| import OAI.Geometry.IsometricImmersion.Caps.EllipticC1 |
| import OAI.Geometry.IsometricImmersion.Caps.EllipticRatio |
| import OAI.Geometry.IsometricImmersion.Caps.EllipticCutoffEdge |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped ContDiff Topology Interval |
|
|
| namespace SmoothLocal.Weighted |
|
|
| open SmoothLocal.Geometry |
|
|
| def threeEdgeDerivativeConstant (l0 l1 r1 r0 a0 a1 : ℝ) : ℝ := |
| transitionDerivativeBound / (l1 - l0) + transitionDerivativeBound / (r0 - r1) + |
| transitionDerivativeBound / (a1 - a0) |
|
|
| theorem threeEdgeDerivativeConstant_nonneg {l0 l1 r1 r0 a0 a1 : ℝ} |
| (hL : l0 < l1) (hR : r1 < r0) (hS : a0 < a1) : |
| 0 ≤ threeEdgeDerivativeConstant l0 l1 r1 r0 a0 a1 := by |
| have ht := transitionDerivativeBound_pos.le |
| have hl := sub_pos.mpr hL |
| have hr := sub_pos.mpr hR |
| have hs := sub_pos.mpr hS |
| unfold threeEdgeDerivativeConstant |
| positivity |
|
|
| theorem threeEdgeCutoff_partial_common {l0 l1 r1 r0 a0 a1 : ℝ} |
| (hL : l0 < l1) (hR : r1 < r0) (hS : a0 < a1) (p : Coord) (i : Fin 2) : |
| |coordPartial i (threeEdgeCutoff l0 l1 r1 r0 a0 a1) p| ≤ |
| threeEdgeDerivativeConstant l0 l1 r1 r0 a0 a1 := by |
| have hb := threeEdgeCutoff_partial_bounds hL hR hS p |
| have ht := transitionDerivativeBound_pos.le |
| have hl : 0 ≤ transitionDerivativeBound / (l1 - l0) := div_nonneg ht (sub_pos.mpr hL).le |
| have hr : 0 ≤ transitionDerivativeBound / (r0 - r1) := div_nonneg ht (sub_pos.mpr hR).le |
| have hs : 0 ≤ transitionDerivativeBound / (a1 - a0) := div_nonneg ht (sub_pos.mpr hS).le |
| unfold threeEdgeDerivativeConstant |
| fin_cases i < |
|
|
| theorem ellipticWeightC1_errors_eq {b c : ℝ} (A B C chi K : Coord → ℝ) |
| {p : Coord} (hp : p 1 < b) : |
| ellipticErrorT B (ellipticWeightC1 b c chi K) p = |
| ellipticErrorT B (ellipticWeight b c chi K) p ∧ |
| ellipticErrorS A C (ellipticWeightC1 b c chi K) p = |
| ellipticErrorS A C (ellipticWeight b c chi K) p := by |
| have he : (fun q => A q * ellipticWeightC1 b c chi K q) =ᶠ[𝓝 p] |
| (fun q => A q * ellipticWeight b c chi K q) := by |
| filter_upwards [(isOpen_lt (continuous_apply 1) continuous_const).mem_nhds hp] with q hq |
| rw [ellipticWeightC1_eq hq.le] |
| have hderiv : coordPartial 1 (fun q => A q * ellipticWeightC1 b c chi K q) p = |
| coordPartial 1 (fun q => A q * ellipticWeight b c chi K q) p := by |
| unfold coordPartial |
| rw [he.fderiv_eq] |
| constructor |
| · rw [ellipticErrorT, ellipticErrorT, ellipticWeightC1_partial_eq hp 0, |
| ellipticWeightC1_eq hp.le] |
| · rw [ellipticErrorS, ellipticErrorS, hderiv, ellipticWeightC1_eq hp.le] |
|
|
| theorem threeEdge_ellipticWeight_boundary {l0 l1 r1 r0 a0 a1 b c : ℝ} |
| (hL : l0 < l1) (hR : r1 < r0) (hS : a0 < a1) (K : Coord → ℝ) : |
| zeroRectangleBoundary l0 r0 a0 b |
| (ellipticWeightC1 b c (threeEdgeCutoff l0 l1 r1 r0 a0 a1) K) := by |
| constructor |
| · intro s hs |
| have hl := threeEdgeCutoff_zero hL hR hS (boxPoint l0 s) |
| (Or.inl (by simp [boxPoint])) |
| have hr := threeEdgeCutoff_zero hL hR hS (boxPoint r0 s) |
| (Or.inr (Or.inl (by simp [boxPoint]))) |
| constructor < |
| · intro t ht |
| have hb := threeEdgeCutoff_zero hL hR hS (boxPoint t a0) |
| (Or.inr (Or.inr (by simp [boxPoint]))) |
| constructor |
| · simp [ellipticWeightC1, hb] |
| · exact ellipticWeightC1_zero (by simp [boxPoint]) |
|
|
| variable {U : Set Coord} {K G1 B C u f : Coord → ℝ} |
| {l0 l1 r1 r0 a0 a1 b c MK g0 M : ℝ} |
|
|
| theorem threeEdge_elliptic_energy_bound |
| (hL : l0 < l1) (hR : r1 < r0) (hS : a0 < a1) |
| (ht : l0 ≤ r0) (hs : a0 ≤ b) (hc : 0 < c) |
| (hMK : 0 ≤ MK) (hg0 : 0 < g0) (hM : 0 ≤ M) |
| (hU : IsOpen U) (hK : ContDiffOn ℝ ∞ K U) (hG1 : ContDiffOn ℝ ∞ G1 U) |
| (hB : ContDiffOn ℝ ∞ B U) (hC : ContDiffOn ℝ ∞ C U) (hu : ContDiffOn ℝ ∞ u U) |
| (hbox : closedRectangle l0 r0 a0 b ⊆ U) |
| (hEq : ∀ p ∈ U, multiplierOperator (fun q => G1 q * K q) B C u p = f p) |
| (hKi : ∀ p ∈ closedRectangle l0 r0 a0 b, ∀ i : Fin 2, |coordPartial i K p| ≤ MK) |
| (hGlow : ∀ p ∈ closedRectangle l0 r0 a0 b, g0 ≤ G1 p) |
| (hAhi : ∀ p ∈ closedRectangle l0 r0 a0 b, |G1 p * K p| ≤ M) |
| (hAs : ∀ p ∈ closedRectangle l0 r0 a0 b, |
| |coordPartial 1 (fun q => G1 q * K q) p| ≤ M) |
| (hBb : ∀ p ∈ closedRectangle l0 r0 a0 b, |B p| ≤ M) |
| (hCb : ∀ p ∈ closedRectangle l0 r0 a0 b, |C p| ≤ M) : |
| rectangleIntegral l0 r0 a0 b |
| (ellipticEnergy (fun q => G1 q * K q) |
| (ellipticWeight b c (threeEdgeCutoff l0 l1 r1 r0 a0 a1) K) u) ≤ |
| (b - a0) ^ 6 * rectangleIntegral l0 r0 a0 b (fun p => (f p) ^ 2) + |
| ((b - a0) ^ 6 + ellipticRatioConstant (b - a0) (g0 * c / 2) |
| (ellipticCutoffDerivativeConstant c MK (threeEdgeDerivativeConstant l0 l1 r1 r0 a0 a1)) M) * |
| rectangleIntegral l0 r0 a0 b (fun p => (u p) ^ 2) := by |
| let chi := threeEdgeCutoff l0 l1 r1 r0 a0 a1 |
| let v := ellipticWeightC1 b c chi K |
| let X := threeEdgeDerivativeConstant l0 l1 r1 r0 a0 a1 |
| let H := ellipticRatioConstant (b - a0) (g0 * c / 2) (ellipticCutoffDerivativeConstant c MK X) M |
| have hX : 0 ≤ X := threeEdgeDerivativeConstant_nonneg hL hR hS |
| have hchi : ContDiffOn ℝ ∞ chi U := (threeEdgeCutoff_contDiff _ _ _ _ _ _).contDiffOn |
| have hv : ContDiffOn ℝ 1 v U := ellipticWeightC1_contDiffOn hc |
| (hchi.of_le (by simp)) (hK.of_le (by simp)) |
| have hA : ContDiffOn ℝ ∞ (fun q => G1 q * K q) U := hG1.mul hK |
| have hH : 0 ≤ H := ellipticRatioConstant_nonneg (sub_nonneg.mpr hs) (by positivity) hM |
| have hposD (p : Coord) (hpv : 0 < v p) : p 1 < b := by |
| by_contra hn |
| have hz : v p = 0 := ellipticWeightC1_zero (le_of_not_gt hn) |
| linarith |
| have hweightEq (p : Coord) (hp : p ∈ closedRectangle l0 r0 a0 b) : |
| v p = ellipticWeight b c chi K p := ellipticWeightC1_eq hp.2.2 |
| have hposA (p : Coord) (hp : p ∈ closedRectangle l0 r0 a0 b) (hpv : 0 < v p) : |
| 0 < G1 p * K p := by |
| have hd := hposD p hpv |
| have hcurv := ellipticWeight_pos_implies_curvature hc hd |
| (show 0 < ellipticWeight b c chi K p by simpa only [← hweightEq p hp] using hpv) |
| have hKpos : 0 < K p := (mul_pos (by positivity : 0 < c / 2) (sub_pos.mpr hd)).trans hcurv |
| exact mul_pos (hg0.trans_le (hGlow p hp)) hKpos |
| have hratio (p : Coord) (hp : p ∈ closedRectangle l0 r0 a0 b) (hpv : 0 < v p) : |
| (ellipticErrorT B v p) ^ 2 / v p + |
| (ellipticErrorS (fun q => G1 q * K q) C v p) ^ 2 / (G1 p * K p * v p) ≤ H := by |
| have hd := hposD p hpv |
| have herrors := ellipticWeightC1_errors_eq (c := c) (fun q => G1 q * K q) B C chi K hd |
| change (ellipticErrorT B (ellipticWeightC1 b c chi K) p) ^ 2 / v p + |
| (ellipticErrorS (fun q => G1 q * K q) C (ellipticWeightC1 b c chi K) p) ^ 2 / |
| (G1 p * K p * v p) ≤ H |
| rw [herrors.1, herrors.2, hweightEq p hp] |
| exact ellipticCutoff_actual_ratio hc (sub_pos.mpr hd) |
| (by unfold edgeDistance |
| ((hchi.contDiffAt (hU.mem_nhds (hbox hp))).differentiableAt (by simp)) |
| ((hK.contDiffAt (hU.mem_nhds (hbox hp))).differentiableAt (by simp)) |
| ((hG1.contDiffAt (hU.mem_nhds (hbox hp))).differentiableAt (by simp)) |
| (threeEdgeCutoff_range _ _ _ _ _ _ p) |
| (threeEdgeCutoff_partial_common hL hR hS p) (hKi p hp) |
| (show 0 < ellipticWeight b c chi K p by simpa only [← hweightEq p hp] using hpv) |
| (hGlow p hp) (hAhi p hp) (hAs p hp) (hBb p hp) (hCb p hp) |
| have hestimate := integrated_elliptic_energy_bound_C1 ht hs hU hA hB hC hv hu hbox |
| (threeEdge_ellipticWeight_boundary hL hR hS K) hEq ((b - a0) ^ 6) H |
| (by positivity) hH (fun p _ => ellipticWeightC1_nonneg _ _ _ _ p) |
| (fun p hp => (ellipticWeightC1_le_distance_six _ _ _ _ p |
| (threeEdgeCutoff_range _ _ _ _ _ _ p)).trans |
| (pow_le_pow_left₀ (sub_nonneg.mpr hp.2.2) |
| (by linarith [hp.2.1]) 6)) |
| hposA hratio |
| have henergyEq : rectangleIntegral l0 r0 a0 b |
| (ellipticEnergy (fun q => G1 q * K q) v u) = |
| rectangleIntegral l0 r0 a0 b |
| (ellipticEnergy (fun q => G1 q * K q) (ellipticWeight b c chi K) u) := by |
| apply rectangleIntegral_congr ht hs |
| intro p hp |
| change v p * _ = ellipticWeight b c chi K p * _ |
| rw [hweightEq p hp] |
| rw [henergyEq] at hestimate |
| exact hestimate |
|
|
| end SmoothLocal.Weighted |
|
|
| end |
|
|
| end OAI |
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|
