| import OAI.Analysis.RCDHeat.InnerPath |
| import OAI.Analysis.RCDHeat.KantorovichRadius |
|
|
| namespace OAI |
|
|
| /-! # Compact contact completion by two quadratic transforms |
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| The completion preserves every contact of the original compact potential. |
| It also keeps an explicit compact support radius for the right potential. |
| -/ |
|
|
| open Set Metric |
| open scoped NNReal |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [ProperSpace X] |
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|
| theorem compact_contact_completion (hgeo : IsGeodesicSpace X) |
| (p : X) {φ : X → ℝ} {L : ℝ≥0} (hφ : LipschitzWith L φ) |
| {R : ℝ} (hzero : ∀ x, R < dist p x → φ x = 0) : |
| ∃ (ψ : X → ℝ) (r : ℝ), LipschitzWith L ψ ∧ HasCompactSupport ψ ∧ |
| (∀ x, r < dist p x → ψ x = 0) ∧ |
| LipschitzWith L (hopfLax (-ψ) 1) ∧ HasCompactSupport (hopfLax (-ψ) 1) ∧ |
| (∀ x y, hopfLax (-ψ) 1 x + ψ y ≤ dist x y ^ 2 / 2) ∧ |
| (∀ x, φ x ≤ hopfLax (-ψ) 1 x) ∧ |
| ∀ x y b, (∀ z, φ z + b ≤ dist z y ^ 2 / 2) → |
| φ x + b = dist x y ^ 2 / 2 → |
| hopfLax (-ψ) 1 x = φ x ∧ ψ y = b := by |
| let ψ := hopfLax (-φ) 1 |
| have hpos : (0 : ℝ) < 1 := by norm_num |
| have hψ : LipschitzWith L ψ := hopfLax_lipschitz hφ.neg hgeo hpos |
| have hnegzero (x : X) (hx : R < dist p x) : (-φ) x = 0 := by |
| simp only [Pi.neg_apply,hzero x hx,neg_zero] |
| have hψs : HasCompactSupport ψ := hopfLax_hasCompactSupport hφ.neg p R hnegzero hpos |
| have hψzero (x : X) (hx : R+2*(L:ℝ) < dist p x) : ψ x = 0 := by |
| apply hopfLax_zero_outside hφ.neg p R hnegzero hpos |
| simpa only [mul_one] using hx |
| have hψnegzero (x : X) (hx : R+2*(L:ℝ) < dist p x) : (-ψ) x = 0 := by |
| simp only [Pi.neg_apply,hψzero x hx,neg_zero] |
| have hχ : LipschitzWith L (hopfLax (-ψ) 1) := hopfLax_lipschitz hψ.neg hgeo hpos |
| have hχs : HasCompactSupport (hopfLax (-ψ) 1) := |
| hopfLax_hasCompactSupport hψ.neg p (R+2*(L:ℝ)) hψnegzero hpos |
| have had (x y : X) : φ x + ψ y ≤ dist x y ^ 2 / 2 := by |
| have hh := hopfLax_le_cost hφ.neg hpos y x |
| simp only [Pi.neg_apply,mul_one,dist_comm y x] at hh |
| dsimp only [ψ] |
| linarith |
| refine ⟨ψ,R+2*(L:ℝ),hψ,hψs,hψzero,hχ,hχs,?_,?_,?_⟩ |
| · intro x y |
| have hh := hopfLax_le_cost hψ.neg hpos x y |
| simp only [Pi.neg_apply,mul_one] at hh |
| linarith |
| · intro x |
| apply le_csInf ⟨_,mem_range_self x⟩ |
| rintro _ ⟨y,rfl⟩ |
| have hh := had x y |
| simp only [Pi.neg_apply,mul_one] |
| linarith |
| · intro x y b hb hxy |
| have hby : b ≤ ψ y := by |
| apply le_csInf ⟨_,mem_range_self x⟩ |
| rintro _ ⟨z,rfl⟩ |
| have hh := hb z |
| simp only [Pi.neg_apply,mul_one,dist_comm y z] |
| linarith |
| have hyb : ψ y ≤ b := by linarith [had x y] |
| have he : ψ y = b := le_antisymm hyb hby |
| refine ⟨(hopfLax_of_contact hψ had (x := x) (y := y) ?_).1,he⟩ |
| rwa [he] |
|
|
| end RegularPointCoordinates |
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| end |
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| end OAI |
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|
