File size: 2,986 Bytes
785c727 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 | import OAI.Geometry.EinsteinFour.AngularDerivativeMoments
namespace OAI
/-! Integrate the exact angular identity and the classical Weyl equation.
The integrated Weitzenboeck equations are explicit premises; the derivative
energy inequality required by Jacobi comparison is derived from them.
The published input is Gursky--LeBrun (1999), equation (12), p.8 of the
arXiv version math/9807055: integrate that identity with scalar curvature
12 and determinant `2 * sigma * v^3` to obtain the normalization below. -/
namespace PositiveEinsteinFour.Angular
open MeasureTheory Filter Matrix
noncomputable section
variable {B : Type*} [MeasurableSpace B] {μ : Measure B}
theorem cubic_energy_le {v σ : ℝ} (hv : 0 ≤ v) (hv2 : v ≤ 2) (hσ : σ ≤ 1) :
σ * v ^ 3 - v ^ 2 ≤ v ^ 2 := by
have hs := mul_le_mul_of_nonneg_right hσ (pow_nonneg hv 3)
have ht := mul_le_mul_of_nonneg_right hv2 (sq_nonneg v)
nlinarith
theorem integrated_gradient_energy_le {v σ E : B → ℝ}
(hv : ∀ᵐ c ∂μ, 0 ≤ v c ∧ v c ≤ 2 ∧ σ c ≤ 1)
(hvi : Integrable (fun c => v c ^ 2) μ)
(hci : Integrable (fun c => σ c * v c ^ 3 - v c ^ 2) μ)
(hweitzenbock : (∫ c, E c ∂μ) =
36 * (∫ c, σ c * v c ^ 3 - v c ^ 2 ∂μ)) :
(∫ c, E c ∂μ) ≤ 36 * (∫ c, v c ^ 2 ∂μ) := by
rw [hweitzenbock]
apply mul_le_mul_of_nonneg_left _ (by norm_num)
apply integral_mono_ae hci hvi
filter_upwards [hv] with c hc
exact cubic_energy_le hc.1 hc.2.1 hc.2.2
/-- Equation (8)'s derivative estimate after base integration. The only
geometric differential premises are the two classical integrated equations. -/
theorem integral_normalDerivative_energy_le
(h : FollandMoments) (Z W : B → Gradient) (v b σ τ : B → ℝ)
(hZs : ∀ c a, (Z c a)ᵀ = Z c a)
(hZt : ∀ c a, (Z c a).trace = 0)
(hv : ∀ᵐ c ∂μ, 0 ≤ v c ∧ v c ≤ 2 ∧ σ c ≤ 1)
(hb : ∀ᵐ c ∂μ, 0 ≤ b c ∧ b c ≤ 2 ∧ τ c ≤ 1)
(hvi : Integrable (fun c => v c ^ 2) μ)
(hbi : Integrable (fun c => b c ^ 2) μ)
(hci : Integrable (fun c => σ c * v c ^ 3 - v c ^ 2) μ)
(hdi : Integrable (fun c => τ c * b c ^ 3 - b c ^ 2) μ)
(hZi : Integrable (fun c => gradientSquareNorm (Z c)) μ)
(hWi : Integrable (fun c => gradientSquareNorm (W c)) μ)
(hZ : (∫ c, gradientSquareNorm (Z c) ∂μ) =
36 * (∫ c, σ c * v c ^ 3 - v c ^ 2 ∂μ))
(hW : (∫ c, gradientSquareNorm (W c) ∂μ) =
36 * (∫ c, τ c * b c ^ 3 - b c ^ 2 ∂μ)) :
(∫ c, (∫ q, squareNorm (normalDerivative (Z c) (W c) q) ∂sphereMeasure) ∂μ) ≤
(9 / 4) * (∫ c, v c ^ 2 + b c ^ 2 ∂μ) := by
simp_rw [integral_normalDerivative_squareNorm_exact h _ _ (hZs _) (hZt _)]
rw [integral_div, integral_add hZi hWi, integral_add hvi hbi]
have hz := integrated_gradient_energy_le hv hvi hci hZ
have hw := integrated_gradient_energy_le hb hbi hdi hW
linarith
end
end PositiveEinsteinFour.Angular
end OAI
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