| import OAI.Analysis.AncientFlat.Profile.SphereCore |
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|
| 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 |
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| end OAI |
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