| import OAI.Geometry.IsometricImmersion.Energy.MultiplierIntegral |
| import OAI.Geometry.IsometricImmersion.Caps.EllipticAbsorption |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter |
| open scoped ContDiff Topology |
|
|
| namespace SmoothLocal.Weighted |
| open SmoothLocal.Geometry |
|
|
| def multiplierCutoffError (A chi m n u : Coord → ℝ) (p : Coord) : ℝ := |
| coordPartial 0 chi p * multiplierFluxT A m n u p + |
| coordPartial 1 chi p * multiplierFluxS A m n u p |
|
|
| theorem multiplier_divergence_quadratic |
| {A B C m n u : Coord → ℝ} {U : Set Coord} {p : Coord} |
| (hA : DifferentiableAt ℝ A p) (hm : DifferentiableAt ℝ m p) (hn : DifferentiableAt ℝ n p) |
| (hu : ContDiffOn ℝ ∞ u U) (hU : IsOpen U) (hp : p ∈ U) : |
| multiplierOperator A B C u p * multiplierTest m n u p = |
| coordPartial 0 (multiplierFluxT A m n u) p + coordPartial 1 (multiplierFluxS A m n u) p + |
| multiplierQuadratic A B C m n u p := by |
| have hh := multiplier_divergence_identity (B := B) (C := C) hA hm hn hu hU hp |
| unfold multiplierQuadratic |
| linear_combination hh |
|
|
| theorem multiplierQuadratic_cutoff_split |
| {A B C chi m n u : Coord → ℝ} {U : Set Coord} {p : Coord} |
| (hU : IsOpen U) (hA : ContDiffOn ℝ ∞ A U) (hchi : ContDiffOn ℝ ∞ chi U) |
| (hm : ContDiffOn ℝ ∞ m U) (hn : ContDiffOn ℝ ∞ n U) (hu : ContDiffOn ℝ ∞ u U) |
| (hp : p ∈ U) : |
| multiplierQuadratic A B C (fun q => chi q * m q) (fun q => chi q * n q) u p = |
| chi p * multiplierQuadratic A B C m n u p - multiplierCutoffError A chi m n u p := by |
| have hAd := (hA.contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp) |
| have hmd := (hm.contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp) |
| have hnd := (hn.contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp) |
| have hcd := (hchi.contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp) |
| have hbase := multiplier_divergence_quadratic (B := B) (C := C) hAd hmd hnd hu hU hp |
| have hcut := multiplier_divergence_quadratic (B := B) (C := C) hAd (hcd.mul hmd) (hcd.mul hnd) hu hU hp |
| change multiplierOperator A B C u p * |
| multiplierTest (fun q => chi q * m q) (fun q => chi q * n q) u p = |
| coordPartial 0 (multiplierFluxT A (fun q => chi q * m q) (fun q => chi q * n q) u) p + |
| coordPartial 1 (multiplierFluxS A (fun q => chi q * m q) (fun q => chi q * n q) u) p + |
| multiplierQuadratic A B C (fun q => chi q * m q) (fun q => chi q * n q) u p at hcut |
| have htest : multiplierTest (fun q => chi q * m q) (fun q => chi q * n q) u p = |
| chi p * multiplierTest m n u p := by unfold multiplierTest; ring |
| have hFt : multiplierFluxT A (fun q => chi q * m q) (fun q => chi q * n q) u = |
| (fun q => chi q * multiplierFluxT A m n u q) := by funext q |
| have hFs : multiplierFluxS A (fun q => chi q * m q) (fun q => chi q * n q) u = |
| (fun q => chi q * multiplierFluxS A m n u q) := by funext q |
| obtain ⟨hFts, hFss⟩ := multiplierFluxes_contDiffOn hU hA hm hn hu |
| rw [htest, hFt, hFs, |
| HessianCalculus.coordPartial_mul_at (f := chi) (h := multiplierFluxT A m n u) hcd |
| ((hFts.contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp)) 0, |
| HessianCalculus.coordPartial_mul_at (f := chi) (h := multiplierFluxS A m n u) hcd |
| ((hFss.contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp)) 1] at hcut |
| unfold multiplierCutoffError |
| linear_combination chi p * hbase - hcut |
|
|
| theorem multiplierCutoffError_contDiffOn |
| {A chi m n u : Coord → ℝ} {U : Set Coord} |
| (hU : IsOpen U) (hA : ContDiffOn ℝ ∞ A U) (hchi : ContDiffOn ℝ ∞ chi U) |
| (hm : ContDiffOn ℝ ∞ m U) (hn : ContDiffOn ℝ ∞ n U) (hu : ContDiffOn ℝ ∞ u U) : |
| ContDiffOn ℝ ∞ (multiplierCutoffError A chi m n u) U := by |
| obtain ⟨hFt, hFs⟩ := multiplierFluxes_contDiffOn hU hA hm hn hu |
| exact ((partial_contDiffOn hchi hU 0).mul hFt).add ((partial_contDiffOn hchi hU 1).mul hFs) |
|
|
| theorem multiplierCutoffError_zero (A chi m n u : Coord → ℝ) (p : Coord) |
| (ht : coordPartial 0 chi p = 0) (hs : coordPartial 1 chi p = 0) : |
| multiplierCutoffError A chi m n u p = 0 := by simp [multiplierCutoffError, ht, hs] |
|
|
| theorem directed_source_young_scalar {w f ut us epsilon t R c : ℝ} |
| (hw : 0 ≤ w) (hc : 0 < c) (heps : 0 ≤ epsilon) (heps1 : epsilon ≤ 1) |
| (ht : |t| ≤ R) : |
| |f * (w * (-us + epsilon * t * ut))| ≤ |
| c / 2 * w * (ut^2 + us^2) + ((1 + R^2) / (2 * c)) * w * f^2 := by |
| have hR : 0 ≤ R := (abs_nonneg _).trans ht |
| have het : |epsilon * t| ≤ R := by |
| rw [abs_mul, abs_of_nonneg heps] |
| exact (mul_le_mul_of_nonneg_left ht heps).trans |
| (by have hh := mul_le_mul_of_nonneg_right heps1 hR |
| have hsquare : (epsilon * t)^2 ≤ R^2 := by |
| simpa only [sq_abs] using pow_le_pow_left₀ (abs_nonneg (epsilon * t)) het 2 |
| have hyu := weighted_young c us f hc |
| have hyt := weighted_young c ut (epsilon * t * f) hc |
| have hraw : |f * (-us + epsilon * t * ut)| ≤ |
| c / 2 * (ut^2 + us^2) + (1 + R^2) / (2 * c) * f^2 := by |
| have htriangle : |f * (-us + epsilon * t * ut)| ≤ |us * f| + |ut * (epsilon * t * f)| := by |
| calc |
| _ = |-(us * f) + ut * (epsilon * t * f)| := by congr 1 |
| _ ≤ |us * f| + |ut * (epsilon * t * f)| := by simpa only [abs_neg] using abs_add_le (-(us * f)) (ut * (epsilon * t * f)) |
| have hproduct := mul_le_mul_of_nonneg_right hsquare (sq_nonneg f) |
| have hdiv := div_le_div_of_nonneg_right hproduct (show 0 ≤ 2 * c by positivity) |
| calc |
| _ ≤ (c * us^2 + f^2 / c) / 2 + (c * ut^2 + (epsilon * t * f)^2 / c) / 2 := |
| htriangle.trans (add_le_add hyu hyt) |
| _ = c / 2 * (ut^2 + us^2) + f^2 / (2 * c) + (epsilon * t)^2 * f^2 / (2 * c) := by |
| field_simp [hc.ne'] |
| ring |
| _ ≤ c / 2 * (ut^2 + us^2) + f^2 / (2 * c) + R^2 * f^2 / (2 * c) := |
| add_le_add le_rfl hdiv |
| _ = _ := by ring |
| have hmul := mul_le_mul_of_nonneg_left hraw hw |
| calc |
| _ = w * |f * (-us + epsilon * t * ut)| := by |
| rw [show f * (w * (-us + epsilon * t * ut)) = w * (f * (-us + epsilon * t * ut)) by ring, |
| abs_mul, abs_of_nonneg hw] |
| _ ≤ w * (c / 2 * (ut^2 + us^2) + (1 + R^2) / (2 * c) * f^2) := hmul |
| _ = _ := by ring |
| |
| end SmoothLocal.Weighted |
| |
| end |
| |
| end OAI |
| |
