File size: 3,773 Bytes
9655094 | 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 | import OAI.Analysis.CircleDomains.Modulus.DistinguishedStokesBudget
namespace OAI
/-! 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. -/
open Set Function Filter MeasureTheory Metric
open scoped Topology BigOperators
noncomputable section
namespace CircleDomainRigidity
open FiniteTransfer
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)
end CircleDomainRigidity
end
end OAI
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