| import OAI.Analysis.AncientFlat.Profile.RadialHolderOn |
| import OAI.Analysis.AncientFlat.Regularity.HolderProducts |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter |
| open scoped ContDiff |
| namespace ParabolicCore |
| open ParabolicInterior ParabolicAnnular |
| variable {V : Type} [NormedAddCommGroup V] |
| def outer : Set (V × ℝ) := {p | ‖p.1‖<4 ∧ -257<p.2 ∧ p.2≤-1/2} |
| def inner : Set (V × ℝ) := {p | ‖p.1‖<2 ∧ -(33/2)<p.2 ∧ p.2≤-1/2} |
| def rectangle : Set (ℝ × ℝ) := Icc 0 4 ×ˢ Icc (1/2) 257 |
| def argument (z : ℝ × ℝ) : ℝ := z.1^2/Real.sqrt z.2 |
| def principal (G : ℝ → ℝ) (z : ℝ × ℝ) : ℝ := G (argument z)^2/16 |
| def potential (G : ℝ → ℝ) (z : ℝ × ℝ) : ℝ := |
| -z.2⁻¹*(argument z*derivWithin G (Ici 0) (argument z)/G (argument z)) |
| lemma argument_nonneg {z : ℝ × ℝ} (_hz : z∈rectangle) : 0≤argument z := by unfold argument; positivity |
| lemma argument_smooth : ContDiffOn ℝ ∞ argument rectangle := by |
| apply (contDiff_fst.contDiffOn.pow 2).div |
| (contDiff_snd.contDiffOn.sqrt (fun z hz => ne_of_gt (by linarith [hz.2.1] : 0<z.2))) |
| intro z hz |
| exact (Real.sqrt_pos.mpr (by linarith [hz.2.1] : 0<z.2)).ne' |
| lemma coeff_smooth (G : ℝ → ℝ) (κ : ℝ) (hκ : 0<κ) |
| (hG : ContDiffOn ℝ ∞ G (Ici 0)) (hpos : ∀ y : ℝ, 0≤y → κ≤G y) : |
| ContDiffOn ℝ 1 (principal G) rectangle ∧ ContDiffOn ℝ 1 (potential G) rectangle := by |
| have harg : ContDiffOn ℝ 1 argument rectangle := argument_smooth.of_le (by simp) |
| have hGV : ContDiffOn ℝ 1 (fun z => G (argument z)) rectangle := |
| (hG.of_le (show (1:ℕ∞ω)≤∞ by simp)).comp harg (fun _ hz => argument_nonneg hz) |
| have hGD : ContDiffOn ℝ 1 (derivWithin G (Ici 0)) (Ici 0) := |
| hG.derivWithin (uniqueDiffOn_Ici 0) (by |
| norm_num only [one_add_one_eq_two] |
| exact ENat.natCast_le_of_coe_top_le_withTop le_rfl 2) |
| refine ⟨hGV.pow 2 |>.div contDiffOn_const (by norm_num),?_⟩ |
| have hT : ContDiffOn ℝ 1 (fun z : ℝ × ℝ => z.2) rectangle := contDiff_snd.contDiffOn |
| have hi : ContDiffOn ℝ 1 (fun z : ℝ × ℝ => -z.2⁻¹) rectangle := |
| (hT.inv (fun z hz => ne_of_gt (by linarith [hz.2.1] : 0<z.2))).neg |
| exact hi.mul ((harg.mul (hGD.comp harg (fun _ hz => argument_nonneg hz))).div hGV |
| (fun z hz => (hκ.trans_le (hpos _ (argument_nonneg hz))).ne')) |
| lemma potential_eq (G : ℝ → ℝ) (p : V × ℝ) : |
| potential G (‖p.1‖,-p.2)=profilePotential G p := by |
| unfold potential profilePotential argument |
| by_cases hy : ‖p.1‖^2/Real.sqrt (-p.2)=0 |
| · simp only [hy,zero_mul,zero_div,mul_zero] |
| · have hpos : 0<‖p.1‖^2/Real.sqrt (-p.2) := lt_of_le_of_ne (by positivity) (Ne.symm hy) |
| rw [derivWithin_of_mem_nhds (Ici_mem_nhds hpos)] |
| lemma profile_core_holder (G : ℝ → ℝ) (κ α : ℝ) (hκ : 0<κ) (hα : 0<α) (hα1 : α<1) |
| (hG : ContDiffOn ℝ ∞ G (Ici 0)) (hpos : ∀ y : ℝ, 0≤y → κ≤G y) : |
| ∃ K : ℝ, 0<K ∧ HolderBound α outer (profilePrincipal (V := V) G) K ∧ |
| HolderBound α outer (fun p : V × ℝ => (profilePotential G p : ℂ)) K ∧ |
| ∀ p : V × ℝ, p∈outer → κ^2/16≤profilePrincipal G p := by |
| have hk : IsCompact rectangle := isCompact_Icc.prod isCompact_Icc |
| have hkc : Convex ℝ rectangle := (convex_Icc _ _).prod (convex_Icc _ _) |
| have hm : ∀ p∈(outer : Set (V × ℝ)), (‖p.1‖,-p.2)∈rectangle := by |
| intro p hp |
| exact ⟨⟨norm_nonneg _,hp.1.le⟩,⟨by linarith [hp.2.2],by linarith [hp.2.1]⟩⟩ |
| obtain ⟨K1,hK1,ha⟩ := compact_radial_holder_on α hα hα1.le (principal G) rectangle hk hkc |
| (coeff_smooth G κ hκ hG hpos).1 outer hm |
| obtain ⟨K2,hK2,hb⟩ := compact_radial_holder_on α hα hα1.le (potential G) rectangle hk hkc |
| (coeff_smooth G κ hκ hG hpos).2 outer hm |
| have ha' : HolderBound α outer (profilePrincipal (V := V) G) K1 := ha |
| have hb' : HolderBound α outer (fun p : V × ℝ => (profilePotential G p : ℂ)) K2 := by |
| constructor |
| · intro p hp |
| simpa only [←potential_eq G p,Complex.norm_real] using hb.1 p hp |
| · intro p hp q hq |
| simpa only [←potential_eq G p,←potential_eq G q,←Complex.ofReal_sub,Complex.norm_real] using hb.2 p hp q hq |
| refine ⟨K1+K2,by positivity,ha'.mono (by linarith),hb'.mono (by linarith),?_⟩ |
| intro p hp |
| have hGp := hpos (‖p.1‖^2/Real.sqrt (-p.2)) (by positivity) |
| dsimp [profilePrincipal] |
| exact div_le_div_of_nonneg_right ((sq_le_sq₀ hκ.le (hκ.le.trans hGp)).mpr hGp) (by norm_num) |
| end ParabolicCore |
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| end |
|
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| end OAI |
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