openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralAnnulusFluxPrimitive.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralLocalFluxPrimitive | |
| namespace OAI | |
| /-! The weak-flux primitive localized to an exterior annulus. -/ | |
| open Set MeasureTheory | |
| open scoped ContDiff | |
| namespace DefocusingNLS | |
| theorem spectralAnnulusFlux_primitive (L R a b : ℝ) (ha : L < a) (hab : a < b) (hb : b < R) | |
| (F G : ℝ → ℂ) (hF : LocallyIntegrableOn F (Ioo L R)) | |
| (hG : ContinuousOn G (Ioo L R)) | |
| (hw : ∀ φ : ℝ → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ → | |
| tsupport φ ⊆ Ioo L R → (∫ x, deriv φ x • F x) = -(∫ x, φ x • G x)) : | |
| ∃ P : ℝ → ℂ, (∀ x ∈ Ioo a b, HasDerivAt P (G x) x) ∧ | |
| ∀ᵐ x, x ∈ Ioo a b → F x = P x := by | |
| have hsub : Icc a b ⊆ Ioo L R := fun x hx => ⟨ha.trans_le hx.1,hx.2.trans_lt hb⟩ | |
| let clamp := fun x : ℝ => max a (min b x) | |
| have hclamp (x : ℝ) : clamp x ∈ Icc a b := | |
| ⟨le_max_left _ _,max_le hab.le (min_le_left _ _)⟩ | |
| have hfix (x : ℝ) (hx : x ∈ Icc a b) : clamp x=x := by | |
| dsimp only [clamp] | |
| rw [min_eq_right hx.2,max_eq_right hx.1] | |
| let F₀ := (Icc a b).indicator F | |
| let G₀ := fun x => G (clamp x) | |
| have hF₀ : LocallyIntegrable F₀ := by | |
| apply Integrable.locallyIntegrable | |
| exact (integrable_indicator_iff measurableSet_Icc).mpr | |
| (hF.integrableOn_compact_subset hsub isCompact_Icc) | |
| have hG₀ : Continuous G₀ := | |
| hG.comp_continuous (by fun_prop) (fun x => hsub (hclamp x)) | |
| obtain ⟨P,hP,he⟩ := spectralWeakFlux_primitive a b hab F₀ G₀ hF₀ hG₀ (by | |
| intro φ hφ hφc hφs | |
| have hFtest : (fun x => deriv φ x • F₀ x)=(fun x => deriv φ x • F x) := by | |
| funext x | |
| by_cases hx : x ∈ Icc a b | |
| · simp only [F₀,indicator_of_mem hx] | |
| · have hzero : deriv φ x=0 := image_eq_zero_of_notMem_tsupport | |
| (fun h => hx (hφs (tsupport_deriv_subset h))) | |
| simp only [hzero,zero_smul] | |
| have hGtest : (fun x => φ x • G₀ x)=(fun x => φ x • G x) := by | |
| funext x | |
| by_cases hx : x ∈ Icc a b | |
| · simp only [G₀,hfix x hx] | |
| · have hzero : φ x=0 := image_eq_zero_of_notMem_tsupport (fun h => hx (hφs h)) | |
| simp only [hzero,zero_smul] | |
| rw [hFtest,hGtest] | |
| exact hw φ hφ hφc (hφs.trans hsub)) | |
| refine ⟨P,fun x hx => ?_,?_⟩ | |
| · simpa only [G₀,hfix x ⟨hx.1.le,hx.2.le⟩] using hP x | |
| · filter_upwards [he] with x hx hxab | |
| have h := hx hxab | |
| simpa only [F₀,indicator_of_mem (show x ∈ Icc a b from ⟨hxab.1.le,hxab.2.le⟩)] using h | |
| end DefocusingNLS | |
| end OAI | |