Gaussian Carleman inequality on a spatial ball #
Compact support permits restricting the smooth Gaussian estimate to the
normalized ball used in lem:uc-gaussian.
theorem
ESS.uc_gaussian_weak_carleman_ball_with_constant
(ρ c₀ : ℝ)
(hgaussRaw :
∀ (a : ℝ),
0 < a →
∀ v ∈ CKN.spaceTimeTestFunction Set.univ (Set.Ioo 0 2),
∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet Set.univ (Set.Ioo 0 2), (z.2 * Real.exp ((1 - z.2) / 3)) ^ (-2 * a) * Real.exp (-CKN.Foundation.Parabolic.vec3EuclideanNorm z.1 ^ 2 / (4 * z.2)) * (a / z.2 * CKN.Foundation.Parabolic.vec3EuclideanNorm (v z) ^ 2 + CKN.spatialGradientSq v (CKN.spatialGradient v) z) ≤ c₀ * ∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet Set.univ (Set.Ioo 0 2), (z.2 * Real.exp ((1 - z.2) / 3)) ^ (-2 * a) * Real.exp (-CKN.Foundation.Parabolic.vec3EuclideanNorm z.1 ^ 2 / (4 * z.2)) * (CKN.Foundation.Parabolic.vec3EuclideanNorm fun (i : Fin 3) =>
CKN.timePartial (fun (y : CKN.Foundation.Parabolic.ParabolicPoint) => v y i) z + ∑ j : Fin 3,
CKN.spatialSecondPartial (fun (y : CKN.Foundation.Parabolic.ParabolicPoint) => v y i) j j z) ^ 2)
{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}
(hderiv : CKN.HasSpaceTimeWeakDerivs (CKN.Foundation.Parabolic.vec3Ball 0 ρ) (Set.Ioo 0 2) w Dw D2w Dtw)
(hcompact : HasCompactSupport w)
(htsupport : tsupport w ⊆ ucCylinder ρ)
(hL2 :
∫⁻ (z : CKN.Foundation.Parabolic.ParabolicPoint) in ucCylinder ρ, ‖w z‖ₑ ^ 2 + ‖Dw z‖ₑ ^ 2 + ‖D2w z‖ₑ ^ 2 + ‖Dtw z‖ₑ ^ 2 < ⊤)
(a : ℝ)
:
0 < a →
∫ (z : CKN.Foundation.Parabolic.Vec3 × ℝ) in CKN.Foundation.Parabolic.vec3Ball 0 ρ ×ˢ Set.Ioo 0 2, ucGaussianWeight a z * (a / z.2) * CKN.Foundation.Parabolic.vec3EuclideanNorm (w (CKN.Foundation.Parabolic.parabolicHomeomorph.symm z)) ^ 2 + ucGaussianWeight a z * ∑ i : Fin 3, ∑ j : Fin 3, Dw (CKN.Foundation.Parabolic.parabolicHomeomorph.symm z) i j ^ 2 ≤ c₀ * ∫ (z : CKN.Foundation.Parabolic.Vec3 × ℝ) in CKN.Foundation.Parabolic.vec3Ball 0 ρ ×ˢ Set.Ioo 0 2, ucGaussianWeight a z * (CKN.Foundation.Parabolic.vec3EuclideanNorm fun (i : Fin 3) =>
Dtw (CKN.Foundation.Parabolic.parabolicHomeomorph.symm z) i + ∑ j : Fin 3, D2w (CKN.Foundation.Parabolic.parabolicHomeomorph.symm z) i j j) ^ 2
A fixed smooth Gaussian constant passes unchanged to compactly supported weak fields on a spatial ball.