| import OAI.Geometry.IsometricImmersion.Darboux.QEnergyApplication |
| import OAI.Geometry.IsometricImmersion.Energy.MovingEnergyGradient |
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| namespace OAI |
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| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped ContDiff Topology Interval ENNReal NNReal |
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| namespace SmoothLocal.HighEquation |
| open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE SmoothLocal.Hyperbolic |
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| def spatialEnergyNormBudget (s0 : ℝ) (A : ℝ≥0) : ℝ≥0 := |
| ⟨Real.sqrt (2 * (1 + 1 / s0)) * (A : ℝ), mul_nonneg (Real.sqrt_nonneg _) A.2⟩ |
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| theorem real_slice_eLpNorm_two_le_of_square_integral |
| {f : ℝ → ℝ} {left right : ℝ} {B : ℝ≥0} |
| (hlr : left ≤ right) (hf : MemLp f 2 (volume.restrict (Icc left right))) |
| (hB : (∫ x in left..right, f x ^ 2) ≤ (B : ℝ)^2) : |
| eLpNorm f 2 (volume.restrict (Icc left right)) ≤ (B : ℝ≥0∞) := by |
| rw [intervalIntegral.integral_of_le hlr, restrict_Ioc_eq_restrict_Icc, |
| integral_square_eq_L2_norm_square hf, Lp.norm_def, eLpNorm_congr_ae hf.coeFn_toLp] at hB |
| apply (ENNReal.toReal_le_toReal hf.eLpNorm_ne_top (by simp)).mp |
| change (eLpNorm f 2 (volume.restrict (Icc left right))).toReal ≤ (B : ℝ) |
| exact (sq_le_sq₀ ENNReal.toReal_nonneg B.2).mp hB |
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| theorem spatialFirstJet_slice_eLpNorm_two_of_energy |
| {z S : Coord → ℝ} {U : Set Coord} {left right theta s0 : ℝ} |
| (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U) (hS : ContDiffOn ℝ ∞ S U) |
| (hseg : ∀ x ∈ Icc left right, coordinatePoint x theta ∈ U) |
| (hlr : left ≤ right) (hs0 : 0 < s0) |
| (hfloor : ∀ x ∈ Icc left right, s0 ≤ S (coordinatePoint x theta)) |
| (ell : ℕ) (A : ℝ≥0) |
| (henergy : Real.sqrt (∫ x in left..right, |
| energyDensity S (horizontalJet z ell) (coordinatePoint x theta)) ≤ (A : ℝ)) |
| (i : Fin 2) : |
| eLpNorm (fun x => spatialJet (spatialFirstJet z i) ell (coordinatePoint x theta)) 2 |
| (volume.restrict (Icc left right)) ≤ (spatialEnergyNormBudget s0 A : ℝ≥0∞) := by |
| let u := horizontalJet z ell |
| let E := ∫ x in left..right, energyDensity S u (coordinatePoint x theta) |
| let C := 2 * (1 + 1 / s0) |
| have hC : 0 ≤ C := by dsimp [C]; positivity |
| have hu := horizontalJet_contDiffOn hU hz ell |
| have hd (j : Fin 2) := partial_contDiffOn hu hU j |
| have he : ContDiffOn ℝ ∞ (energyDensity S u) U := |
| (((hd 1).pow 2).add (hS.mul ((hd 0).pow 2))).div_const 2 |
| have hEA : E ≤ (A : ℝ)^2 := (Real.sqrt_le_iff.mp henergy).2 |
| have hfiCont : ContinuousOn (fun x => coordPartial i u (coordinatePoint x theta)^2) (Icc left right) := |
| ((hd i).pow 2).continuousOn.comp (horizontalPoint_contDiff theta).continuous.continuousOn hseg |
| have heCont : ContinuousOn (fun x => C * energyDensity S u (coordinatePoint x theta)) (Icc left right) := |
| (continuousOn_const.mul he.continuousOn).comp (horizontalPoint_contDiff theta).continuous.continuousOn hseg |
| have hiInt : (∫ x in left..right, coordPartial i u (coordinatePoint x theta)^2) ≤ C * E := by |
| have hh : (∫ x in left..right, coordPartial i u (coordinatePoint x theta)^2) ≤ |
| ∫ x in left..right, C * energyDensity S u (coordinatePoint x theta) := |
| intervalIntegral.integral_mono_on hlr |
| (hfiCont.intervalIntegrable_of_Icc hlr) (heCont.intervalIntegrable_of_Icc hlr) |
| (fun x hx => show coordPartial i u (coordinatePoint x theta)^2 ≤ |
| C * energyDensity S u (coordinatePoint x theta) from by |
| have hscalar := scalar_gradient_le_energy hs0 (hfloor x hx) |
| (a := coordPartial 1 u (coordinatePoint x theta)) (b := coordPartial 0 u (coordinatePoint x theta)) |
| have hind : coordPartial i u (coordinatePoint x theta)^2 ≤ |
| coordPartial 1 u (coordinatePoint x theta)^2 + coordPartial 0 u (coordinatePoint x theta)^2 := by |
| fin_cases i |
| · change coordPartial 0 u (coordinatePoint x theta)^2 ≤ _ |
| nlinarith [sq_nonneg (coordPartial 1 u (coordinatePoint x theta))] |
| · change coordPartial 1 u (coordinatePoint x theta)^2 ≤ _ |
| nlinarith [sq_nonneg (coordPartial 0 u (coordinatePoint x theta))] |
| exact hind.trans hscalar) |
| simpa only [intervalIntegral.integral_const_mul] using hh |
| have hiBudget : (∫ x in left..right, coordPartial i u (coordinatePoint x theta)^2) ≤ |
| (spatialEnergyNormBudget s0 A : ℝ)^2 := by |
| have hbound := hiInt.trans (mul_le_mul_of_nonneg_left hEA hC) |
| change _ ≤ (Real.sqrt C * (A : ℝ))^2 |
| rw [mul_pow, Real.sq_sqrt hC] |
| exact hbound |
| have hiLp := real_slice_eLpNorm_two_le_of_square_integral hlr |
| (smooth_slice_memLp_two (hd i).continuousOn hseg) hiBudget |
| have heq : (fun x => spatialJet (spatialFirstJet z i) ell (coordinatePoint x theta)) =ᵐ[ |
| volume.restrict (Icc left right)] (fun x => coordPartial i u (coordinatePoint x theta)) := by |
| filter_upwards [ae_restrict_mem measurableSet_Icc] with x hx |
| rw [← horizontalJet_eq_spatialJet] |
| exact horizontalJet_coordPartial hU hz i ell _ (hseg x hx) |
| rw [eLpNorm_congr_ae heq] |
| exact hiLp |
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| end SmoothLocal.HighEquation |
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| end |
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| end OAI |
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