| import OAI.Geometry.NodalSets.Elliptic.CompactSourceOperator |
| import OAI.Geometry.NodalSets.Elliptic.UniformSmoothBounds |
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| namespace OAI |
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| namespace Yau.Geometry |
| open Yau.Jets Set Filter |
| open scoped ContDiff Topology |
| noncomputable section |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
|
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| end |
| end Yau.Geometry |
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| end OAI |
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