Gradient energy bounds for the two-reflection extension #
The chain rule and the operator-norm estimates from SeeleyBounds are
combined with the pullback estimates from SeeleyEnergy.
theorem
CKN.seeleyExtension_gradient_energy_le
(v : Vec 3 → ℝ)
(hv : ContDiff ℝ 1 v)
:
∫⁻ (x : Vec 3) in seeleyOuterAnnulus, ENNReal.ofReal (‖classicalGradient (seeleyExtension v) x‖ ^ 2) ≤ (2 * 675 ^ 2 * 64 + 2 * 3042 ^ 2 * 648) * ∫⁻ (y : Vec 3) in euclideanClosedBall 0 1, ENNReal.ofReal (‖classicalGradient v y‖ ^ 2)