| import OAI.Geometry.IsometricImmersion.Caps.CapCutoffGeometry |
| import OAI.Geometry.IsometricImmersion.Caps.EllipticGradient |
| import OAI.Geometry.IsometricImmersion.Energy.DirectedEdgeDefect |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter |
| open scoped ContDiff Topology |
|
|
| namespace SmoothLocal.Weighted |
| open SmoothLocal.Geometry |
|
|
| def capOverlapThreshold (kappa c1 D : ℝ) : ℝ := min c1 (kappa / (D + 1)) |
|
|
| theorem capOverlapThreshold_pos {kappa c1 D : ℝ} |
| (hk : 0 < kappa) (hc1 : 0 < c1) (hD : 0 ≤ D) : |
| 0 < capOverlapThreshold kappa c1 D := by |
| exact lt_min hc1 (div_pos hk (by linarith)) |
|
|
| theorem capOverlapThreshold_le (kappa c1 D : ℝ) : |
| capOverlapThreshold kappa c1 D ≤ c1 := min_le_left _ _ |
|
|
| theorem capOverlapThreshold_distance_le {kappa c1 D d : ℝ} |
| (hk : 0 < kappa) (hc1 : 0 < c1) (hD : 0 ≤ D) (_ : 0 ≤ d) (hdD : d ≤ D) : |
| capOverlapThreshold kappa c1 D * d ≤ kappa := by |
| have hc := (capOverlapThreshold_pos hk hc1 hD).le |
| have hD1 : 0 < D + 1 := by linarith |
| have hcd : capOverlapThreshold kappa c1 D * (D + 1) ≤ kappa := |
| (le_div_iff₀ hD1).mp (min_le_right _ _) |
| exact (mul_le_mul_of_nonneg_left (show d ≤ D + 1 by linarith) hc).trans hcd |
|
|
| theorem capOuterElliptic_gradient_pointwise |
| {L R A b c g0 D : ℝ} {K G1 u : Coord → ℝ} {p : Coord} |
| (hL : -2 < L) (hR : R < 2) (hA : -2 < A) |
| (hc : 0 < c) (hg0 : 0 < g0) (hd : 0 < edgeDistance b p) |
| (hdD : edgeDistance b p ≤ D) (hchi : capSpatialCutoff L R A p ≠ 0) |
| (hK : c * edgeDistance b p ≤ K p) (hG1 : g0 ≤ G1 p) : |
| weightedGradient b u p ≤ (D + 1 / (g0 * c)) * |
| ellipticEnergy (fun q => G1 q * K q) |
| (ellipticWeightC1 b c (capOuterSpatialCutoff L R A) K) u p := by |
| have houter := capOuterSpatialCutoff_one_of_nonzero hL hR hA hchi |
| have hh := ellipticCutoff_gradient_pointwise (u := u) hc le_rfl hg0 hd hdD houter hK hG1 |
| have hp : p 1 ≤ b := (sub_pos.mp hd).le |
| simpa only [ellipticEnergy, ellipticWeightC1_eq hp] using hh |
|
|
| theorem directedCoercivityDefect_nonzero_cutoff |
| {A B C chi I u : Coord → ℝ} {b lambda epsilon c2 : ℝ} {p : Coord} |
| (hdef : directedCoercivityDefect A B C chi I u b lambda epsilon c2 p ≠ 0) : chi p ≠ 0 := by |
| intro hz |
| apply hdef |
| simp only [directedCoercivityDefect, hz, zero_mul] |
|
|
| theorem capOuterElliptic_gradient_on_defect |
| {L R A0 b c1 c2 kappa D g0 lambda epsilon : ℝ} |
| {A B C K G1 I u : Coord → ℝ} {p : Coord} |
| (hL : -2 < L) (hR : R < 2) (hA0 : -2 < A0) |
| (hk : 0 < kappa) (hc1 : 0 < c1) (hD : 0 ≤ D) (hg0 : 0 < g0) |
| (hd : 0 < edgeDistance b p) (hdD : edgeDistance b p ≤ D) (hG1 : g0 ≤ G1 p) |
| (hcoercive : K p ≤ c1 * edgeDistance b p → |
| c2 * directedWeight b lambda I p * ((coordPartial 0 u p)^2 + (coordPartial 1 u p)^2) ≤ |
| multiplierQuadratic A B C (directedM b lambda I) (directedN b lambda epsilon I) u p) |
| (hdef : directedCoercivityDefect A B C (capSpatialCutoff L R A0) I u b lambda epsilon c2 p ≠ 0) : |
| weightedGradient b u p ≤ (D + 1 / (g0 * capOverlapThreshold kappa c1 D)) * |
| ellipticEnergy (fun q => G1 q * K q) |
| (ellipticWeightC1 b (capOverlapThreshold kappa c1 D) (capOuterSpatialCutoff L R A0) K) u p := by |
| have hK : c1 * edgeDistance b p < K p := by |
| by_contra h |
| exact hdef (directedCoercivityDefect_zero _ _ _ _ _ _ _ _ _ _ _ (hcoercive (le_of_not_gt h))) |
| apply capOuterElliptic_gradient_pointwise hL hR hA0 (capOverlapThreshold_pos hk hc1 hD) |
| hg0 hd hdD (directedCoercivityDefect_nonzero_cutoff hdef) _ hG1 |
| exact (mul_le_mul_of_nonneg_right (capOverlapThreshold_le kappa c1 D) hd.le).trans hK.le |
|
|
| end SmoothLocal.Weighted |
|
|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.Weighted |
|
|
| theorem capOuterElliptic_gradient_on_actual_cutoff_support |
| {g0 eta : MetricField} {U : Set Coord} {Y : ℝ → ℝ → ℝ} |
| {L R A b kappa c1 D glow : ℝ} {G1 u : Coord → ℝ} {p : Coord} {i : Fin 2} |
| (hL : -2 < L) (hR : R < 2) (hA : -2 < A) (hk : 0 < kappa) |
| (hc1 : 0 < c1) (hD : 0 ≤ D) (hg : 0 < glow) |
| (hbackground : ∀ q ∈ U, gaussianCurvature g0 q = SmoothLocal.Model.modelCurvature kappa q) |
| (hsupport : tsupport eta ⊆ patchBox) (hpU : capChart Y p ∈ U) |
| (hdisp : |capFlowHeight Y p - p 1| ≤ (1 : ℝ) / 50) |
| (hd : 0 < edgeDistance b p) (hdD : edgeDistance b p ≤ D) (hG1 : glow ≤ G1 p) |
| (hpartial : coordPartial i (capSpatialCutoff L R A) p ≠ 0) : |
| weightedGradient b u p ≤ (D + 1 / (glow * capOverlapThreshold kappa c1 D)) * |
| ellipticEnergy (fun q => G1 q * capPullback Y (gaussianCurvature (g0 + eta)) q) |
| (ellipticWeightC1 b (capOverlapThreshold kappa c1 D) (capOuterSpatialCutoff L R A) |
| (capPullback Y (gaussianCurvature (g0 + eta)))) u p := by |
| have hK := capSpatialCutoff_partial_actual_curvature_lower hk hL hR hA hbackground hsupport |
| hpU hdisp hpartial |
| apply capOuterElliptic_gradient_pointwise hL hR hA (capOverlapThreshold_pos hk hc1 hD) |
| hg hd hdD (capSpatialCutoff_nonzero_of_partial hpartial) _ hG1 |
| exact (capOverlapThreshold_distance_le hk hc1 hD hd.le hdD).trans hK.le |
|
|
| end SmoothLocal.Flow |
|
|
| end |
|
|
| end OAI |
|
|
