All done.

This commit is contained in:
Stefan Kebekus 2024-06-03 13:35:53 +02:00
parent 40659c2f17
commit c2c730e510
2 changed files with 37 additions and 11 deletions

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@ -407,11 +407,7 @@ theorem log_normSq_of_holomorphicOn_is_harmonicOn
-- HarmonicOn (Complex.log ∘ ⇑(starRingEnd ) ∘ f) s
apply holomorphicOn_is_harmonicOn hs
apply?
exact h₁ z
exact DifferentiableOn.clog h₁ h₃
theorem log_normSq_of_holomorphic_is_harmonic
{f : }

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@ -65,22 +65,52 @@ theorem laplace_add_ContDiffOn
simp
intro x hx
have hf₁ : ∀ z ∈ s, DifferentiableAt f₁ z := by
intro z hz
convert DifferentiableOn.differentiableAt _ (IsOpen.mem_nhds hs hz)
apply ContDiffOn.differentiableOn h₁ one_le_two
have hf₂ : ∀ z ∈ s, DifferentiableAt f₂ z := by
intro z hz
convert DifferentiableOn.differentiableAt _ (IsOpen.mem_nhds hs hz)
apply ContDiffOn.differentiableOn h₂ one_le_two
have : partialDeriv 1 (f₁ + f₂) =ᶠ[nhds x] (partialDeriv 1 f₁) + (partialDeriv 1 f₂) := by
sorry
apply Filter.eventuallyEq_iff_exists_mem.2
use s
constructor
· exact IsOpen.mem_nhds hs hx
· intro z hz
apply partialDeriv_add₂_differentiableAt
exact hf₁ z hz
exact hf₂ z hz
rw [partialDeriv_eventuallyEq this]
have t₁ : DifferentiableAt (partialDeriv 1 f₁) x := by
sorry
let B₀ := (h₁ x hx).contDiffAt (IsOpen.mem_nhds hs hx)
let A₀ := partialDeriv_contDiffAt B₀ 1
exact A₀.differentiableAt (Submonoid.oneLE.proof_2 ℕ∞)
have t₂ : DifferentiableAt (partialDeriv 1 f₂) x := by
sorry
let B₀ := (h₂ x hx).contDiffAt (IsOpen.mem_nhds hs hx)
let A₀ := partialDeriv_contDiffAt B₀ 1
exact A₀.differentiableAt (Submonoid.oneLE.proof_2 ℕ∞)
rw [partialDeriv_add₂_differentiableAt t₁ t₂]
have : partialDeriv Complex.I (f₁ + f₂) =ᶠ[nhds x] (partialDeriv Complex.I f₁) + (partialDeriv Complex.I f₂) := by
sorry
apply Filter.eventuallyEq_iff_exists_mem.2
use s
constructor
· exact IsOpen.mem_nhds hs hx
· intro z hz
apply partialDeriv_add₂_differentiableAt
exact hf₁ z hz
exact hf₂ z hz
rw [partialDeriv_eventuallyEq this]
have t₃ : DifferentiableAt (partialDeriv Complex.I f₁) x := by
sorry
let B₀ := (h₁ x hx).contDiffAt (IsOpen.mem_nhds hs hx)
let A₀ := partialDeriv_contDiffAt B₀ Complex.I
exact A₀.differentiableAt (Submonoid.oneLE.proof_2 ℕ∞)
have t₄ : DifferentiableAt (partialDeriv Complex.I f₂) x := by
sorry
let B₀ := (h₂ x hx).contDiffAt (IsOpen.mem_nhds hs hx)
let A₀ := partialDeriv_contDiffAt B₀ Complex.I
exact A₀.differentiableAt (Submonoid.oneLE.proof_2 ℕ∞)
rw [partialDeriv_add₂_differentiableAt t₃ t₄]
-- I am super confused at this point because the tactic 'ring' does not work.