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Stefan Kebekus 2024-06-04 10:27:46 +02:00
parent 4d3332b15d
commit 6eea56e788
3 changed files with 55 additions and 49 deletions
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18
Nevanlinna/complexHarmonic.examples.lean
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@ -249,6 +249,18 @@ theorem log_normSq_of_holomorphicOn_is_harmonicOn
(h₂ : ∀ z ∈ s, f z ≠ 0) :
HarmonicOn (Real.log ∘ Complex.normSq ∘ f) s := by
have slitPlaneLemma {z : ℂ} (hz : z ≠ 0) : z ∈ Complex.slitPlane ∨ -z ∈ Complex.slitPlane := by
rw [Complex.mem_slitPlane_iff]
rw [Complex.mem_slitPlane_iff]
simp at hz
rw [Complex.ext_iff] at hz
push_neg at hz
simp at hz
simp
by_contra contra
push_neg at contra
exact hz (le_antisymm contra.1.1 contra.2.1) contra.1.2
let s₁ : Set ℂ := { z | f z ∈ Complex.slitPlane} ∩ s
have hs₁ : IsOpen s₁ := by
@ -318,10 +330,8 @@ theorem log_normSq_of_holomorphicOn_is_harmonicOn
· exact hs₂
· constructor
· constructor
· simp
rw [Complex.mem_slitPlane_iff]
rw [Complex.mem_slitPlane_iff] at hfz
simp at hfz
· apply Or.resolve_left (slitPlaneLemma (h₂ z hz)) hfz
· exact hz
· have : s₂ = s ∩ s₂ := by
apply Set.right_eq_inter.mpr
exact Set.inter_subset_right {z | -f z ∈ Complex.slitPlane} s

9
Nevanlinna/complexHarmonic.lean
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@ -23,11 +23,12 @@ variable {G₁ : Type*} [NormedAddCommGroup G₁] [NormedSpace ℂ G₁] [Comple
def Harmonic (f : ℂ → F) : Prop :=
(ContDiff ℝ 2 f) ∧ (∀ z, Complex.laplace f z = 0)
(ContDiff ℝ 2 f) ∧ (∀ z, Δ f z = 0)
def HarmonicOn (f : ℂ → F) (s : Set ℂ) : Prop :=
(ContDiffOn ℝ 2 f s) ∧ (∀ z ∈ s, Complex.laplace f z = 0)
(ContDiffOn ℝ 2 f s) ∧ (∀ z ∈ s, Δ f z = 0)
theorem HarmonicOn_of_locally_HarmonicOn {f : ℂ → F} {s : Set ℂ} (h : ∀ x ∈ s, ∃ (u : Set ℂ), IsOpen u ∧ x ∈ u ∧ HarmonicOn f (s ∩ u)) :
@ -135,7 +136,7 @@ theorem harmonic_comp_CLM_is_harmonic {f : ℂ → F₁} {l : F₁ →L[ℝ] G}
apply ContDiff.comp
exact ContinuousLinearMap.contDiff l
exact h.1
· rw [laplace_compContLin]
· rw [laplace_compCLM]
simp
intro z
rw [h.2 z]
@ -152,7 +153,7 @@ theorem harmonicOn_comp_CLM_is_harmonicOn {f : ℂ → F₁} {s : Set ℂ} {l :
exact h.1
· -- Vanishing of Laplace
intro z zHyp
rw [laplace_compContLinAt]
rw [laplace_compCLMAt]
simp
rw [h.2 z]
simp

75
Nevanlinna/laplace.lean
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@ -20,18 +20,25 @@ variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
variable {G : Type*} [NormedAddCommGroup G] [NormedSpace ℝ G]
noncomputable def Complex.laplace : (ℂ → F) → (ℂ → F) :=
fun f ↦ partialDeriv ℝ 1 (partialDeriv ℝ 1 f) + partialDeriv ℝ Complex.I (partialDeriv ℝ Complex.I f)
noncomputable def Complex.laplace (f : ℂ → F) : ℂ → F :=
partialDeriv ℝ 1 (partialDeriv ℝ 1 f) + partialDeriv ℝ Complex.I (partialDeriv ℝ Complex.I f)
notation "Δ" => Complex.laplace
theorem laplace_eventuallyEq {f₁ f₂ : ℂ → F} {x : ℂ} (h : f₁ =ᶠ[nhds x] f₂) : Complex.laplace f₁ x = Complex.laplace f₂ x := by
theorem laplace_eventuallyEq {f₁ f₂ : ℂ → F} {x : ℂ} (h : f₁ =ᶠ[nhds x] f₂) : Δ f₁ x = Δ f₂ x := by
unfold Complex.laplace
simp
rw [partialDeriv_eventuallyEq ℝ (partialDeriv_eventuallyEq' ℝ h 1) 1]
rw [partialDeriv_eventuallyEq ℝ (partialDeriv_eventuallyEq' ℝ h Complex.I) Complex.I]
theorem laplace_add {f₁ f₂ : ℂ → F} (h₁ : ContDiff ℝ 2 f₁) (h₂ : ContDiff ℝ 2 f₂): Complex.laplace (f₁ + f₂) = (Complex.laplace f₁) + (Complex.laplace f₂) := by
theorem laplace_add
{f₁ f₂ : ℂ → F}
(h₁ : ContDiff ℝ 2 f₁)
(h₂ : ContDiff ℝ 2 f₂) :
Δ (f₁ + f₂) = (Δ f₁) + (Δ f₂) := by
unfold Complex.laplace
rw [partialDeriv_add₂]
rw [partialDeriv_add₂]
@ -59,7 +66,7 @@ theorem laplace_add_ContDiffOn
(hs : IsOpen s)
(h₁ : ContDiffOn ℝ 2 f₁ s)
(h₂ : ContDiffOn ℝ 2 f₂ s) :
∀ x ∈ s, Complex.laplace (f₁ + f₂) x = (Complex.laplace f₁) x + (Complex.laplace f₂) x := by
∀ x ∈ s, Δ (f₁ + f₂) x = (Δ f₁) x + (Δ f₂) x := by
unfold Complex.laplace
simp
@ -124,39 +131,21 @@ theorem laplace_add_ContDiffOn
rw [add_comm]
theorem laplace_smul {f : ℂ → F} : ∀ v : ℝ, Complex.laplace (v • f) = v • (Complex.laplace f) := by
theorem laplace_smul {f : ℂ → F} : ∀ v : ℝ, Δ (v • f) = v • (Δ f) := by
intro v
unfold Complex.laplace
rw [partialDeriv_smul₂]
rw [partialDeriv_smul₂]
rw [partialDeriv_smul₂]
repeat
rw [partialDeriv_smul₂]
simp
theorem laplace_compContLin {f : ℂ → F} {l : F →L[ℝ] G} (h : ContDiff ℝ 2 f) :
Complex.laplace (l ∘ f) = l ∘ (Complex.laplace f) := by
unfold Complex.laplace
rw [partialDeriv_compContLin]
rw [partialDeriv_compContLin]
rw [partialDeriv_compContLin]
rw [partialDeriv_compContLin]
simp
theorem laplace_compCLMAt {f : ℂ → F} {l : F →L[ℝ] G} {x : ℂ} (h : ContDiffAt ℝ 2 f x) :
Δ (l ∘ f) x = (l ∘ (Δ f)) x := by
exact (partialDeriv_contDiff ℝ h Complex.I).differentiable le_rfl
exact h.differentiable one_le_two
exact (partialDeriv_contDiff ℝ h 1).differentiable le_rfl
exact h.differentiable one_le_two
theorem laplace_compContLinAt {f : ℂ → F} {l : F →L[ℝ] G} {x : ℂ} (h : ContDiffAt ℝ 2 f x) :
Complex.laplace (l ∘ f) x = (l ∘ (Complex.laplace f)) x := by
have A₂ : ∃ v ∈ nhds x, (IsOpen v) ∧ (x ∈ v) ∧ (ContDiffOn ℝ 2 f v) := by
have : ∃ u ∈ nhds x, ContDiffOn ℝ 2 f u := by
obtain ⟨v, hv₁, hv₂, hv₃, hv₄⟩ : ∃ v ∈ nhds x, (IsOpen v) ∧ (x ∈ v) ∧ (ContDiffOn ℝ 2 f v) := by
obtain ⟨u, hu₁, hu₂⟩ : ∃ u ∈ nhds x, ContDiffOn ℝ 2 f u := by
apply ContDiffAt.contDiffOn h
rfl
obtain ⟨u, hu₁, hu₂⟩ := this
obtain ⟨v, hv₁, hv₂, hv₃⟩ := mem_nhds_iff.1 hu₁
use v
constructor
@ -166,7 +155,6 @@ theorem laplace_compContLinAt {f : ℂ → F} {l : F →L[ℝ] G} {x : ℂ} (h :
constructor
· exact hv₃
· exact ContDiffOn.congr_mono hu₂ (fun x => congrFun rfl) hv₁
obtain ⟨v, hv₁, hv₂, hv₃, hv₄⟩ := A₂
have D : ∀ w : ℂ, partialDeriv ℝ w (l ∘ f) =ᶠ[nhds x] l ∘ partialDeriv ℝ w (f) := by
intro w
@ -192,17 +180,24 @@ theorem laplace_compContLinAt {f : ℂ → F} {l : F →L[ℝ] G} {x : ℂ} (h :
simp
-- DifferentiableAt ℝ (partialDeriv ℝ Complex.I f) x
apply ContDiffAt.differentiableAt (partialDeriv_contDiffAt ℝ (ContDiffOn.contDiffAt hv₄ hv₁) Complex.I)
rfl
apply ContDiffAt.differentiableAt (partialDeriv_contDiffAt ℝ (ContDiffOn.contDiffAt hv₄ hv₁) Complex.I) le_rfl
-- DifferentiableAt ℝ (partialDeriv ℝ 1 f) x
apply ContDiffAt.differentiableAt (partialDeriv_contDiffAt ℝ (ContDiffOn.contDiffAt hv₄ hv₁) 1)
rfl
apply ContDiffAt.differentiableAt (partialDeriv_contDiffAt ℝ (ContDiffOn.contDiffAt hv₄ hv₁) 1) le_rfl
theorem laplace_compCLE {f : ℂ → F} {l : F ≃L[ℝ] G} (h : ContDiff ℝ 2 f) :
Complex.laplace (l ∘ f) = l ∘ (Complex.laplace f) := by
let l' := (l : F →L[ℝ] G)
have : Complex.laplace (l' ∘ f) = l' ∘ (Complex.laplace f) := by
exact laplace_compContLin h
exact this
theorem laplace_compCLM
{f : ℂ → F}
{l : F →L[ℝ] G}
(h : ContDiff ℝ 2 f) :
Δ (l ∘ f) = l ∘ (Δ f) := by
funext z
exact laplace_compCLMAt h.contDiffAt
theorem laplace_compCLE {f : ℂ → F} {l : F ≃L[ℝ] G} :
Δ (l ∘ f) = l ∘ (Δ f) := by
unfold Complex.laplace
repeat
rw [partialDeriv_compCLE]
simp