| import Mathlib |
| import OAI.Analysis.CoulombIonization.FormDomain.WeakDerivative |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace Convolution |
| open Filter |
| open scoped Topology InnerProductSpace |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace ContDiff |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace Convolution |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace ENNReal |
| namespace CoulombAtom |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasureSpace E] [BorelSpace E] |
| [IsLocallyFiniteMeasure (volume : Measure E)] |
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| abbrev SmoothRealTest (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E] := |
| {φ : E → ℝ |
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| lemma countable_weak_derivative_tests (v : E) : |
| ∃ S : Set (SmoothRealTest E), S.Countable ∧ |
| ∀ (f g : E → ℂ), MemLp f 2 → MemLp g 2 → |
| (∀ φ ∈ S, (∫ x, f x * Complex.ofReal (lineDeriv ℝ φ.val x v)) = |
| -(∫ x, g x * (φ.val x : ℂ))) → IsWeakDerivative v f g := by |
| let : Fact ((2 : ℝ≥0∞) ≠ ⊤) := ⟨by norm_num⟩ |
| let V := WeakGraphAmbient E Unit |
| let L : SmoothRealTest E → V →L[ℂ] ℂ := fun φ => |
| weakGraphTest (fun _ : Unit => v) () φ.val φ.property.1 φ.property.2 |
| obtain ⟨S, hS, hcover⟩ := TopologicalSpace.isOpen_iUnion_countable |
| (fun φ : SmoothRealTest E => ((L φ).ker : Set V)ᶜ) (fun φ => (L φ).isClosed_ker.isOpen_compl) |
| refine ⟨S, hS, ?_⟩ |
| intro f g hf hg htest |
| let A : V := WithLp.toLp 2 (fun j : Option Unit => |
| match j with | none => hf.toLp f | some _ => hg.toLp g) |
| have hAS (φ : SmoothRealTest E) (hφ : φ ∈ S) : A ∈ (L φ).ker := by |
| change l2Test _ _ (hf.toLp f) + l2Test _ _ (hg.toLp g) = 0 |
| rw [l2Test_apply, l2Test_apply] |
| have h1 := integral_congr_ae (hf.coeFn_toLp.mono fun x hx => |
| congrArg (· * Complex.ofReal (lineDeriv ℝ φ.val x v)) hx) |
| have h2 := integral_congr_ae (hg.coeFn_toLp.mono fun x hx => |
| congrArg (· * (φ.val x : ℂ)) hx) |
| rw [h1, h2, htest φ hφ, neg_add_cancel] |
| have hAll (φ : SmoothRealTest E) : A ∈ (L φ).ker := by |
| by_contra hn |
| have hm : A ∈ ⋃ φ : SmoothRealTest E, ((L φ).ker : Set V)ᶜ := Set.mem_iUnion.mpr ⟨φ, hn⟩ |
| rw [← hcover] at hm |
| obtain ⟨ψ, hψ⟩ := Set.mem_iUnion.mp hm |
| obtain ⟨hψS, hψA⟩ := Set.mem_iUnion.mp hψ |
| exact hψA (hAS ψ hψS) |
| intro φ hφ hcφ |
| have hh := hAll ⟨φ, hφ, hcφ⟩ |
| change l2Test _ _ (hf.toLp f) + l2Test _ _ (hg.toLp g) = 0 at hh |
| rw [l2Test_apply, l2Test_apply] at hh |
| have h1 := integral_congr_ae (hf.coeFn_toLp.mono fun x hx => |
| congrArg (· * Complex.ofReal (lineDeriv ℝ φ x v)) hx) |
| have h2 := integral_congr_ae (hg.coeFn_toLp.mono fun x hx => |
| congrArg (· * (φ x : ℂ)) hx) |
| rw [h1, h2] at hh |
| exact eq_neg_of_add_eq_zero_left hh |
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| end CoulombAtom |
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| end |
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| end OAI |
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