| import OAI.Analysis.AllenCahn.ConeDerivatives |
| import OAI.Analysis.AllenCahn.CompactCone |
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| namespace OAI |
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| noncomputable section |
| namespace DeGiorgi |
| open Filter Set Topology |
| open scoped NNReal |
| open scoped SchwartzMap Laplacian ContDiff |
| open FourierTransform |
| open Filter Topology |
| open scoped ContDiff |
| open DeGiorgiCompactness (JetSpace CoherentJets pdeJetCone) |
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| def smoothJet {n : ℕ} (f : Space n → ℝ) (hf : ContDiff ℝ ∞ f) : JetSpace n := |
| fun l => ⟨coordinateJet f l, (coordinateJet_smooth hf l).continuous⟩ |
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| lemma smoothJet_coherent {n : ℕ} {f : Space n → ℝ} (hf : ContDiff ℝ ∞ f) : |
| CoherentJets (smoothJet f hf) := by |
| intro l |
| exact ⟨(coordinateJet_smooth hf l).differentiable (by simp), fun _ _ => rfl⟩ |
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| lemma smoothJet_mem {n : ℕ} {P f : Space n → ℝ} (hf : f ∈ positiveLinearCone P) : |
| smoothJet f hf.1 ∈ pdeJetCone P := by |
| refine ⟨smoothJet_coherent hf.1, hf.2.1, ?_, hf.2.2.2⟩ |
| intro x |
| change (∑ i : Fin n, coordDeriv (coordDeriv f i) i x) = P x*f x |
| rw [← laplacian_eq_sum_coordDeriv (smooth_C2 hf.1)] |
| exact hf.2.2.1 x |
|
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| lemma jetCone_base_mem {n : ℕ} {P : Space n → ℝ} {g : JetSpace n} |
| (hg : g ∈ pdeJetCone P) : (g [] : Space n → ℝ) ∈ positiveLinearCone P := by |
| have hs := DeGiorgiCompactness.coherentJets_smooth hg.1 [] |
| refine ⟨hs, hg.2.1, ?_, hg.2.2.2⟩ |
| intro x |
| rw [laplacian_eq_sum_coordDeriv (smooth_C2 hs), ← hg.2.2.1 x] |
| apply Finset.sum_congr rfl |
| intro i hi |
| have he : coordDeriv (g [] : Space n → ℝ) i = (g [i] : Space n → ℝ) := |
| funext ((hg.1 []).2 i) |
| rw [he] |
| exact (hg.1 [i]).2 i x |
|
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| lemma coherentJets_eq_smoothJet {n : ℕ} {g : JetSpace n} (hg : CoherentJets g) : |
| g = smoothJet (g [] : Space n → ℝ) (DeGiorgiCompactness.coherentJets_smooth hg []) := |
| DeGiorgiCompactness.coherentJets_eq_of_base hg (smoothJet_coherent _) rfl |
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| lemma jetCone_jet_eq {n : ℕ} {g : JetSpace n} (hg : CoherentJets g) |
| (l : List (Fin n)) (x : Space n) : g l x = coordinateJet (g [] : Space n → ℝ) l x := by |
| conv_lhs => rw [coherentJets_eq_smoothJet hg] |
| rfl |
|
|
| lemma conePotential_harnack_base {n : ℕ} {ν : Space n} (hν : ‖ν‖=1) |
| {f : Space n → ℝ} (hf : f ∈ positiveLinearCone (conePotential ν)) (x : Space n) : |
| f x ≤ Real.exp (Real.sqrt (linearLogGradientBound n 2)*‖x‖) := by |
| have H := positive_linear_harnack hf.1 (conePotential_smooth ν) |
| (positiveLinearCone_pos (fun y => (conePotential_bounds ν y).2) hf) (by norm_num : (0:ℝ)≤2) |
| (fun y => by rw [abs_of_nonneg (by linarith [(conePotential_bounds ν y).1] : 0≤conePotential ν y)]; exact (conePotential_bounds ν y).2) |
| (conePotential_coord_bound hν) hf.2.2.1 0 x |
| simpa only [hf.2.2.2, mul_one, sub_zero] using H |
|
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| lemma conePotential_jetCone_compact {n : ℕ} {ν : Space n} (hν : ‖ν‖=1) : |
| IsCompact (pdeJetCone (conePotential ν)) := by |
| apply DeGiorgiCompactness.pdeJetCone_compact_of_local_bounds |
| intro l R hR |
| obtain ⟨M,hM,hMb⟩ := conePotential_relative_jets hν l |
| choose N hN hNb using fun i : Fin n => conePotential_relative_jets hν (i::l) |
| let B := Real.exp (Real.sqrt (linearLogGradientBound n 2)*R) |
| have hB : 0 ≤ B := (Real.exp_pos _).le |
| have hsum : 0 ≤ ∑ i, N i := Finset.sum_nonneg (fun i _ => hN i) |
| let L := (M+‖DeGiorgiCompactness.coordinateDual n‖*(∑ i, N i))*B |
| have hnorm : 0 ≤ ‖DeGiorgiCompactness.coordinateDual n‖ := by positivity |
| have hL : 0 ≤ L := by dsimp [L]; positivity |
| refine ⟨⟨L,hL⟩, ?_⟩ |
| intro g hg x hx |
| have hf := jetCone_base_mem hg |
| have hb : g [] x ≤ B := (conePotential_harnack_base hν hf x).trans |
| (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hx (Real.sqrt_nonneg _))) |
| have hval : |g l x| ≤ M*B := by |
| rw [jetCone_jet_eq hg.1] |
| exact (hMb _ hf x).trans (mul_le_mul_of_nonneg_left hb hM) |
| have hder : ‖fderiv ℝ (g l : Space n → ℝ) x‖ ≤ |
| ‖DeGiorgiCompactness.coordinateDual n‖*(∑ i, N i)*B := by |
| calc ‖fderiv ℝ (g l : Space n → ℝ) x‖ ≤ |
| ‖DeGiorgiCompactness.coordinateDual n‖*((∑ i, N i)*B) := by |
| apply DeGiorgiCompactness.fderiv_norm_le_coord_bound _ _ (by positivity) |
| intro i |
| rw [(hg.1 l).2, jetCone_jet_eq hg.1] |
| exact (hNb i _ hf x).trans ((mul_le_mul_of_nonneg_left hb (hN i)).trans |
| (mul_le_mul_of_nonneg_right (Finset.single_le_sum (fun j _ => hN j) (Finset.mem_univ i)) hB)) |
| _ = _ := by ring |
| change |g l x| ≤ L ∧ ‖fderiv ℝ (g l : Space n → ℝ) x‖ ≤ L |
| constructor |
| · exact hval.trans (mul_le_mul_of_nonneg_right (le_add_of_nonneg_right (mul_nonneg hnorm hsum)) hB) |
| · exact hder.trans (mul_le_mul_of_nonneg_right (le_add_of_nonneg_left hM) hB) |
|
|
| end DeGiorgi |
| end |
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| end OAI |
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