working…

This commit is contained in:
Stefan Kebekus 2024-08-08 12:36:06 +02:00
parent acb1f34879
commit 3e924d5b4a
3 changed files with 151 additions and 73 deletions

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@ -1,6 +1,6 @@
import Nevanlinna.complexHarmonic
import Nevanlinna.holomorphicAt
import Nevanlinna.holomorphic_primitive
import Nevanlinna.holomorphic_primitive2
import Nevanlinna.mathlibAddOn
@ -108,103 +108,145 @@ function.
theorem harmonic_is_realOfHolomorphic
{f : }
(hf : ∀ z, HarmonicAt f z) :
∃ F : , (∀ z, HolomorphicAt F z) ∧ (Complex.reCLM ∘ F = f) := by
{z : }
{R : }
(hR : 0 < R)
(hf : ∀ x ∈ Metric.ball z R, HarmonicAt f x) :
∃ F : , (∀ z ∈ Metric.ball z R, HolomorphicAt F z) ∧ (Set.EqOn (Complex.reCLM ∘ F) f (Metric.ball z R)) := by
let f_1 : := Complex.ofRealCLM ∘ (partialDeriv 1 f)
let f_I : := Complex.ofRealCLM ∘ (partialDeriv Complex.I f)
let g : := f_1 - Complex.I • f_I
have reg₂f : ContDiff 2 f := by
apply contDiff_iff_contDiffAt.mpr
intro z
exact (hf z).1
have contDiffOn_if_contDiffAt
{f' : }
{z' : }
{R' : }
{n' : }
(hf' : ∀ x ∈ Metric.ball z' R', ContDiffAt n' f' x) :
ContDiffOn n' f' (Metric.ball z' R') := by
intro z hz
apply ContDiffAt.contDiffWithinAt
exact hf' z hz
have reg₁f_1 : ContDiff 1 f_1 := by
apply contDiff_iff_contDiffAt.mpr
intro z
have reg₂f : ContDiffOn 2 f (Metric.ball z R) := by
apply contDiffOn_if_contDiffAt
intro x hx
exact (hf x hx).1
have contDiffOn_if_contDiffAt'
{f' : }
{z' : }
{R' : }
{n' : }
(hf' : ∀ x ∈ Metric.ball z' R', ContDiffAt n' f' x) :
ContDiffOn n' f' (Metric.ball z' R') := by
intro z hz
apply ContDiffAt.contDiffWithinAt
exact hf' z hz
have reg₁f_1 : ContDiffOn 1 f_1 (Metric.ball z R) := by
apply contDiffOn_if_contDiffAt'
intro z hz
dsimp [f_1]
apply ContDiffAt.continuousLinearMap_comp
exact partialDeriv_contDiffAt (hf z).1 1
exact partialDeriv_contDiffAt (hf z hz).1 1
have reg₁f_I : ContDiff 1 f_I := by
apply contDiff_iff_contDiffAt.mpr
intro z
have reg₁f_I : ContDiffOn 1 f_I (Metric.ball z R) := by
apply contDiffOn_if_contDiffAt'
intro z hz
dsimp [f_I]
apply ContDiffAt.continuousLinearMap_comp
exact partialDeriv_contDiffAt (hf z).1 Complex.I
exact partialDeriv_contDiffAt (hf z hz).1 Complex.I
have reg₁g : ContDiff 1 g := by
have reg₁g : ContDiffOn 1 g (Metric.ball z R) := by
dsimp [g]
apply ContDiff.sub
apply ContDiffOn.sub
exact reg₁f_1
apply ContDiff.const_smul'
have : Complex.I • f_I = fun x ↦ Complex.I • f_I x := by rfl
rw [this]
apply ContDiffOn.const_smul
exact reg₁f_I
have reg₁ : Differentiable g := by
intro z
have reg₁ : DifferentiableOn g (Metric.ball z R) := by
intro x hx
apply DifferentiableAt.differentiableWithinAt
apply CauchyRiemann₇.2
constructor
· apply Differentiable.differentiableAt
apply ContDiff.differentiable
exact reg₁g
rfl
· apply DifferentiableWithinAt.differentiableAt (reg₁g.differentiableOn le_rfl x hx)
apply IsOpen.mem_nhds Metric.isOpen_ball hx
· dsimp [g]
rw [partialDeriv_sub₂, partialDeriv_sub₂]
simp
rw [partialDeriv_sub₂_differentiableAt, partialDeriv_sub₂_differentiableAt]
dsimp [f_1, f_I]
rw [partialDeriv_smul'₂, partialDeriv_smul'₂]
rw [partialDeriv_compContLin, partialDeriv_compContLin, partialDeriv_compContLin, partialDeriv_compContLin]
rw [partialDeriv_compContLinAt, partialDeriv_compContLinAt]
simp
rw [partialDeriv_compContLinAt, partialDeriv_compContLinAt]
rw [mul_sub]
simp
rw [← mul_assoc]
simp
rw [add_comm, sub_eq_add_neg]
congr 1
· rw [partialDeriv_comm reg₂f Complex.I 1]
· let A := Filter.EventuallyEq.eq_of_nhds (hf z).2
· rw [partialDeriv_commOn _ reg₂f Complex.I 1]
exact hx
exact Metric.isOpen_ball
· let A := Filter.EventuallyEq.eq_of_nhds (hf x hx).2
simp at A
unfold Complex.laplace at A
conv =>
right
right
rw [← sub_zero (partialDeriv 1 (partialDeriv 1 f) z)]
rw [← sub_zero (partialDeriv 1 (partialDeriv 1 f) x)]
rw [← A]
simp
--Differentiable (partialDeriv _ f)
--DifferentiableAt (partialDeriv _ f)
repeat
apply ContDiff.differentiable
apply contDiff_iff_contDiffAt.mpr
exact fun w ↦ partialDeriv_contDiffAt (hf w).1 _
apply ContDiffAt.differentiableAt
apply partialDeriv_contDiffAt (hf x hx).1
apply le_rfl
-- Differentiable f_1
exact reg₁f_1.differentiable le_rfl
-- Differentiable (Complex.I • f_I)
apply Differentiable.const_smul'
exact reg₁f_I.differentiable le_rfl
-- Differentiable f_1
exact reg₁f_1.differentiable le_rfl
-- Differentiable (Complex.I • f_I)
apply Differentiable.const_smul'
exact reg₁f_I.differentiable le_rfl
let F := fun z ↦ (primitive 0 g) z + f 0
-- DifferentiableAt f_1 x
apply (reg₁f_1.differentiableOn le_rfl).differentiableAt
apply IsOpen.mem_nhds Metric.isOpen_ball hx
have regF : Differentiable F := by
apply Differentiable.add
apply primitive_differentiable reg₁
-- DifferentiableAt (Complex.I • f_I)
have : Complex.I • f_I = fun x ↦ Complex.I • f_I x := by rfl
rw [this]
apply DifferentiableAt.const_smul
apply (reg₁f_I.differentiableOn le_rfl).differentiableAt
apply IsOpen.mem_nhds Metric.isOpen_ball hx
-- Differentiable f_1
apply (reg₁f_1.differentiableOn le_rfl).differentiableAt
apply IsOpen.mem_nhds Metric.isOpen_ball hx
-- Differentiable (Complex.I • f_I)
have : Complex.I • f_I = fun x ↦ Complex.I • f_I x := by rfl
rw [this]
apply DifferentiableAt.const_smul
apply (reg₁f_I.differentiableOn le_rfl).differentiableAt
apply IsOpen.mem_nhds Metric.isOpen_ball hx
let F := fun z' ↦ (primitive z g) z' + f z
have regF : DifferentiableOn F (Metric.ball z R) := by
apply DifferentiableOn.add
apply primitive_differentiableOn reg₁
simp
have pF'' : ∀ x, (fderiv F x) = ContinuousLinearMap.lsmul (g x) := by
intro x
rw [DifferentiableAt.fderiv_restrictScalars (regF x)]
have pF'' : ∀ x ∈ Metric.ball z R, (fderiv F x) = ContinuousLinearMap.lsmul (g x) := by
intro x hx
have : DifferentiableAt F x := by
apply (regF x hx).differentiableAt
apply IsOpen.mem_nhds Metric.isOpen_ball hx
rw [DifferentiableAt.fderiv_restrictScalars this]
dsimp [F]
rw [fderiv_add_const]
rw [primitive_fderiv']
rw [primitive_fderiv' reg₁ x hx]
exact rfl
exact reg₁
use F

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@ -781,12 +781,11 @@ theorem primitive_hasDerivAt
apply hasDerivAt_const
theorem primitive_differentiable
theorem primitive_differentiableOn
{E : Type u} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E]
{f : → E}
(z₀ : )
(R : )
{z₀ : }
{R : }
(hf : DifferentiableOn f (Metric.ball z₀ R))
:
DifferentiableOn (primitive z₀ f) (Metric.ball z₀ R) := by

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@ -102,20 +102,6 @@ theorem partialDeriv_add₂ {f₁ f₂ : E → F} (h₁ : Differentiable 𝕜 f
simp
theorem partialDeriv_sub₂ {f₁ f₂ : E → F} (h₁ : Differentiable 𝕜 f₁) (h₂ : Differentiable 𝕜 f₂) : ∀ v : E, partialDeriv 𝕜 v (f₁ - f₂) = (partialDeriv 𝕜 v f₁) - (partialDeriv 𝕜 v f₂) := by
unfold partialDeriv
intro v
have : f₁ - f₂ = fun y ↦ f₁ y - f₂ y := by rfl
rw [this]
conv =>
left
intro w
left
rw [fderiv_sub (h₁ w) (h₂ w)]
funext w
simp
theorem partialDeriv_add₂_differentiableAt
{f₁ f₂ : E → F}
{v : E}
@ -153,6 +139,57 @@ theorem partialDeriv_add₂_contDiffAt
exact (hf₂' x (Set.mem_of_mem_inter_right hx)).differentiableAt
theorem partialDeriv_sub₂ {f₁ f₂ : E → F} (h₁ : Differentiable 𝕜 f₁) (h₂ : Differentiable 𝕜 f₂) : ∀ v : E, partialDeriv 𝕜 v (f₁ - f₂) = (partialDeriv 𝕜 v f₁) - (partialDeriv 𝕜 v f₂) := by
unfold partialDeriv
intro v
have : f₁ - f₂ = fun y ↦ f₁ y - f₂ y := by rfl
rw [this]
conv =>
left
intro w
left
rw [fderiv_sub (h₁ w) (h₂ w)]
funext w
simp
theorem partialDeriv_sub₂_differentiableAt
{f₁ f₂ : E → F}
{v : E}
{x : E}
(h₁ : DifferentiableAt 𝕜 f₁ x)
(h₂ : DifferentiableAt 𝕜 f₂ x) :
partialDeriv 𝕜 v (f₁ - f₂) x = (partialDeriv 𝕜 v f₁) x - (partialDeriv 𝕜 v f₂) x := by
unfold partialDeriv
have : f₁ - f₂ = fun y ↦ f₁ y - f₂ y := by rfl
rw [this]
rw [fderiv_sub h₁ h₂]
rfl
theorem partialDeriv_sub₂_contDiffAt
{f₁ f₂ : E → F}
{v : E}
{x : E}
(h₁ : ContDiffAt 𝕜 1 f₁ x)
(h₂ : ContDiffAt 𝕜 1 f₂ x) :
partialDeriv 𝕜 v (f₁ - f₂) =ᶠ[nhds x] (partialDeriv 𝕜 v f₁) - (partialDeriv 𝕜 v f₂) := by
obtain ⟨f₁', u₁, hu₁, _, hf₁'⟩ := contDiffAt_one_iff.1 h₁
obtain ⟨f₂', u₂, hu₂, _, hf₂'⟩ := contDiffAt_one_iff.1 h₂
apply Filter.eventuallyEq_iff_exists_mem.2
use u₁ ∩ u₂
constructor
· exact Filter.inter_mem hu₁ hu₂
· intro x hx
simp
apply partialDeriv_sub₂_differentiableAt 𝕜
exact (hf₁' x (Set.mem_of_mem_inter_left hx)).differentiableAt
exact (hf₂' x (Set.mem_of_mem_inter_right hx)).differentiableAt
theorem partialDeriv_compContLin
{f : E → F}
{l : F →L[𝕜] G}