| 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 |
|
|
