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12dee50 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 | import OAI.Analysis.AncientFlat.Kernel.ClassicalBase
import OAI.Analysis.AncientFlat.Evolution.MovingUnique
import OAI.Analysis.AncientFlat.Evolution.DilationCore
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped ContDiff ZeroAtInfty BoundedContinuousFunction SchwartzMap Laplacian Convolution
namespace BiharmonicKernel
variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
[FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V]
lemma evolution_bounded_jets (t : ℝ) (ht : 0<t) (f : C₀(V,ℂ)) (n : ℕ) :
∃ M : ℝ, ∀ x, ‖iteratedFDeriv ℝ n (fun y => evolution t f y) x‖≤M := by
let g := heatKernel (V := V) t ht
let g' := SchwartzDerivatives.iterated g n
have he : (fun y => evolution t f y)=
ScaledSchwartzConvolution.value (fun _ : V => (1:ℝ))
(ContinuousLinearMap.lsmul ℝ ℂ) f.toBCF g 0 := by
rw [evolution_coe_convolution t ht]
funext x
simp only [ScaledSchwartzConvolution.value,convolution_def,inv_one,one_pow,
one_smul,ContinuousLinearMap.lsmul_apply,ContinuousLinearMap.mul_apply',smul_eq_mul,g]
refine ⟨‖f‖*(∫ y : V, ‖g' y‖),fun x => ?_⟩
rw [he,ScaledSchwartzConvolution.iteratedFDeriv_value_smul_apply
zero_lt_one (fun _ : V => (1:ℝ)) continuous_const (fun _ => ⟨le_rfl,le_rfl⟩)]
simp only [inv_one,one_pow,one_smul]
have hi : Integrable (fun y : V => ‖f‖*‖g' (x-y)‖) :=
(g'.integrable.norm.comp_sub_left x).const_mul ‖f‖
calc
_ ≤ ∫ y : V, ‖f‖*‖g' (x-y)‖ := by
apply norm_integral_le_of_norm_le hi
filter_upwards with y
rw [norm_smul]
rw [←SchwartzDerivatives.iterated_apply]
exact mul_le_mul_of_nonneg_right (f.toBCF.norm_coe_le_norm y) (norm_nonneg (g' (x-y)))
_ = _ := by rw [integral_const_mul,integral_sub_left_eq_self (fun y : V => ‖g' y‖) volume x]
lemma evolution_smul (t : ℝ) (ht : 0<t) (a : ℂ) (f : C₀(V,ℂ)) :
evolution t (a • f)=a • evolution t f := by
rw [evolution_eq_convolve t ht,evolution_eq_convolve t ht,C0Convolution.convolve_smul]
end BiharmonicKernel
namespace FlatSimilarity
open BiharmonicKernel ParametrixDilation
variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
[FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V]
def constantTheta (c t : ℝ) : ℝ := c^2/16*(1-Real.exp (-t))
lemma constantTheta_zero (c : ℝ) : constantTheta c 0=0 := by simp [constantTheta]
lemma constantTheta_pos {c t : ℝ} (hc : 0<c) (ht : 0<t) : 0<constantTheta c t := by
have he : Real.exp (-t)<1 := by simpa using Real.exp_lt_one_iff.mpr (neg_neg_of_pos ht)
unfold constantTheta
positivity
lemma constantTheta_deriv (c t : ℝ) :
HasDerivAt (constantTheta c) (Real.exp (-t)/16*c^2) t := by
have he : HasDerivAt (fun r : ℝ => Real.exp (-r)) (-Real.exp (-t)) t := by
convert (hasDerivAt_id t).neg.exp using 1 <;> simp
exact ((he.const_sub 1).const_mul (c^2/16)).congr_deriv (by ring)
def constantMoving (c : ℝ) (d : C₀(V,ℂ)) (t : ℝ) : C₀(V,ℂ) :=
(Real.exp (-t/2) : ℝ) • evolution (constantTheta c t) d
def toBCFReal : C₀(V,ℂ) →L[ℝ] (V →ᵇ ℂ) := LinearMap.mkContinuous
{ toFun := ZeroAtInftyContinuousMap.toBCF
map_add' := fun _ _ => rfl
map_smul' := fun _ _ => rfl } 1 (fun f => by
change ‖f.toBCF‖≤1*‖f‖
simp)
omit [MeasurableSpace V] [BorelSpace V] in
lemma bilaplacian_real_smul {f : V → ℂ} (hf : ContDiff ℝ 4 f) (a : ℝ) :
Δ (Δ (fun y => a • f y))=fun y => a • Δ (Δ f) y := by
have h2 : ContDiff ℝ 2 f := hf.of_le (by norm_num)
have h1 : Δ (fun y => a • f y)=fun y => a • Δ f y :=
funext (fun y => InnerProductSpace.laplacian_smul a (h2.contDiffAt (x := y)))
rw [h1]
exact funext (fun y => InnerProductSpace.laplacian_smul a
((ClassicalIBP.contDiff_laplacian f 2 hf).contDiffAt (x := y)))
lemma constantMoving_deriv (c : ℝ) (hc : 0<c) (d : C₀(V,ℂ)) (t : ℝ) (ht : 0<t) :
∃ p : V →ᵇ ℂ, HasDerivAt (fun r => (constantMoving c d r).toBCF) p t ∧
∀ x, p x=-((Real.exp (-t)/16*c^2 : ℝ) : ℂ)*
Δ (Δ (fun y => constantMoving c d t y)) x-(1/2+(0:ℂ))*constantMoving c d t x := by
let θ := constantTheta c t
have hθ : 0<θ := constantTheta_pos hc ht
let v := -C0Convolution.convolve (Δ (Δ (heatKernel (V := V) θ hθ) : SchwartzMap V ℂ) : SchwartzMap V ℂ) d
have hv := (evolution_hasDerivAt θ hθ d).scomp t (constantTheta_deriv c t)
have he : HasDerivAt (fun r : ℝ => Real.exp (-r/2)) (-Real.exp (-t/2)/2) t := by
convert ((hasDerivAt_id t).neg.div_const 2).exp using 1 <;>
simp [Pi.neg_apply,div_eq_mul_inv]
have hu := he.smul hv
let p : C₀(V,ℂ) := Real.exp (-t/2) • ((Real.exp (-t)/16*c^2) • v)+
(-Real.exp (-t/2)/2) • evolution θ d
refine ⟨p.toBCF,?_,fun x => ?_⟩
· exact (toBCFReal (V := V)).hasFDerivAt.comp_hasDerivAt t hu
· have hvx : v x=-Δ (Δ (fun y => evolution θ d y)) x := by
have h1 := (evolution_heat_equation θ hθ d x)
have h2 := ((C0Convolution.eval x).restrictScalars ℝ).hasFDerivAt.comp_hasDerivAt θ
(evolution_hasDerivAt θ hθ d)
exact h2.unique h1
dsimp only [p]
simp only [ZeroAtInftyContinuousMap.toBCF_apply,ZeroAtInftyContinuousMap.add_apply,
ZeroAtInftyContinuousMap.smul_apply,hvx]
change _=-((Real.exp (-t)/16*c^2 : ℝ) : ℂ)*
Δ (Δ (fun y => Real.exp (-t/2) • evolution θ d y)) x-
(1/2+(0:ℂ))*(Real.exp (-t/2) • evolution θ d x)
have hsp : ContDiff ℝ 4 (fun y => evolution θ d y) :=
(evolution_smooth θ hθ d).of_le (by exact WithTop.coe_le_coe.mpr (le_top : (4:ℕ∞)≤⊤))
rw [bilaplacian_real_smul hsp]
simp only [Complex.real_smul,Complex.ofReal_div,Complex.ofReal_mul,
Complex.ofReal_neg,Complex.ofReal_ofNat,add_zero]
ring
lemma constantMoving_classical (c : ℝ) (hc : 0<c) (d : C₀(V,ℂ)) :
MovesClassically (fun _ => c^2) (fun _ => 0) d (constantMoving c d) := by
refine ⟨?_,?_,?_,?_,?_⟩
· simp [constantMoving,constantTheta_zero,evolution_zero]
· exact ((Real.continuous_exp.comp (continuous_neg.div_const 2)).smul
((evolution_continuous d).comp (by unfold constantTheta; fun_prop))).continuousOn
· intro t ht
exact contDiff_const.smul ((evolution_smooth _ (constantTheta_pos hc ht.1) d).of_le (by exact WithTop.coe_le_coe.mpr (le_top : (4:ℕ∞)≤⊤)))
· intro t ht n hn
obtain ⟨M,hM⟩ := evolution_bounded_jets (constantTheta c t) (constantTheta_pos hc ht.1) d n
refine ⟨|Real.exp (-t/2)| * M,fun x => ?_⟩
change ‖iteratedFDeriv ℝ n (fun y => Real.exp (-t/2) • evolution (constantTheta c t) d y) x‖≤_
rw [iteratedFDeriv_const_smul_apply'
((evolution_smooth _ (constantTheta_pos hc ht.1) d).of_le
(by exact_mod_cast (le_top : (n:ℕ∞)≤⊤))).contDiffAt,norm_smul,Real.norm_eq_abs]
exact mul_le_mul_of_nonneg_left (hM x) (abs_nonneg _)
· intro t ht
exact constantMoving_deriv c hc d t ht.1
end FlatSimilarity
end
end OAI
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