Localized Equation Gradient Measurability #
Part of the Caffarelli–Kohn–Nirenberg partial regularity proof.
theorem
CKN.Core.Step4.localized_gradient_source_aemeasurable_of_sws
{Ω : Set Foundation.Parabolic.Vec3}
{I : Set ℝ}
{q : ℝ}
{u : Foundation.Parabolic.ParabolicPoint → Foundation.Parabolic.Vec3}
{Du : Foundation.Parabolic.ParabolicPoint → Fin 3 → Foundation.Parabolic.Vec3}
{p : Foundation.Parabolic.ParabolicPoint → ℝ}
{f : Foundation.Parabolic.ParabolicPoint → Foundation.Parabolic.Vec3}
(hsol : IsSuitableWeakSolutionIntegrable Ω I q u Du p f)
{φ : Foundation.Parabolic.Vec3 × ℝ → ℝ}
(hφ : φ ∈ spaceTimeTestFunction Ω I)
{Ω' : Set Foundation.Parabolic.Vec3}
{J : Set ℝ}
(hbox : localBox Ω I Ω' J)
(hφbox : tsupport φ ⊆ Ω' ×ˢ J)
{Dp : Foundation.Parabolic.ParabolicPoint → Foundation.Parabolic.Vec3}
(hDp :
∀ (i : Fin 3),
AEMeasurable (fun (z : Foundation.Parabolic.ParabolicPoint) => Dp z i)
(MeasureTheory.volume.restrict (spaceTimeSet Ω' J)))
:
(∀ (i : Fin 3),
AEMeasurable (fun (z : Foundation.Parabolic.ParabolicPoint) => localizedGradientSourceG φ u Du f Dp z i)
MeasureTheory.volume) ∧ ∀ (j i : Fin 3),
AEMeasurable (fun (z : Foundation.Parabolic.ParabolicPoint) => -localizedGradientSourceH φ u j z i)
MeasureTheory.volume