Integrated lower bound for the Carleman energy #
At parameters at least one, the half-space Carleman left side controls the weighted quadratic energy of the compact cutoff field.
theorem
ESS.bu_short_energy_integral_le_left
(scale R ε a : ℝ)
(hscale : 0 < scale)
(hscale1 : scale ≤ 1)
(hR : 0 < R)
(hε : 0 < ε)
(ha : 1 ≤ a)
(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 ({x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} ×ˢ Set.Ico 0 1))
(hweak : CKN.HasSpaceTimeWeakDerivs {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo 0 1) w Dw D2w Dtw)
(hL2 :
∀ S ⊆ CKN.spaceTimeSet {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo 0 1),
Bornology.IsBounded S →
∫⁻ (z : CKN.Foundation.Parabolic.ParabolicPoint) in S, ‖Dw z‖ₑ ^ 2 + ‖D2w z‖ₑ ^ 2 + ‖Dtw z‖ₑ ^ 2 < ⊤)
:
have K := buCutSupportSet (buShortFullCutoff scale R hR ε);
have κ := buShortFullCutoff scale R hR ε;
have v := buAffineField (-scale ^ 2 / 2) scale w;
have Dv := buAffineDw (-scale ^ 2 / 2) scale Dw;
have W := buCutField κ v;
have DW := buCutDw κ v Dv;
∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in K, buShortCarlemanWeight a z * (CKN.Foundation.Parabolic.vec3EuclideanNorm (W z) ^ 2 + CKN.spatialGradientSq W DW z) ≤ ∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in K, buShortCarlemanWeight a z * (a * CKN.Foundation.Parabolic.vec3EuclideanNorm (W z) ^ 2 / z.2 ^ 2 + CKN.spatialGradientSq W DW z / z.2)
The weighted cutoff energy is bounded by the half-space Carleman mass and gradient integral on the compact cutoff support.