File size: 3,412 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 69 70 71 72 73 | import OAI.Analysis.AncientFlat.Evolution.EvolutionCore
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff ZeroAtInfty BoundedContinuousFunction Laplacian
namespace ParametrixEvolution
variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
[FiniteDimensional ℝ V]
variable {α β : ℝ}
lemma bilaplacian_add {f g : V → ℂ} (hf : ContDiff ℝ 4 f) (hg : ContDiff ℝ 4 g) :
Δ (Δ (fun x => f x+g x))=fun x => Δ (Δ f) x+Δ (Δ g) x := by
have hh : Δ (fun x => f x+g x)=fun x => Δ f x+Δ g x :=
funext (fun x => (hf.of_le (by norm_num)).contDiffAt.laplacian_add
(hg.of_le (by norm_num)).contDiffAt)
rw [hh]
exact funext (fun x => (ClassicalIBP.contDiff_laplacian f 2 hf).contDiffAt.laplacian_add
(ClassicalIBP.contDiff_laplacian g 2 hg).contDiffAt)
lemma bilaplacian_smul {f : V → ℂ} (hf : ContDiff ℝ 4 f) (c : ℂ) :
Δ (Δ (fun x => c • f x))=fun x => c • Δ (Δ f) x := by
have hh : Δ (fun x => c • f x)=fun x => c • Δ f x :=
funext (fun x => InnerProductSpace.laplacian_smul c (hf.of_le (by norm_num)).contDiffAt)
rw [hh]
exact funext (fun x => InnerProductSpace.laplacian_smul c
(ClassicalIBP.contDiff_laplacian f 2 hf).contDiffAt)
variable {A : C(ℝ,V →ᵇ Icc α β)} {b : C(ℝ,V →ᵇ ℂ)}
{s T : ℝ} {d e : C₀(V,ℂ)} {u v : ℝ → C₀(V,ℂ)}
lemma Evolves.add (hu : Evolves A b s T d u) (hv : Evolves A b s T e v) :
Evolves A b s T (d+e) (fun t => u t+v t) := by
refine ⟨by simp [hu.1,hv.1],hu.2.1.add hv.2.1,?_,?_,?_⟩
· intro t ht
exact (hu.2.2.1 t ht).add (hv.2.2.1 t ht)
· intro t ht n hn
obtain ⟨M,hM⟩ := hu.2.2.2.1 t ht n hn
obtain ⟨N,hN⟩ := hv.2.2.2.1 t ht n hn
refine ⟨M+N,fun x => ?_⟩
have hcu : ContDiff ℝ n (fun y => u t y) := (hu.2.2.1 t ht).of_le (by exact_mod_cast hn)
have hcv : ContDiff ℝ n (fun y => v t y) := (hv.2.2.1 t ht).of_le (by exact_mod_cast hn)
change ‖iteratedFDeriv ℝ n (fun y => u t y+v t y) x‖≤_
rw [fun_iteratedFDeriv_add hcu hcv]
exact (norm_add_le _ _).trans (add_le_add (hM x) (hN x))
· intro t ht
obtain ⟨p,hp,hpe⟩ := hu.2.2.2.2 t ht
obtain ⟨q,hq,hqe⟩ := hv.2.2.2.2 t ht
refine ⟨p+q,hp.add hq,fun x => ?_⟩
change p x+q x= -((A t x).val : ℂ)*Δ (Δ (fun y => u t y+v t y)) x-b t x*(u t x+v t x)
rw [hpe,hqe,bilaplacian_add (hu.2.2.1 t ht) (hv.2.2.1 t ht)]
ring
lemma Evolves.smul (hu : Evolves A b s T d u) (c : ℂ) :
Evolves A b s T (c•d) (fun t => c•u t) := by
refine ⟨by simp [hu.1],hu.2.1.const_smul c,?_,?_,?_⟩
· intro t ht
exact (hu.2.2.1 t ht).const_smul c
· intro t ht n hn
obtain ⟨M,hM⟩ := hu.2.2.2.1 t ht n hn
refine ⟨‖c‖*M,fun x => ?_⟩
have hc : ContDiff ℝ n (fun y => u t y) := (hu.2.2.1 t ht).of_le (by exact_mod_cast hn)
change ‖iteratedFDeriv ℝ n (fun y => c • u t y) x‖≤_
rw [iteratedFDeriv_const_smul_apply' hc.contDiffAt]
simpa only [Pi.smul_apply,norm_smul] using mul_le_mul_of_nonneg_left (hM x) (norm_nonneg c)
· intro t ht
obtain ⟨p,hp,hpe⟩ := hu.2.2.2.2 t ht
refine ⟨c•p,hp.const_smul c,fun x => ?_⟩
change c • p x= -((A t x).val : ℂ)*Δ (Δ (fun y => c•u t y)) x-b t x*(c•u t x)
rw [hpe,bilaplacian_smul (hu.2.2.1 t ht) c]
simp only [smul_eq_mul]
ring
end ParametrixEvolution
end
end OAI
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