| import OAI.Geometry.RCD.Sobolev.CoerciveQuadraticEnvelope |
| import OAI.Geometry.RCD.Sobolev.CompactCoerciveTruncation |
| import OAI.Geometry.RCD.Transport.CompactContactCompletion |
|
|
| namespace OAI |
|
|
| /-! # Compact Sobolev contact potentials for a local quadratic envelope -/ |
|
|
| open Set Metric |
| open scoped NNReal |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X Y : Type*} [MetricSpace X] [ProperSpace X] |
|
|
| theorem compact_quadratic_contact (hgeo : IsGeodesicSpace X) |
| (p : X) (point : Y → X) (F : Y → ℝ) {R t B A : ℝ} |
| (hR : 0 ≤ R) (ht : 0 < t) (hF : ∀ a, F a ≤ B) |
| (hd : ∀ a, dist p (point a) ≤ R) (Φ : X → ℝ) |
| (hupper : ∀ x a, Φ x ≤ dist x (point a)^2/2-t*F a+t*A) |
| (hexact : ∀ x, ∃ a, Φ x = dist x (point a)^2/2-t*F a+t*A) |
| (hlip : ∀ r : ℝ, 0 < r → ∃ L : ℝ≥0, LipschitzOnWith L Φ (closedBall p r)) |
| {r₀ : ℝ} (hr₀ : 0 ≤ r₀) : |
| ∃ (M r : ℝ) (ψ : X → ℝ) (L : ℝ≥0), |
| 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 ∈ closedBall p r₀, hopfLax (-ψ) 1 x = Φ x-M) ∧ |
| ∀ x ∈ closedBall p r₀, ∀ a, |
| Φ x = dist x (point a)^2/2-t*F a+t*A → |
| hopfLax (-ψ) 1 x+ψ (point a) = dist x (point a)^2/2 := by |
| have hcoerce := quadratic_envelope_coercive p hR ht point F hF hd Φ hexact |
| obtain ⟨M,r,L,_,hf,hfs,hfzero,hlocal⟩ := |
| compact_coercive_truncation p Φ hlip hcoerce hr₀ |
| let f : X → ℝ := fun x => min (Φ x) M-M |
| obtain ⟨ψ,s,hψ,hψs,hψzero,hχ,hχs,had,_,hc⟩ := |
| compact_contact_completion hgeo p hf hfzero |
| have hbound (a : Y) (z : X) : |
| f z+(t*F a-t*A+M) ≤ dist z (point a)^2/2 := by |
| have hm := min_le_left (Φ z) M |
| have hu := hupper z a |
| dsimp [f] |
| linarith |
| have hcontact (x : X) (hx : x ∈ closedBall p r₀) (a : Y) |
| (ha : Φ x = dist x (point a)^2/2-t*F a+t*A) : |
| hopfLax (-ψ) 1 x = Φ x-M ∧ ψ (point a) = t*F a-t*A+M := by |
| have he : f x+(t*F a-t*A+M) = dist x (point a)^2/2 := by |
| change (min (Φ x) M-M)+(t*F a-t*A+M) = _ |
| rw [hlocal x hx,ha] |
| ring |
| obtain ⟨heq,hψeq⟩ := hc x (point a) (t*F a-t*A+M) (hbound a) he |
| exact ⟨heq.trans (hlocal x hx),hψeq⟩ |
| refine ⟨M,s,ψ,L,hψ,hψs,hψzero,hχ,hχs,had,?_,?_⟩ |
| · intro x hx |
| obtain ⟨a,ha⟩ := hexact x |
| exact (hcontact x hx a ha).1 |
| · intro x hx a ha |
| obtain ⟨he,hψeq⟩ := hcontact x hx a ha |
| rw [he,hψeq,ha] |
| ring |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
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|
