File size: 3,439 Bytes
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 | import OAI.Analysis.AncientFlat.Profile.SphereCore
namespace OAI
noncomputable section
open scoped ContDiff Laplacian
namespace CompactCoreFinite
open CompactCore
variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
lemma correction_smooth {n : ℕ} (f : V → ℂ) (hf : ContDiff ℝ (n+2 : ℕ) f) :
ContDiff ℝ n (correction f) := by
have hd : ContDiff ℝ (n+1 : ℕ) (fderiv ℝ f) := hf.fderiv_right (by norm_num [Nat.cast_add, add_assoc])
have hdd : ContDiff ℝ n (fderiv ℝ (fderiv ℝ f)) := hd.fderiv_right (by simp)
change ContDiff ℝ n (fun x => radialJet 2 f x+29*radialJet 1 f x)
simp only [radialJet,iteratedFDeriv_one_apply,iteratedFDeriv_two_apply]
exact ((hdd.clm_apply contDiff_id).clm_apply contDiff_id).add
((contDiff_const : ContDiff ℝ n (fun _ : V => (29:ℂ))).mul
((hd.of_le (by simp)).clm_apply contDiff_id))
lemma laplacian_smooth [FiniteDimensional ℝ V] {n : ℕ} (f : V → ℂ) (hf : ContDiff ℝ (n+2 : ℕ) f) :
ContDiff ℝ n (Δ f) := by
rw [InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis]
apply ContDiff.sum
intro i _
have hd : ContDiff ℝ (n+1 : ℕ) (fderiv ℝ f) := hf.fderiv_right (by norm_num [Nat.cast_add, add_assoc])
have hdd : ContDiff ℝ n (fderiv ℝ (fderiv ℝ f)) := hd.fderiv_right (by simp)
simpa only [iteratedFDeriv_two_apply] using
((hdd.clm_apply contDiff_const).clm_apply contDiff_const)
lemma sphereLap_smooth [FiniteDimensional ℝ V] {n : ℕ} (f : V → ℂ) (hf : ContDiff ℝ (n+2 : ℕ) f) :
ContDiff ℝ n (sphereLap f) := (laplacian_smooth f hf).sub (correction_smooth f hf)
lemma radialJet_scaled (j : ℕ) (f : V → ℂ) (hf : ContDiff ℝ j f) (q : ℝ) (x : V) :
radialJet j (fun y => f (q • y)) x=radialJet j f (q • x) := by
simp only [radialJet,iteratedFDeriv_comp_const_smul q
hf,smul_apply]
rw [(iteratedFDeriv ℝ j f (q • x)).map_smul_univ (fun _ => q) (fun _ => x)]
simp
lemma correction_scaled (f : V → ℂ) (hf : ContDiff ℝ 2 f) (q : ℝ) :
correction (fun y => f (q • y))=fun x => correction f (q • x) := by
funext x
simp only [correction,radialJet_scaled 2 f hf q,
radialJet_scaled 1 f (hf.of_le (by norm_num)) q]
lemma laplacian_scaled [FiniteDimensional ℝ V] (f : V → ℂ) (hf : ContDiff ℝ 2 f) (q : ℝ) :
Δ (fun y => f (q • y))=fun x => (q:ℂ)^2*Δ f (q • x) := by
funext x
rw [InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis]
simp_rw [iteratedFDeriv_comp_const_smul (i := 2) q hf]
simp only [smul_apply,
InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis,Complex.real_smul,
Complex.ofReal_pow,←Finset.mul_sum]
lemma correction_sub (f g : V → ℂ) (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 2 g) :
correction (f-g)=correction f-correction g := by
funext x
simp only [correction,radialJet,iteratedFDeriv_sub (i := 2) hf hg,
iteratedFDeriv_sub (i := 1) (hf.of_le (by norm_num)) (hg.of_le (by norm_num)),
Pi.sub_apply,sub_apply]
ring
lemma correction_smul (f : V → ℂ) (hf : ContDiff ℝ 2 f) (a : ℂ) :
correction (a • f)=a • correction f := by
funext x
simp only [correction,radialJet,
iteratedFDeriv_const_smul_apply (i := 2) (hf.contDiffAt),
iteratedFDeriv_const_smul_apply (i := 1) ((hf.of_le (by norm_num)).contDiffAt),
smul_apply,Pi.smul_apply,smul_eq_mul]
ring
end CompactCoreFinite
end
end OAI
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