| import OAI.Analysis.AncientFlat.Kernel.ClassicalBase |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped SchwartzMap Topology ZeroAtInfty BoundedContinuousFunction Laplacian ContDiff |
| namespace FrozenParametrix |
| open BiharmonicKernel |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] |
| variable {α β : ℝ} |
| lemma c0Time_eq_sourceValue (hα : 0 < α) (a : V →ᵇ Icc α β) |
| (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) (x : V) : |
| c0Time hα a q f x = SourceFrozenRows.value (heatKernel 1 zero_lt_one) |
| (fun y => (a y).val) (q^(1/4 : ℝ)) f.toBCF x := by |
| rw [c0Time_apply hα a q hq f, timeValue_eq_scaled hα a.toContinuousMap q hq f] |
| unfold SourceFrozenRows.value |
| have he : sourceScale a.toContinuousMap q = fun y => q^(1/4 : ℝ)*(a y).val^(1/4 : ℝ) := by |
| funext y |
| unfold sourceScale |
| rw [Real.sqrt_eq_rpow,Real.sqrt_eq_rpow] |
| change ((q*(a y).val)^(1/2 : ℝ))^(1/2 : ℝ) = _ |
| rw [← Real.rpow_mul (mul_nonneg hq.le (hα.trans_le (a y).property.1).le)] |
| norm_num |
| exact Real.mul_rpow hq.le (hα.trans_le (a y).property.1).le |
| rw [he] |
| rfl |
| end FrozenParametrix |
|
|
| namespace ParametrixVolterra |
| section ExtendedA |
| open FrozenParametrix StrongWeightedVolterra BiharmonicKernel |
| variable {V : Type} [NormedAddCommGroup V] |
| variable {T α β : ℝ} |
| variable (A : C(Slab T,V →ᵇ Icc α β)) (hT : 0 ≤ T) |
|
|
| def extendedA (s : ℝ) (x : V) : ℝ := (A (projIcc 0 T hT s) x).val |
|
|
| lemma continuous_extendedA : Continuous (Function.uncurry (extendedA A hT)) := by |
| change Continuous (fun p : ℝ × V => (A (projIcc 0 T hT p.1) p.2).val) |
| have h : Continuous (fun p : ℝ × V => A (projIcc 0 T hT p.1) p.2) := by fun_prop |
| exact continuous_subtype_val.comp h |
|
|
| lemma extendedA_range (s : ℝ) (x : V) : extendedA A hT s x ∈ Icc α β := |
| (A (projIcc 0 T hT s) x).property |
|
|
| lemma extendedA_lipschitz (L : ℝ) |
| (hLip : ∀ s r : Slab T, ∀ x z : V, |
| |(A s x).val-(A r z).val| ≤ L*(‖x-z‖+|s.val-r.val|)) : |
| ∀ s x y, ‖extendedA A hT s x-extendedA A hT s y‖ ≤ L*‖x-y‖ := by |
| intro s x y |
| simpa only [extendedA,Real.norm_eq_abs,sub_self,abs_zero,add_zero] using |
| hLip (projIcc 0 T hT s) (projIcc 0 T hT s) x y |
|
|
| variable [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] |
| variable (hα : 0 < α) |
| lemma duhamel_eq_sourceDuhamel (j : Time T →ᵇ C₀(V,ℂ)) (t : Time T) (x : V) : |
| duhamel hα A j t.val x = SourceFrozenRows.duhamel (heatKernel 1 zero_lt_one) |
| (extendedA A hT) (fun s => (unweighted j s).toBCF) t.val x := by |
| unfold duhamel SourceFrozenRows.duhamel |
| rw [HolderIntegral.intervalIntegral_apply _ (intervalIntegrable_sourceFrozen hα A j t.val t.val |
| ⟨t.property.1.le,t.property.2⟩) x, |
| intervalIntegral.integral_of_le t.property.1.le,integral_Ioc_eq_integral_Ioo] |
| apply setIntegral_congr_fun measurableSet_Ioo |
| intro s hs |
| have hsT : s ∈ Ioc (0:ℝ) T := ⟨hs.1,hs.2.le.trans t.property.2⟩ |
| dsimp only |
| rw [VolterraIntegral.sourceExtend_apply _ _ _ hsT] |
| dsimp [sourceFrozen] |
| rw [c0Time_eq_sourceValue hα _ _ (sub_pos.mpr hs.2)] |
| have he : projIcc 0 T hT s = now (⟨s,hsT⟩ : Time T) := |
| projIcc_of_mem hT ⟨hs.1.le,hsT.2⟩ |
| have heA : extendedA A hT s = fun y => (A (now (⟨s,hsT⟩ : Time T)) y).val := by |
| funext y |
| unfold extendedA |
| rw [he] |
| rw [heA] |
|
|
| include hT in |
| lemma contDiff_duhamel_of_holder (L : ℝ) (hL : 0 ≤ L) |
| (hLip : ∀ s r : Slab T, ∀ x z : V, |
| |(A s x).val-(A r z).val| ≤ L*(‖x-z‖+|s.val-r.val|)) |
| (j : Time T →ᵇ C₀(V,ℂ)) (H : ℝ) (hH : 0 ≤ H) |
| (hj : ∀ s : Time T, ∀ x y, |
| ‖unweighted j s.val x-unweighted j s.val y‖ ≤ H*s.val^(-7/8 : ℝ)*‖x-y‖^(1/4 : ℝ)) |
| (t : Time T) : ContDiff ℝ 4 (fun x => duhamel hα A j t.val x) := by |
| have hf : StronglyMeasurable (fun s => (unweighted j s).toBCF) := |
| ZeroAtInftyContinuousMap.isometry_toBCF.continuous.comp_stronglyMeasurable |
| (stronglyMeasurable_unweighted j) |
| have hn : ∀ s ∈ Ioo 0 t.val, ‖(unweighted j s).toBCF‖ ≤ ‖j‖*s^(-(3/4 : ℝ)) := by |
| intro s hs |
| rw [ZeroAtInftyContinuousMap.norm_toBCF_eq_norm] |
| simpa only [neg_div,mul_comm] using norm_unweighted_le j s hs.1 |
| have hh : ∀ s ∈ Ioo 0 t.val, ∀ x y : V, |
| ‖(unweighted j s).toBCF x-(unweighted j s).toBCF y‖ ≤ |
| (H*s^(-(7/8 : ℝ)))*‖x-y‖^(1/4 : ℝ) := by |
| intro s hs x y |
| change ‖unweighted j s x-unweighted j s y‖ ≤ _ |
| simpa only [neg_div] using hj ⟨s,hs.1,hs.2.le.trans t.property.2⟩ x y |
| have he := funext (duhamel_eq_sourceDuhamel A hT hα j t) |
| rw [he] |
| exact WeightedSourceFrozenRows.contDiff_duhamel (heatKernel 1 zero_lt_one) (extendedA A hT) |
| (continuous_extendedA A hT) α β hα (extendedA_range A hT) L hL (extendedA_lipschitz A hT L hLip) |
| (fun s => (unweighted j s).toBCF) hf (3/4) (7/8) ‖j‖ H (1/4) t.val |
| (by norm_num) (by norm_num) (norm_nonneg j) hH t.property.1 (by norm_num) (by norm_num) hn hh |
| end ExtendedA |
|
|
| open FrozenParametrix StrongWeightedVolterra |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] |
| variable {T α β : ℝ} (hα : 0 < α) (A : C(Slab T,V →ᵇ Icc α β)) |
| (b : C(Slab T,V →ᵇ ℂ)) |
|
|
| lemma contDiff_mildTime_of_correction (hT₀ : 0 ≤ T) (hT : T ≤ 1) (L B : ℝ) |
| (hL : 0 ≤ L) (hB : 0 ≤ B) (hb : ∀ s, ‖b s‖ ≤ B) |
| (hLip : ∀ s r : Slab T, ∀ x z : V, |
| |(A s x).val-(A r z).val| ≤ L*(‖x-z‖+|s.val-r.val|)) |
| (hLipb : ∀ s : Slab T, ∀ x z : V, ‖b s x-b s z‖ ≤ L*‖x-z‖) |
| (d : C₀(V,ℂ)) (j : Time T →ᵇ C₀(V,ℂ)) |
| (hi : ∀ t : Time T, IntervalIntegrable |
| (fun r => residualKernel hα A b hT₀ t r (unweighted j r)) volume 0 t.val) |
| (hj : ∀ t : Time T, unweighted j t.val + ∫ r in 0..t.val, |
| residualKernel hα A b hT₀ t r (unweighted j r) = -actualResidual hα A b t (initial hT₀) d) |
| (t : Time T) : ContDiff ℝ 4 (fun x => mildTime hα A hT₀ d j t.val x) := by |
| obtain ⟨C,hC,hR⟩ := exists_actualResidual_holder_bound hα A b hT L B hL hB hb hLip hLipb |
| (1/4) (by norm_num) (by norm_num) |
| have hR' : ∀ (t : Time T) (s : Slab T), s.val<t.val → ∀ (f : C₀(V,ℂ)) (x y : V), |
| ‖actualResidual hα A b t s f x-actualResidual hα A b t s f y‖ ≤ |
| C*‖f‖*(t.val-s.val)^(-7/8 : ℝ)*‖x-y‖^(1/4 : ℝ) := by |
| convert hR using 1; norm_num |
| obtain ⟨H,hH,hholder⟩ := correction_holder hα A b hT₀ hT C hC hR' d 0 j 0 le_rfl |
| (by intro t x y; simp) hi (by simpa using hj) |
| have h0 : ContDiff ℝ 4 (fun x => c0Time hα (A (initial hT₀)) t.val d x) := by |
| have he := funext (c0Time_apply hα (A (initial hT₀)) t.val t.property.1 d) |
| rw [he] |
| exact contDiff_infty.mp (contDiff_timeValue hα (A (initial hT₀)).toContinuousMap t.val t.property.1 d) 4 |
| exact h0.add (contDiff_duhamel_of_holder A hT₀ hα L hL hLip j H hH hholder t) |
| end ParametrixVolterra |
|
|
| end |
|
|
| end OAI |
|
|
