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59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 | 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
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