOrthonormalBasis ℝ E).repr (fderiv ℝ e x v) j : ℂ)) e.source := by
apply Complex.ofRealCLM.contDiff.comp_contDiffOn
apply ((EuclideanSpace.proj j).contDiff.comp (stdOrthonormalBasis ℝ E).repr.contDiff).comp_contDiffOn
exact (he.fderiv_of_isOpen e.open_source (by simp)).clm_apply contDiffOn_const
def transportCoefficient (κ : ChartCutoff e.source) (v : E) (j : Fin (Module.finrank ℝ E)) :
ChartCutoff e.source := κ.mulOn e.open_source (transportCoefficient_smooth e he v j)
@[simp] lemma transportCoefficient_apply (κ : ChartCutoff e.source) (v : E)
(j : Fin (Module.finrank ℝ E)) (x : E) :
transportCoefficient e he κ v j x =
κ x * ((stdOrthonormalBasis ℝ E).repr (fderiv ℝ e x v) j : ℂ) := rfl
omit [FiniteDimensional ℝ E] in
lemma chartTransport_tsupport (κ : ChartCutoff e.source) (f : 𝓢(E, ℂ)) :
tsupport (chartTransport e he κ f : E → ℂ) ⊆ tsupport (κ : E → ℂ) :=
κ.product_tsupport e.open_source ((f.smooth ⊤).comp_contDiffOn he)
lemma chartTransport_deriv (κ : ChartCutoff e.source) (v : E) (f : 𝓢(E, ℂ)) :
∂_{v} (chartTransport e he κ f) = chartTransport e he (κ.deriv v) f +
∑ j, chartTransport e he (transportCoefficient e he κ v j)
(∂_{stdOrthonormalBasis ℝ E j} f) := by
ext x
by_cases hx : x ∈ e.source
· have hed := (he.contDiffAt (e.open_source.mem_nhds hx)).differentiableAt (by simp)
have hfd := f.differentiableAt.comp x hed
simp only [SchwartzMap.lineDerivOp_apply_eq_fderiv, chartTransport_apply,
_root_.add_apply, _root_.sum_apply, transportCoefficient_apply]
change fderiv ℝ (fun y => κ y * f (e y)) x v =
fderiv ℝ κ.val x v * f (e x) + _
have hm := fderiv_fun_mul κ.val.differentiableAt hfd
have hc := fderiv_comp x f.differentiableAt hed
dsimp only [Function.comp_def] at hm hc
rw [hm, hc]
simp only [_root_.add_apply, _root_.smul_apply,
ContinuousLinearMap.comp_apply, smul_eq_mul]
have hb := congrArg (fun z => fderiv ℝ f (e x) z)
((stdOrthonormalBasis ℝ E).sum_repr (fderiv ℝ e x v))
simp only [map_sum, map_smul, Complex.real_smul] at hb
rw [← hb, Finset.mul_sum]
ring_nf
· have hk : x ∉ tsupport (κ : E → ℂ) := fun ht => hx (κ.support_sub ht)
have ht : x ∉ tsupport (chartTransport e he κ f : E → ℂ) :=
fun ht => hk (chartTransport_tsupport e he κ f ht)
have hder : (∂_{v} (chartTransport e he κ f) : 𝓢(E, ℂ)) x = 0 := by
rw [SchwartzMap.lineDerivOp_apply_eq_fderiv,
(notMem_tsupport_iff_eventuallyEq.mp ht).fderiv_eq]
simp
have hk' : κ.deriv v x = 0 := image_eq_zero_of_notMem_tsupport
(fun h => hk (κ.deriv_tsupport v h))
simp only [_root_.add_apply, _root_.sum_apply, chartTransport_apply,
transportCoefficient_apply, hder, image_eq_zero_of_notMem_tsupport hk, hk', zero_mul,
Finset.sum_const_zero, add_zero]
variable [MeasurableSpace E] [BorelSpace E]
lemma word_deriv_comm (ws : List E) (v : E) (f : 𝓢(E, ℂ)) :
schwartzWord ws (∂_{v} f) = ∂_{v} (schwartzWord ws f) := by
induction ws with
| nil => rfl
| cons w ws ih =>
change ∂_{w} (schwartzWord ws (∂_{v} f)) = ∂_{v} (∂_{w} (schwartzWord ws f))
rw [ih, schwartz_deriv_comm]
include he in
lemma chartTransport_word_bound (hs : ContDiffOn ℝ ∞ e.symm e.target) (ws : List E)
(κ : ChartCutoff e.source) {s : ℝ} (hws : (ws.length : ℝ) ≤ s) :
CoreBound s 0 (fun f => schwartzWord ws (chartTransport e he κ f)) := by
induction ws generalizing κ s with
| nil =>
have hzero : CoreBound 0 0 (chartTransport e he κ) := chartTransport_l2_bound e he hs κ
exact hzero.comp (CoreBound.id (by simpa using hws))
| cons w ws ih =>
have hw : (ws.length : ℝ) + 1 ≤ s := by simpa only [List.length_cons, Nat.cast_add, Nat.cast_one] using hws
have hfirst := ih (κ.deriv w) (s := s) (by linarith)
have hrest (j : Fin (Module.finrank ℝ E)) :=
(ih (transportCoefficient e he κ w j) (s := s - 1) (by linarith)).comp
(coreBound_deriv (stdOrthonormalBasis ℝ E j) (s := s) (t := s - 1) le_rfl)
have hh := hfirst.add (CoreBound.sum Finset.univ _ (fun j _ => hrest j))
convert hh using 1
funext f
change ∂_{w} (schwartzWord ws (chartTransport e he κ f)) = _
rw [← word_deriv_comm, chartTransport_deriv, map_add, map_sum]
rfl
theorem chartTransport_even_bound (hs : ContDiffOn ℝ ∞ e.symm e.target)
(κ : ChartCutoff e.source) (k : ℕ) :
CoreBound (2 * (k : ℝ)) (2 * (k : ℝ)) (chartTransport e he κ) := by
apply coreBound_of_words k
intro ws hw
apply chartTransport_word_bound e he hs ws κ
exact_mod_cast hw
end GlobalElliptic
end
end
end OAI
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