Local energy on an extended short-time cell #
The Caccioppoli estimate applies to a bounded cylinder inside the extended rescaled half-space domain.
theorem
ESS.bu_short_cell_caccioppoli
(X : CKN.Foundation.Parabolic.Vec3)
(t r c : ℝ)
(hr : 0 < r)
(hc : 0 ≤ c)
(hball : CKN.Foundation.Parabolic.vec3Ball X (2 * r) ⊆ {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2})
(htime : Set.Ioo t (t + 4 * r ^ 2) ⊆ Set.Ioo (1 / 2) (3 / 2))
{v : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3}
{Dv : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → CKN.Foundation.Parabolic.Vec3}
{D2v : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → Fin 3 → CKN.Foundation.Parabolic.Vec3}
{Dtv : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3}
(hcont : ContinuousOn v ({x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} ×ˢ Set.Ico (1 / 2) (3 / 2)))
(hweak :
CKN.HasSpaceTimeWeakDerivs {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo (1 / 2) (3 / 2)) v Dv D2v Dtv)
(hlocal :
∀ S ⊆ CKN.spaceTimeSet {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo (1 / 2) (3 / 2)),
Bornology.IsBounded S →
∫⁻ (z : CKN.Foundation.Parabolic.ParabolicPoint) in S, ‖v z‖ₑ ^ 2 + ‖Dv z‖ₑ ^ 2 + ‖D2v z‖ₑ ^ 2 + ‖Dtv z‖ₑ ^ 2 < ⊤)
(hineq :
∀ᵐ (z :
CKN.Foundation.Parabolic.ParabolicPoint) ∂MeasureTheory.volume.restrict
(CKN.spaceTimeSet {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo (1 / 2) (3 / 2))), CKN.Foundation.Parabolic.vec3EuclideanNorm (ucWeakHeatVector D2v Dtv z) ≤ c * (√(CKN.spatialGradientSq v Dv z) + CKN.Foundation.Parabolic.vec3EuclideanNorm (v z)))
:
∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball X r) (Set.Ioo t (t + r ^ 2)), CKN.spatialGradientSq v Dv z ≤ 256 * (1 + c ^ 2 + 1 / r ^ 2) * ∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball X (2 * r)) (Set.Ioo t (t + 4 * r ^ 2)), CKN.Foundation.Parabolic.vec3EuclideanNorm (v z) ^ 2
Caccioppoli bounds the inner gradient energy on any enlarged cell
inside the extended short-time cylinder (lem:caccioppoli).