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ef96466 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 | 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
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