| import OAI.Analysis.CircleDomains.Modulus.DistinguishedStokesBudget |
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| namespace OAI |
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|
| /-! The actual finite model obeys the countable-band Stokes bound with |
| its singular mark removed. Disjointness and containment of its round |
| holes provide the radius budget; the source collar branches provide |
| the boundary oscillations. -/ |
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|
| open Set Function Filter MeasureTheory Metric |
| open scoped Topology BigOperators |
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| noncomputable section |
| namespace CircleDomainRigidity |
| open FiniteTransfer |
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|
| theorem assigned_model_stokes_band_bound |
| (hgreen : FiniteCircleGreenFormula) |
| {I S : Type*} [Fintype I] {T : I → ℂ ≃ₜ ℂ} {f fi : Sphere → Sphere} {V : Set Sphere} |
| (a : TerminalComponentAssignment T f V) |
| (hf : IsSmoothSphereEquivalence f fi (openCompletedRegion T) V) |
| (c : I → ℂ) (r : I → ℝ) (hr : ∀ i, 0 ≤ r i) |
| (hd : Pairwise (Disjoint on fun i => closedBall (c i) (r i))) |
| (hext : ((↑) : ℂ → Sphere) ⁻¹' V = (⋃ i, closedBall (c i) (r i))ᶜ) |
| (htarget : ∀ i, ((↑) : ℂ → Sphere) '' closedBall (c i) (r i) ⊆ |
| connectedComponentIn Vᶜ (a.point i)) |
| (n : ℕ) (κ : Fin n → I) (hκ : Injective κ) (b : Fin n) |
| (H : ℂ ≃ₜ ℂ) (γ : ℝ → ℂ) (hγ : PositiveJordanParametrization H γ) |
| (hin : ∀ j, closedBall (c (κ j)) (r (κ j)) ⊆ jordanInterior H) |
| (O : Set ℂ) (hO : IsOpen O) |
| (hsub : (closure (jordanInterior H) \ ⋃ j, closedBall (c (κ j)) (r (κ j))) ⊆ O) |
| (w : Sphere → UnitAddCircle) |
| (hu : SmoothCircleOn (fun z : ℂ => w (fi (z : Sphere))) O) |
| {L : ℝ} (hL : 0 ≤ L) (hbounded : jordanInterior H ⊆ closedBall 0 L) |
| (henergy : MemLp (circleFDeriv (fun z : ℂ => w (fi (z : Sphere)))) 2 |
| (volume.restrict (jordanInterior H \ ⋃ j, closedBall (c (κ j)) (r (κ j))))) |
| (U : Fin n → Set Sphere) (v : Fin n → Sphere → ℝ) |
| (hU : ∀ j ≠ b, IsOpen (U j)) |
| (hboundary : ∀ j ≠ b, ((↑) : ℂ → Sphere) '' frontier (jordanInterior (T (κ j))) ⊆ U j) |
| (hv : ∀ j ≠ b, ContinuousOn (v j) (U j)) |
| (hproj : ∀ j ≠ b, ∀ x ∈ U j, (v j x : UnitAddCircle) = w x) |
| (A : Fin n → ℝ) (hA : ∀ j ≠ b, ∀ x ∈ frontier (jordanInterior (T (κ j))), |
| ∀ y ∈ frontier (jordanInterior (T (κ j))), |v j (x : Sphere) - v j (y : Sphere)| ≤ A j) |
| (label : I → S) (hlabel : Injective label) |
| (band : S → ℝ) (hband : Summable (fun s => band s ^ 2)) (hb : ∀ s, 0 ≤ band s) |
| {η : ℝ} (hexcess : Real.sqrt |
| (∑ j, max (A j - band (label (κ j))) 0 ^ 2) ≤ η) : |
| |circleFormIntegral (fun z => infDist z (closedBall (c (κ b)) (r (κ b)))) |
| (fun z : ℂ => w (fi (z : Sphere))) γ| ≤ |
| Real.sqrt Real.pi * L * Real.sqrt |
| (∫ z in jordanInterior H \ ⋃ j, closedBall (c (κ j)) (r (κ j)), |
| ‖circleFDeriv (fun z : ℂ => w (fi (z : Sphere))) z‖ ^ 2) + |
| 2 * Real.pi * L * |
| (Real.sqrt (∑' s : {s : S // s ≠ label (κ b)}, band s ^ 2) + η) := by |
| classical |
| have hnonneg (j : Fin n) (hj : j ≠ b) : 0 ≤ A j := by |
| obtain ⟨x, hx⟩ := (isConnected_frontier_jordanInterior (T (κ j))).nonempty |
| simpa only [sub_self, abs_zero] using hA j hj x hx x hx |
| have hradius : (∑ j, r (κ j) ^ 2) ≤ L ^ 2 := |
| sum_radii_sq_le_of_disjoint_closed_disks (c ∘ κ) (r ∘ κ) |
| (fun j => hr (κ j)) hL (fun j k hjk => hd (fun h => hjk (hκ h))) |
| (fun j => (hin j).trans hbounded) |
| exact circle_stokes_bound_off_distinguished (label ∘ κ) (hlabel.comp hκ) b band hband |
| (r ∘ κ) A hnonneg hb hL hradius hexcess |
| (assigned_model_stokes_bound hgreen a hf c r hr hd hext htarget n κ hκ b H γ hγ |
| hin O hO hsub w hu hL hbounded henergy U v hU hboundary hv hproj A hA) |
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| end CircleDomainRigidity |
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| end |
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| end OAI |
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