| import OAI.Geometry.EinsteinFour.AngularDerivativeMoments |
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| namespace OAI |
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| /-! 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 |
| arXiv version math/9807055: integrate that identity with scalar curvature |
| 12 and determinant `2 * sigma * v^3` to obtain the normalization below. -/ |
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| namespace PositiveEinsteinFour.Angular |
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| open MeasureTheory Filter Matrix |
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| noncomputable section |
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| variable {B : Type*} [MeasurableSpace B] {μ : Measure B} |
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| 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 |
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| 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 |
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| / |
| 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 |
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| end |
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| end PositiveEinsteinFour.Angular |
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| end OAI |
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