| import OAI.Analysis.AncientFlat.Geometry.InteriorCutoff |
| import OAI.Analysis.AncientFlat.Evolution.TraceFour |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter Metric |
| open scoped ContDiff ZeroAtInfty Laplacian |
| namespace ParabolicLocalization |
| open ParabolicInterior |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| omit [FiniteDimensional ℝ V] in |
| lemma HolderBound.of_jet_zero {α K : ℝ} {Ω : Set (V × ℝ)} (u : ℝ → V → E) |
| (h : HolderBound α Ω (fun p => iteratedFDeriv ℝ 0 (u p.2) p.1) K) : |
| HolderBound α Ω (fun p => u p.2 p.1) K := by |
| refine ⟨fun p hp => by simpa only [norm_iteratedFDeriv_zero] using h.1 p hp,?_⟩ |
| intro p hp q hq |
| have hh := h.2 p hp q hq |
| dsimp only at hh |
| simpa only [iteratedFDeriv_zero_eq_comp,Function.comp_apply,←map_sub,LinearIsometryEquiv.norm_map] using hh |
| lemma cutoff_support (t : ℝ) : tsupport (cutoff t : V → ℝ)⊆closedBall 0 (3/2) := by |
| have h := tsupport_smul_subset_right (fun _ : V => baseBump ℝ t) (baseBump V) |
| change tsupport (cutoff t : V → ℝ) ⊆ tsupport (baseBump V) at h |
| rw [(baseBump V).tsupport_eq] at h |
| exact h |
| lemma cutoff_product_support (t : ℝ) (u : V → E) : |
| tsupport (fun x => cutoff t x • u x)⊆closedBall 0 (3/2) := |
| (tsupport_smul_subset_left (cutoff t) u).trans (cutoff_support t) |
| lemma cutoff_product_jet_zero (n : ℕ) (t : ℝ) (u : V → E) (x : V) (hx : 7/4≤‖x‖) : |
| iteratedFDeriv ℝ n (fun y => cutoff t y • u y) x=0 := by |
| apply Function.notMem_support.mp |
| intro h |
| have hh := cutoff_product_support t u (support_iteratedFDeriv_subset n h) |
| simp only [mem_closedBall,dist_zero_right] at hh |
| linarith |
| lemma cutoff_product_compact (t : ℝ) (u : V → E) : |
| HasCompactSupport (fun x => cutoff t x • u x) := |
| (cutoff_compact t).smul_right |
| lemma cutoff_time_zero (t : ℝ) (x : V) (hx : 3/2≤‖x‖) : cutoffTime t x=0 := by |
| unfold cutoffTime |
| rw [(baseBump V).zero_of_le_dist (by simpa only [baseBump,dist_zero_right] using hx),mul_zero] |
| lemma cutoff_time_compact (t : ℝ) : HasCompactSupport (cutoffTime t : V → ℝ) := |
| (baseBump V).hasCompactSupport.mul_left |
| def cutoffValue (u : ℝ → V → E) (t : ℝ) (x : V) : E := cutoff t x • u t x |
| def cutoffDerivative (u ut : ℝ → V → E) (t : ℝ) (x : V) : E := |
| cutoff t x • ut t x+cutoffTime t x • u t x |
| lemma cutoffDerivative_compact (u ut : ℝ → V → E) (t : ℝ) : |
| HasCompactSupport (cutoffDerivative u ut t) := by |
| have h1 : HasCompactSupport (fun x => cutoff t x • ut t x) := cutoff_product_compact t _ |
| have h2 : HasCompactSupport (fun x => cutoffTime t x • u t x) := (cutoff_time_compact t).smul_right |
| exact h1.add h2 |
| lemma cutoffDerivative_zero (u ut : ℝ → V → E) (t : ℝ) (x : V) (hx : 3/2≤‖x‖) : |
| cutoffDerivative u ut t x=0 := by |
| simp only [cutoffDerivative,cutoff_zero_space t hx,cutoff_time_zero t x hx,zero_smul,add_zero] |
| lemma cutoffValue_contDiff (u : ℝ → V → E) (t : ℝ) (hs : ContDiff ℝ 4 (u t)) : |
| ContDiff ℝ 4 (cutoffValue u t) := |
| ((cutoff_contDiff t).of_le (by change ((4:ℕ∞):WithTop ℕ∞)≤((⊤:ℕ∞):WithTop ℕ∞); exact WithTop.coe_le_coe.mpr le_top)).smul hs |
| def cutoffC0 (u : ℝ → V → ℂ) (hs : ∀ t, ContDiff ℝ 4 (u t)) (t : ℝ) : C₀(V,ℂ) := |
| ⟨⟨cutoffValue u t,(cutoffValue_contDiff u t (hs t)).continuous⟩, |
| (cutoff_product_compact t (u t)).is_zero_at_infty⟩ |
| @[simp] lemma cutoffC0_apply (u : ℝ → V → ℂ) (hs : ∀ t, ContDiff ℝ 4 (u t)) (t : ℝ) (x : V) : |
| cutoffC0 u hs t x=cutoffValue u t x := rfl |
| lemma cutoffValue_pointwise_derivative (u ut : ℝ → V → E) |
| (hd : ∀ p∈cylinder 2, HasDerivAt (fun t => u t p.1) (ut p.2 p.1) p.2) |
| (t : ℝ) (ht : t∈Ioo (-2) 0) (x : V) : |
| HasDerivAt (fun s => cutoffValue u s x) (cutoffDerivative u ut t x) t := by |
| by_cases hx : ‖x‖<2 |
| · have hz : (x,t)∈cylinder 2 := by unfold cylinder; dsimp; norm_num; exact ⟨hx,by linarith [ht.1],ht.2.le⟩ |
| simpa only [cutoffValue,cutoffDerivative] using (cutoff_hasDerivAt t x).fun_smul (hd (x,t) hz) |
| · have hx' : 3/2≤‖x‖ := by linarith [le_of_not_gt hx] |
| have he : (fun s => cutoffValue u s x)=fun _ => (0:E) := by |
| funext s; simp only [cutoffValue,cutoff_zero_space s hx',zero_smul] |
| rw [he,cutoffDerivative_zero u ut t x hx'] |
| exact hasDerivAt_const t 0 |
| end ParabolicLocalization |
| |
| end |
| |
| end OAI |
| |
