Update complexHarmonic.lean

This commit is contained in:
Stefan Kebekus 2024-05-17 12:31:36 +02:00
parent 82fdc5ac37
commit 348492cc94
1 changed files with 40 additions and 163 deletions

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@ -26,7 +26,7 @@ def Harmonic (f : → F) : Prop :=
(ContDiff 2 f) ∧ (∀ z, Complex.laplace f z = 0)
theorem harmonic_add_harmonic_is_harmonic {f₁ f₂ : → F} (h₁ : Harmonic f₁) (h₂ : Harmonic f₂) :
theorem harmonic_add_harmonic_is_harmonic {f₁ f₂ : → F} (h₁ : Harmonic f₁) (h₂ : Harmonic f₂) :
Harmonic (f₁ + f₂) := by
constructor
· exact ContDiff.add h₁.1 h₂.1
@ -37,6 +37,25 @@ theorem harmonic_add_harmonic_is_harmonic {f₁ f₂ : → F₁} (h₁ : Har
simp
theorem harmonic_smul_const_is_harmonic {f : → F} {c : } (h : Harmonic f) :
Harmonic (c • f) := by
constructor
· exact ContDiff.const_smul c h.1
· rw [laplace_smul h.1]
dsimp
intro z
rw [h.2 z]
simp
theorem harmonic_iff_smul_const_is_harmonic {f : → F} {c : } (hc : c ≠ 0) :
Harmonic f ↔ Harmonic (c • f) := by
constructor
· exact harmonic_smul_const_is_harmonic
· nth_rewrite 2 [((eq_inv_smul_iff₀ hc).mpr rfl : f = c⁻¹ • c • f)]
exact fun a => harmonic_smul_const_is_harmonic a
theorem harmonic_comp_CLM_is_harmonic {f : → F₁} {l : F₁ →L[] G} (h : Harmonic f) :
Harmonic (l ∘ f) := by
@ -149,31 +168,17 @@ theorem log_normSq_of_holomorphic_is_harmonic
(h₃ : ∀ z, f z ∈ Complex.slitPlane) :
Harmonic (Real.log ∘ Complex.normSq ∘ f) := by
-- Suffices to show Harmonic (⇑Complex.ofRealCLM ∘ Real.log ∘ ⇑Complex.normSq ∘ f)
let F := Real.log ∘ Complex.normSq ∘ f
have : Harmonic (Complex.ofRealCLM ∘ F) → Harmonic F := by
intro hyp
have t₁ : Harmonic (Complex.reCLM ∘ Complex.ofRealCLM ∘ F) := harmonic_comp_CLM_is_harmonic hyp
have t₂ : Complex.reCLM ∘ Complex.ofRealCLM ∘ F = F := rfl
rw [t₂] at t₁
exact t₁
apply this
dsimp [F]
suffices hyp : Harmonic (⇑Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f) from
(harmonic_comp_CLM_is_harmonic hyp : Harmonic (Complex.reCLM ∘ Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f))
-- Suffices to show Harmonic (Complex.log ∘ Complex.ofRealCLM ∘ Complex.normSq ∘ f)
have : Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f = Complex.log ∘ Complex.ofRealCLM ∘ Complex.normSq ∘ f := by
unfold Function.comp
funext z
apply Complex.ofReal_log
exact Complex.normSq_nonneg (f z)
rw [this]
-- Suffices to show Harmonic (Complex.log ∘ (⇑(starRingEnd ) ∘ f * f))
have : Complex.ofRealCLM ∘ ⇑Complex.normSq ∘ f = ((starRingEnd ) ∘ f) * f := by
suffices hyp : Harmonic (Complex.log ∘ (((starRingEnd ) ∘ f) * f)) from by
have : Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f = Complex.log ∘ (((starRingEnd ) ∘ f) * f) := by
funext z
simp
exact Complex.normSq_eq_conj_mul_self
rw [Complex.ofReal_log (Complex.normSq_nonneg (f z))]
rw [Complex.normSq_eq_conj_mul_self]
rw [this]
exact hyp
-- Suffices to show Harmonic (Complex.log ∘ ⇑(starRingEnd ) ∘ f + Complex.log ∘ f)
@ -224,55 +229,7 @@ theorem logabs_of_holomorphic_is_harmonic
(h₃ : ∀ z, f z ∈ Complex.slitPlane) :
Harmonic (fun z ↦ Real.log ‖f z‖) := by
/- We start with a number of lemmas on regularity of all the functions involved -/
-- The norm square is real C²
have normSq_is_real_C2 : ContDiff 2 Complex.normSq := by
unfold Complex.normSq
simp
conv =>
arg 3
intro x
rw [← Complex.reCLM_apply, ← Complex.imCLM_apply]
apply ContDiff.add
apply ContDiff.mul
apply ContinuousLinearMap.contDiff Complex.reCLM
apply ContinuousLinearMap.contDiff Complex.reCLM
apply ContDiff.mul
apply ContinuousLinearMap.contDiff Complex.imCLM
apply ContinuousLinearMap.contDiff Complex.imCLM
-- f is real C²
have f_is_real_C2 : ContDiff 2 f :=
ContDiff.restrict_scalars (Differentiable.contDiff h₁)
-- Complex.log ∘ f is real C²
have log_f_is_holomorphic : Differentiable (Complex.log ∘ f) := by
intro z
apply DifferentiableAt.comp
exact Complex.differentiableAt_log (h₃ z)
exact h₁ z
-- Real.log |f|² is real C²
have t₄ : ContDiff 2 (Real.log ∘ ⇑Complex.normSq ∘ f) := by
rw [contDiff_iff_contDiffAt]
intro z
apply ContDiffAt.comp
apply Real.contDiffAt_log.mpr
simp
exact h₂ z
apply ContDiff.comp_contDiffAt z normSq_is_real_C2
exact ContDiff.contDiffAt f_is_real_C2
have t₂ : Complex.log ∘ ⇑(starRingEnd ) ∘ f = Complex.conjCLE ∘ Complex.log ∘ f := by
funext z
unfold Function.comp
rw [Complex.log_conj]
rfl
exact Complex.slitPlane_arg_ne_pi (h₃ z)
constructor
· -- logabs f is real C²
-- Suffices: Harmonic (2⁻¹ • Real.log ∘ ⇑Complex.normSq ∘ f)
have : (fun z ↦ Real.log ‖f z‖) = (2 : )⁻¹ • (Real.log ∘ Complex.normSq ∘ f) := by
funext z
simp
@ -283,87 +240,7 @@ theorem logabs_of_holomorphic_is_harmonic
exact Complex.normSq_nonneg (f z)
rw [this]
have : (2 : )⁻¹ • (Real.log ∘ Complex.normSq ∘ f) = (fun z ↦ (2 : )⁻¹ • ((Real.log ∘ ⇑Complex.normSq ∘ f) z)) := by
exact rfl
rw [this]
apply ContDiff.const_smul
exact t₄
-- Suffices: Harmonic (Real.log ∘ ⇑Complex.normSq ∘ f)
apply (harmonic_iff_smul_const_is_harmonic (inv_ne_zero two_ne_zero)).1
· -- Laplace vanishes
have : (fun z ↦ Real.log ‖f z‖) = (2 : )⁻¹ • (Real.log ∘ Complex.normSq ∘ f) := by
funext z
simp
unfold Complex.abs
simp
rw [Real.log_sqrt]
rw [div_eq_inv_mul (Real.log (Complex.normSq (f z))) 2]
exact Complex.normSq_nonneg (f z)
rw [this]
rw [laplace_smul]
simp
have : ∀ (z : ), Complex.laplace (Real.log ∘ ⇑Complex.normSq ∘ f) z = 0 ↔ Complex.laplace (Complex.ofRealCLM ∘ Real.log ∘ ⇑Complex.normSq ∘ f) z = 0 := by
intro z
rw [laplace_compContLin]
simp
-- ContDiff 2 (Real.log ∘ ⇑Complex.normSq ∘ f)
exact t₄
conv =>
intro z
rw [this z]
have : Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f = Complex.log ∘ Complex.ofRealCLM ∘ Complex.normSq ∘ f := by
unfold Function.comp
funext z
apply Complex.ofReal_log
exact Complex.normSq_nonneg (f z)
rw [this]
have : Complex.ofRealCLM ∘ ⇑Complex.normSq ∘ f = ((starRingEnd ) ∘ f) * f := by
funext z
simp
exact Complex.normSq_eq_conj_mul_self
rw [this]
have : Complex.log ∘ (⇑(starRingEnd ) ∘ f * f) = Complex.log ∘ ⇑(starRingEnd ) ∘ f + Complex.log ∘ f := by
unfold Function.comp
funext z
simp
rw [Complex.log_mul_eq_add_log_iff]
have : Complex.arg ((starRingEnd ) (f z)) = - Complex.arg (f z) := by
rw [Complex.arg_conj]
have : ¬ Complex.arg (f z) = Real.pi := by
exact Complex.slitPlane_arg_ne_pi (h₃ z)
simp
tauto
rw [this]
simp
constructor
· exact Real.pi_pos
· exact Real.pi_nonneg
exact (AddEquivClass.map_ne_zero_iff starRingAut).mpr (h₂ z)
exact h₂ z
rw [this]
rw [laplace_add]
rw [t₂, laplace_compCLE]
intro z
simp
rw [(holomorphic_is_harmonic log_f_is_holomorphic).2 z]
simp
-- ContDiff 2 (Complex.log ∘ f)
exact ContDiff.restrict_scalars (Differentiable.contDiff log_f_is_holomorphic)
-- ContDiff 2 (Complex.log ∘ ⇑(starRingEnd ) ∘ f)
rw [t₂]
apply ContDiff.comp
exact ContinuousLinearEquiv.contDiff Complex.conjCLE
exact ContDiff.restrict_scalars (Differentiable.contDiff log_f_is_holomorphic)
-- ContDiff 2 (Complex.log ∘ f)
exact ContDiff.restrict_scalars (Differentiable.contDiff log_f_is_holomorphic)
-- ContDiff 2 (Real.log ∘ ⇑Complex.normSq ∘ f)
exact t₄
exact log_normSq_of_holomorphic_is_harmonic h₁ h₂ h₃