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8fadf68 | 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 | import OAI.Geometry.NodalSets.Elliptic.CompactSourceOperator
import OAI.Geometry.NodalSets.Elliptic.UniformSmoothBounds
namespace OAI
namespace Yau.Geometry
open Yau.Jets Set Filter
open scoped ContDiff Topology
noncomputable section
lemma compact_finite_derivativeBound {ι : Type*} [Fintype ι]
(f : ι → Coord → ℂ) (hf : ∀ i, ContDiff ℝ ∞ (f i))
{Q : Set Coord} (hQ : IsCompact Q) (k : ℕ) :
∃ C > 0, ∀ i x, x ∈ Q → DerivativeBound k (f i) x C := by
have hb (i : ι) (j : Fin (k+1)) : ∃ C > 0, ∀ x ∈ Q,
‖iteratedFDeriv ℝ j.val (f i) x‖ ≤ C := by
obtain ⟨C,hC,hb⟩ := (hQ.image ((hf i).continuous_iteratedFDeriv
(by exact_mod_cast (show (j.val:ℕ∞) ≤ ⊤ from le_top)))).isBounded.exists_pos_norm_le
exact ⟨C,hC,fun x hx ↦ hb _ ⟨x,hx,rfl⟩⟩
choose b hb hbound using hb
let C : ℝ := 1+∑ i, ∑ j, b i j
have hsum (i : ι) : 0 ≤ ∑ j, b i j := Finset.sum_nonneg (fun j _ ↦ (hb i j).le)
have hC : 0 < C := by
have hh := Finset.sum_nonneg (fun i (_ : i ∈ Finset.univ) ↦ hsum i)
dsimp [C]
linarith
refine ⟨C,hC,?_⟩
intro i x hx j hj
refine (hbound i ⟨j,by omega⟩ x hx).trans ?_
have h1 := Finset.single_le_sum (fun j _ ↦ (hb i j).le) (Finset.mem_univ ⟨j,by omega⟩)
have h2 := Finset.single_le_sum (fun i _ ↦ hsum i) (Finset.mem_univ i)
dsimp [C]
linarith
lemma smoothFluxOperator_derivativeBound
(v : Coord → ℂ) (F : Fin 4 → Fin 4 → Coord → ℂ)
(hv : ContDiff ℝ ∞ v) (hF : ∀ i j, ContDiff ℝ ∞ (F i j))
(u : Coord → ℂ) (hu : ContDiff ℝ ∞ u) (k : ℕ) (x : Coord) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hvb : DerivativeBound k v x A)
(hFb : ∀ i j, DerivativeBound (k+1) (F i j) x A)
(hub : DerivativeBound (k+2) u x B) :
DerivativeBound k (smoothFluxOperator v F u) x
((2^k*A*(4*(4*(2^(k+1)*A))))*B) := by
let flux : Fin 4 → Coord → ℂ := fun i z ↦ ∑ j, F i j z*coordPartial j u z
have hs (i : Fin 4) : ContDiff ℝ ∞ (flux i) :=
ContDiff.sum (fun j _ ↦ (hF i j).mul (coordPartial_contDiff hu j))
have hb (i : Fin 4) : DerivativeBound (k+1) (flux i) x (4*(2^(k+1)*A*B)) := by
have h := DerivativeBound.sum Finset.univ
(fun j _ ↦ (hF i j).mul (coordPartial_contDiff hu j))
(fun j _ ↦ (hFb i j).mul (hF i j) (coordPartial_contDiff hu j) hA hB
(hub.coordPartial hu j))
simpa [flux] using h
have hd : DerivativeBound k (fun z ↦ ∑ i, coordPartial i (flux i) z) x
(4*(4*(2^(k+1)*A*B))) := by
have h := DerivativeBound.sum Finset.univ
(fun i _ ↦ coordPartial_contDiff (hs i) i)
(fun i _ ↦ (hb i).coordPartial (hs i) i)
simpa using h
have h := hvb.mul hv (ContDiff.sum (fun i _ ↦ coordPartial_contDiff (hs i) i)) hA
(by positivity) hd
convert h using 1 <;> first | rfl | ring
theorem source_operator_compact_derivative_bound {Q U : Set Coord}
(hQ : IsCompact Q) (hU : IsOpen U) (hQU : Q ⊆ U) (hUn : U.Nonempty)
(g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) (hg : ContDiffOn ℝ ∞ g U)
(hs : ∀ x ∈ U, ∀ u v, g x u v = g x v u)
(hp : ∀ x ∈ U, ∀ v, v ≠ 0 → 0 < g x v v)
(w : Coord → ℝ) (hw : ContDiffOn ℝ ∞ w U) (hwp : ∀ x ∈ U, 0 < w x)
(k : ℕ) :
∃ C > 0, ∀ u : Coord → ℂ, ContDiff ℝ ∞ u → ∀ x ∈ Q, ∀ B : ℝ, 0 ≤ B →
DerivativeBound (k+2) u x B → DerivativeBound k (sourceWeightedOperator g w u) x (C*B) := by
obtain ⟨v,F,hv,hF,he⟩ := compact_source_operator_coefficients hQ hU hQU hUn g hg hs hp w hw hwp
let f : Option (Fin 4 × Fin 4) → Coord → ℂ := fun i ↦ match i with
| none => v
| some (i,j) => F i j
have hf (i : Option (Fin 4 × Fin 4)) : ContDiff ℝ ∞ (f i) := by
cases i with
| none => exact hv
| some ij => exact hF ij.1 ij.2
obtain ⟨A,hA,hbound⟩ := compact_finite_derivativeBound f hf hQ (k+1)
refine ⟨2^k*A*(4*(4*(2^(k+1)*A))),by positivity,?_⟩
intro u hu x hx B hB hub j hj
rw [((he x hx u).iteratedFDeriv ℝ j).eq_of_nhds]
exact smoothFluxOperator_derivativeBound v F hv hF u hu k x hA.le hB
((hbound none x hx).mono (by omega)) (fun i j ↦ hbound (some (i,j)) x hx) hub j hj
theorem source_residual_compact_derivative_bound {Q U : Set Coord}
(hQ : IsCompact Q) (hU : IsOpen U) (hQU : Q ⊆ U) (hUn : U.Nonempty)
(g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) (hg : ContDiffOn ℝ ∞ g U)
(hs : ∀ x ∈ U, ∀ u v, g x u v = g x v u)
(hp : ∀ x ∈ U, ∀ v, v ≠ 0 → 0 < g x v v)
(w : Coord → ℝ) (hw : ContDiffOn ℝ ∞ w U) (hwp : ∀ x ∈ U, 0 < w x)
(k : ℕ) :
∃ C > 0, ∀ u : Coord → ℂ, ContDiff ℝ ∞ u → ∀ x ∈ Q, ∀ B : ℝ, 0 ≤ B →
DerivativeBound (k+2) u x B → ∀ lam : ℂ,
DerivativeBound k (fun z ↦ sourceWeightedOperator g w u z+lam*u z) x ((C+‖lam‖)*B) := by
obtain ⟨C,hC,hbound⟩ := source_operator_compact_derivative_bound hQ hU hQU hUn g hg hs hp w hw hwp k
obtain ⟨v,F,hv,hF,he⟩ := compact_source_operator_coefficients hQ hU hQU hUn g hg hs hp w hw hwp
refine ⟨C,hC,?_⟩
intro u hu x hx B hB hub lam j hj
have hL : ContDiffAt ℝ ∞ (sourceWeightedOperator g w u) x :=
(smoothFluxOperator_smooth v F hv hF u hu).contDiffAt.congr_of_eventuallyEq (he x hx u)
have hlu : ContDiff ℝ ∞ (lam • u) := contDiff_const.mul hu
change ‖iteratedFDeriv ℝ j (sourceWeightedOperator g w u+lam • u) x‖ ≤ _
rw [iteratedFDeriv_add_apply (hL.of_le (by exact_mod_cast (show (j:ℕ∞) ≤ ⊤ from le_top)))
(hlu.of_le (by exact_mod_cast (show (j:ℕ∞) ≤ ⊤ from le_top))).contDiffAt,
iteratedFDeriv_const_smul_apply (hu.of_le
(by exact_mod_cast (show (j:ℕ∞) ≤ ⊤ from le_top))).contDiffAt]
apply (norm_add_le _ _).trans
rw [norm_smul]
have hh := add_le_add (hbound u hu x hx B hB hub j hj)
(mul_le_mul_of_nonneg_left (hub j (by omega)) (norm_nonneg lam))
simpa only [add_mul] using hh
end
end Yau.Geometry
end OAI
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