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Stefan Kebekus
parent
637c0cf175
commit
40659c2f17
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@ -4,12 +4,12 @@ import Mathlib.Analysis.Calculus.LineDeriv.Basic
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import Mathlib.Analysis.Calculus.ContDiff.Defs
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import Mathlib.Analysis.Calculus.FDeriv.Basic
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import Mathlib.Analysis.Calculus.FDeriv.Symmetric
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import Mathlib.Analysis.RCLike.Basic
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import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
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import Mathlib.Data.Complex.Module
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import Mathlib.Data.Complex.Order
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import Mathlib.Data.Complex.Exponential
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import Mathlib.Data.Fin.Tuple.Basic
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import Mathlib.Analysis.RCLike.Basic
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import Mathlib.Topology.Algebra.InfiniteSum.Module
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import Mathlib.Topology.Instances.RealVectorSpace
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import Nevanlinna.cauchyRiemann
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@ -254,13 +254,6 @@ theorem holomorphicOn_is_harmonicOn {f : ℂ → F₁} {s : Set ℂ} (hs : IsOpe
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have f_is_real_C2 : ContDiffOn ℝ 2 f s :=
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ContDiffOn.restrict_scalars ℝ (DifferentiableOn.contDiffOn h hs)
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have fI_is_real_differentiable : DifferentiableOn ℝ (partialDeriv ℝ 1 f) s := by
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intro z hz
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apply DifferentiableAt.differentiableWithinAt
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let ZZ := (f_is_real_C2 z hz).contDiffAt (IsOpen.mem_nhds hs hz)
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let AA := partialDeriv_contDiffAt ℝ ZZ 1
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exact AA.differentiableAt (by rfl)
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constructor
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· -- f is two times real continuously differentiable
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exact f_is_real_C2
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@ -269,8 +262,6 @@ theorem holomorphicOn_is_harmonicOn {f : ℂ → F₁} {s : Set ℂ} (hs : IsOpe
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unfold Complex.laplace
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intro z hz
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simp
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have : ∀ z ∈ s, partialDeriv ℝ Complex.I f z = Complex.I • partialDeriv ℝ 1 f z := by
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sorry
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have : partialDeriv ℝ Complex.I f =ᶠ[nhds z] Complex.I • partialDeriv ℝ 1 f := by
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unfold Filter.EventuallyEq
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unfold Filter.Eventually
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@ -289,17 +280,33 @@ theorem holomorphicOn_is_harmonicOn {f : ℂ → F₁} {s : Set ℂ} (hs : IsOpe
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rw [partialDeriv_eventuallyEq ℝ this Complex.I]
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rw [partialDeriv_smul'₂]
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rw [partialDeriv_comm f_is_real_C2 Complex.I 1]
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rw [CauchyRiemann₄ h]
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simp
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rw [partialDeriv_commOn hs f_is_real_C2 Complex.I 1 z hz]
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have : Complex.I • partialDeriv ℝ 1 (partialDeriv ℝ Complex.I f) z = Complex.I • (partialDeriv ℝ 1 (partialDeriv ℝ Complex.I f) z) := by
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rfl
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rw [this]
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have : partialDeriv ℝ Complex.I f =ᶠ[nhds z] Complex.I • partialDeriv ℝ 1 f := by
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unfold Filter.EventuallyEq
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unfold Filter.Eventually
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simp
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refine mem_nhds_iff.mpr ?_
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use s
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constructor
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· intro x hx
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simp
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apply CauchyRiemann₅
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apply DifferentiableOn.differentiableAt h
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exact IsOpen.mem_nhds hs hx
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· constructor
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· exact hs
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· exact hz
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rw [partialDeriv_eventuallyEq ℝ this 1]
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rw [partialDeriv_smul'₂]
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simp
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rw [← smul_assoc]
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simp
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-- Subgoals coming from the application of 'this'
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-- Differentiable ℝ (Real.partialDeriv 1 f)
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exact fI_is_real_differentiable
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-- Differentiable ℝ (Real.partialDeriv 1 f)
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exact fI_is_real_differentiable
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theorem re_of_holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
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@ -367,11 +374,13 @@ theorem log_normSq_of_holomorphicOn_is_harmonicOn
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rw [HarmonicOn_congr hs this]
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simp
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apply harmonicOn_add_harmonicOn_is_harmonicOn
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exact hs
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apply harmonicOn_add_harmonicOn_is_harmonicOn hs
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have : (fun x => Complex.log ((starRingEnd ℂ) (f x))) = (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f) := by
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rfl
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rw [this]
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-- HarmonicOn (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f) s
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have : ∀ z ∈ s, (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f) z = (Complex.conjCLE ∘ Complex.log ∘ f) z := by
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intro z hz
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unfold Function.comp
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@ -383,9 +392,24 @@ theorem log_normSq_of_holomorphicOn_is_harmonicOn
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rw [← harmonicOn_iff_comp_CLE_is_harmonicOn]
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apply holomorphicOn_is_harmonicOn
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intro z
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exact hs
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intro z hz
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apply DifferentiableAt.differentiableWithinAt
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apply DifferentiableAt.comp
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exact Complex.differentiableAt_log (h₃ z)
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exact Complex.differentiableAt_log (h₃ z hz)
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apply DifferentiableOn.differentiableAt h₁ -- (h₁ z hz)
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exact IsOpen.mem_nhds hs hz
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exact hs
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-- HarmonicOn (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f) s
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apply holomorphicOn_is_harmonicOn hs
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apply?
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exact h₁ z
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@ -1,6 +1,8 @@
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import Mathlib.Analysis.Calculus.FDeriv.Basic
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import Mathlib.Analysis.Calculus.FDeriv.Symmetric
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import Mathlib.Analysis.Calculus.ContDiff.Basic
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import Mathlib.Topology.Basic
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import Mathlib.Topology.Defs.Filter
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variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
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@ -168,6 +170,23 @@ lemma partialDeriv_fderiv {f : E → F} (hf : ContDiff 𝕜 2 f) (z a b : E) :
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· simp
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lemma partialDeriv_fderivOn
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{s : Set E}
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{f : E → F}
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(hs : IsOpen s)
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(hf : ContDiffOn 𝕜 2 f s) (a b : E) :
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∀ z ∈ s, fderiv 𝕜 (fderiv 𝕜 f) z b a = partialDeriv 𝕜 b (partialDeriv 𝕜 a f) z := by
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intro z hz
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unfold partialDeriv
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rw [fderiv_clm_apply]
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· simp
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· convert DifferentiableOn.differentiableAt _ (IsOpen.mem_nhds hs hz)
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apply ContDiffOn.differentiableOn _ (Submonoid.oneLE.proof_2 ℕ∞)
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exact ((contDiffOn_succ_iff_fderiv_of_isOpen hs).1 hf).2
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· simp
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theorem partialDeriv_eventuallyEq {f₁ f₂ : E → F} {x : E} (h : f₁ =ᶠ[nhds x] f₂) : ∀ v : E, partialDeriv 𝕜 v f₁ x = partialDeriv 𝕜 v f₂ x := by
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unfold partialDeriv
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rw [Filter.EventuallyEq.fderiv_eq h]
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@ -268,3 +287,42 @@ theorem partialDeriv_comm
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rw [← partialDeriv_fderiv ℝ h z v₂ v₁]
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rw [derivSymm]
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rw [partialDeriv_fderiv ℝ h z v₁ v₂]
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theorem partialDeriv_commOn
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{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
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{F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
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{s : Set E}
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{f : E → F}
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(hs : IsOpen s)
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(h : ContDiffOn ℝ 2 f s) :
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∀ v₁ v₂ : E, ∀ z ∈ s, partialDeriv ℝ v₁ (partialDeriv ℝ v₂ f) z = partialDeriv ℝ v₂ (partialDeriv ℝ v₁ f) z := by
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intro v₁ v₂ z hz
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have derivSymm :
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(fderiv ℝ (fun w => fderiv ℝ f w) z) v₁ v₂ = (fderiv ℝ (fun w => fderiv ℝ f w) z) v₂ v₁ := by
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let f' := fderiv ℝ f
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have h₀ : ∀ y ∈ s, HasFDerivAt f (f' y) y := by
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intro y hy
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apply DifferentiableAt.hasFDerivAt
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apply DifferentiableOn.differentiableAt _ (IsOpen.mem_nhds hs hy)
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apply h.differentiableOn one_le_two
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let f'' := (fderiv ℝ f' z)
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have h₁ : HasFDerivAt f' f'' z := by
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apply DifferentiableAt.hasFDerivAt
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apply DifferentiableOn.differentiableAt _ (IsOpen.mem_nhds hs hz)
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apply ContDiffOn.differentiableOn _ (Submonoid.oneLE.proof_2 ℕ∞)
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exact ((contDiffOn_succ_iff_fderiv_of_isOpen hs).1 h).2
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have h₀' : ∀ᶠ (y : E) in nhds z, HasFDerivAt f (f' y) y := by
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apply eventually_nhds_iff.mpr
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use s
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exact second_derivative_symmetric_of_eventually h₀' h₁ v₁ v₂
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rw [← partialDeriv_fderivOn ℝ hs h v₂ v₁ z hz]
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rw [derivSymm]
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rw [← partialDeriv_fderivOn ℝ hs h v₁ v₂ z hz]
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