Propagation from a zero neighborhood

The origin is infinitely flat when the field vanishes on a one-sided space-time neighborhood. Unique continuation then fills the spatial ball.

theorem ESS.bu_short_uc_propagate_local_zero (R T c₁ : ℝ) (hR : 0 < R) (hT : 0 < T) (hc₁ : 0 < c₁) (w : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3) (Dw : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → CKN.Foundation.Parabolic.Vec3) (D2w : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → Fin 3 → CKN.Foundation.Parabolic.Vec3) (Dtw : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3) (hcont : ContinuousOn w (CKN.Foundation.Parabolic.vec3Ball 0 R ×ˢ Set.Ico 0 T)) (hweak : CKN.HasSpaceTimeWeakDerivs (CKN.Foundation.Parabolic.vec3Ball 0 R) (Set.Ioo 0 T) w Dw D2w Dtw) (hL2 : ∫⁻ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 R) (Set.Ioo 0 T), ‖w z‖ₑ ^ 2 + ‖Dw z‖ₑ ^ 2 + ‖D2w z‖ₑ ^ 2 + ‖Dtw z‖ₑ ^ 2 < ⊤) (hineq : ∀ᵐ (z : CKN.Foundation.Parabolic.ParabolicPoint) ∂MeasureTheory.volume.restrict (CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 R) (Set.Ioo 0 T)), CKN.Foundation.Parabolic.vec3EuclideanNorm (ucWeakHeatVector D2w Dtw z) ≤ c₁ * (CKN.Foundation.Parabolic.vec3EuclideanNorm (w z) + √(CKN.spatialGradientSq w Dw z))) (hzero : w (0, 0) = 0) (r₀ : ℝ) (hr₀ : 0 < r₀) (hlocal : ∀ z ∈ CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 r₀) (Set.Ioo 0 (r₀ ^ 2)), w z = 0) (y : CKN.Foundation.Parabolic.Vec3) : y ∈ CKN.Foundation.Parabolic.vec3Ball 0 R → w (y, 0) = 0

A local zero region at the initial time propagates to the whole ball under the weak heat inequality (thm:uc).