This commit is contained in:
Stefan Kebekus 2024-05-30 17:02:15 +02:00
parent b0ab121868
commit c8e1aacb15
3 changed files with 64 additions and 6 deletions

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@ -57,8 +57,13 @@ theorem HarmonicOn_congr {f₁ f₂ : → F} {s : Set } (hs : IsOpen s) (
unfold Filter.EventuallyEq unfold Filter.EventuallyEq
unfold Filter.Eventually unfold Filter.Eventually
simp simp
apply? refine mem_nhds_iff.mpr ?_
sorry use s
constructor
· exact hf₁₂
· constructor
· exact hs
· exact hz
rw [← laplace_eventuallyEq this] rw [← laplace_eventuallyEq this]
exact h₁.2 z hz exact h₁.2 z hz
· intro h₁ · intro h₁
@ -67,7 +72,17 @@ theorem HarmonicOn_congr {f₁ f₂ : → F} {s : Set } (hs : IsOpen s) (
intro x hx intro x hx
exact hf₁₂ x hx exact hf₁₂ x hx
· intro z hz · intro z hz
have : f₁ =ᶠ[nhds z] f₂ := by sorry have : f₁ =ᶠ[nhds z] f₂ := by
unfold Filter.EventuallyEq
unfold Filter.Eventually
simp
refine mem_nhds_iff.mpr ?_
use s
constructor
· exact hf₁₂
· constructor
· exact hs
· exact hz
rw [laplace_eventuallyEq this] rw [laplace_eventuallyEq this]
exact h₁.2 z hz exact h₁.2 z hz
@ -83,6 +98,17 @@ theorem harmonic_add_harmonic_is_harmonic {f₁ f₂ : → F} (h₁ : Harmon
simp simp
theorem harmonicOn_add_harmonicOn_is_harmonicOn {f₁ f₂ : → F} {s : Set } (hs : IsOpen s) (h₁ : HarmonicOn f₁ s) (h₂ : HarmonicOn f₂ s) :
HarmonicOn (f₁ + f₂) s := by
constructor
· exact ContDiffOn.add h₁.1 h₂.1
· rw [laplace_add h₁.1 h₂.1]
simp
intro z
rw [h₁.2 z, h₂.2 z]
simp
theorem harmonic_smul_const_is_harmonic {f : → F} {c : } (h : Harmonic f) : theorem harmonic_smul_const_is_harmonic {f : → F} {c : } (h : Harmonic f) :
Harmonic (c • f) := by Harmonic (c • f) := by
constructor constructor
@ -268,7 +294,8 @@ theorem log_normSq_of_holomorphicOn_is_harmonicOn
· exact Real.pi_nonneg · exact Real.pi_nonneg
exact (AddEquivClass.map_ne_zero_iff starRingAut).mpr (h₂ z hz) exact (AddEquivClass.map_ne_zero_iff starRingAut).mpr (h₂ z hz)
exact h₂ z hz exact h₂ z hz
rw [HarmonicOn_ext this]
rw [HarmonicOn_congr hs this]
simp simp
apply harmonic_add_harmonic_is_harmonic apply harmonic_add_harmonic_is_harmonic

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@ -53,6 +53,37 @@ theorem laplace_add {f₁ f₂ : → F} (h₁ : ContDiff 2 f₁) (h₂
exact h₂.differentiable one_le_two exact h₂.differentiable one_le_two
theorem laplace_add_ContDiffOn {f₁ f₂ : → F} {s : Set } (hs : IsOpen s) (h₁ : ContDiffOn 2 f₁ s) (h₂ : ContDiffOn 2 f₂ s): ∀ x ∈ s, Complex.laplace (f₁ + f₂) x = (Complex.laplace f₁) x + (Complex.laplace f₂) x := by
unfold Complex.laplace
simp
intro x hx
have : partialDeriv 1 (f₁ + f₂) =ᶠ[nhds x] (partialDeriv 1 f₁) + (partialDeriv 1 f₂) := by
sorry
rw [partialDeriv_eventuallyEq this]
rw [partialDeriv_add₂]
rw [partialDeriv_add₂]
rw [partialDeriv_add₂]
rw [partialDeriv_add₂]
exact
add_add_add_comm (partialDeriv 1 (partialDeriv 1 f₁))
(partialDeriv 1 (partialDeriv 1 f₂))
(partialDeriv Complex.I (partialDeriv Complex.I f₁))
(partialDeriv Complex.I (partialDeriv Complex.I f₂))
exact (partialDeriv_contDiff h₁ Complex.I).differentiable le_rfl
exact (partialDeriv_contDiff h₂ Complex.I).differentiable le_rfl
exact h₁.differentiable one_le_two
exact h₂.differentiable one_le_two
exact (partialDeriv_contDiff h₁ 1).differentiable le_rfl
exact (partialDeriv_contDiff h₂ 1).differentiable le_rfl
exact h₁.differentiable one_le_two
exact h₂.differentiable one_le_two
theorem laplace_smul {f : → F} (h : ContDiff 2 f) : ∀ v : , Complex.laplace (v • f) = v • (Complex.laplace f) := by theorem laplace_smul {f : → F} (h : ContDiff 2 f) : ∀ v : , Complex.laplace (v • f) = v • (Complex.laplace f) := by
intro v intro v
unfold Complex.laplace unfold Complex.laplace