| import OAI.Analysis.CircleDomains.Sobolev.CoordinateGradientSize |
| import OAI.Analysis.CircleDomains.Sobolev.PriorSmoothedCompatibility |
|
|
| namespace OAI |
|
|
| /-! The real barrier in the endpoint problem, with its actual physical |
| representative, countable enlarged flags, and finite-torus compatibility. -/ |
|
|
| open Set Function Filter MeasureTheory |
| open scoped ENNReal Topology |
|
|
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| attribute [local instance] sphereMetric sphereBorel |
| local instance : BorelSpace Sphere := ⟨rfl⟩ |
|
|
| variable [MeasurableSpace C(PathTime, Sphere)] |
|
|
| structure CompatibleLevelBarrier (d : PathCodeData Sphere Sphere) (π : Sphere → Sphere) |
| (previous : PriorLevelFamily d.physicalDomain π) (flags EStar : Set Sphere) |
| (regions : ℕ → Set Sphere) (f : d.quotientDomain → Sphere) (W₁ : Set Sphere) where |
| coordinate : PhysicalRealCoordinate d π |
| level : ℕ |
| penalty : Set Sphere |
| penalty_open : IsOpen penalty |
| penalty_inside : closure penalty ⊆ d.quotientDomain |
| penalty_disjoint : Disjoint d.anchor (closure penalty) |
| penalty_avoids : ∀x : d.quotientDomain, x.val ∈ closure penalty → f x ∉ W₁ |
| zero : ∀x ∈ EStar, coordinate.quotient x = 0 |
| one : ∀x ∈ closure (regions level) ∩ d.quotientDomain, coordinate.quotient x = 1 |
| exclusions : Set Sphere |
| exclusions_countable : exclusions.Countable |
| prior_flags : flags ⊆ exclusions |
| weight : Sphere → ℝ≥0∞ |
| weight_finite : (∑'s, weight s ^ 2) < ∞ |
| weight_support : support weight ⊆ exclusions |
| band_bound : ∀s ∈ d.dust, ENNReal.ofReal |
| (limsup coordinate.quotient (𝓝[d.quotientDomain] s) - |
| liminf coordinate.quotient (𝓝[d.quotientDomain] s)) ≤ weight s |
| compatible : ∀n labels, (volume.prod (Measure.pi (fun _ : Fin (n + 1) => |
| (volume : Measure UnitAddCircle)))) |
| (freeComponentFailure (d.dust \ exclusions) |
| (circleBandGraph d.quotientDomain coordinate.quotient) |
| (fun i => previous.graph i) labels) = 0 |
|
|
| namespace CompatibleLevelBarrier |
|
|
| variable {d : PathCodeData Sphere Sphere} {π : Sphere → Sphere} |
| {previous : PriorLevelFamily d.physicalDomain π} {flags EStar : Set Sphere} |
| {regions : ℕ → Set Sphere} {f : d.quotientDomain → Sphere} {W₁ : Set Sphere} |
|
|
| def controlledSize (B : CompatibleLevelBarrier d π previous flags EStar regions f W₁) : ℝ≥0∞ := |
| ENNReal.ofReal (Real.sqrt (∫x in FiniteTransfer.realChartDomain d.physicalDomain \ |
| (fun x : ℝ × ℝ => π (Complex.equivRealProdCLM.symm x : Sphere)) ⁻¹' B.penalty, |
| FiniteTransfer.gradientEnergy B.coordinate.physical |
| (fun z => FiniteTransfer.sphericalMetricFactor z • 1) x)) + |
| countingSquareNorm B.weight |
|
|
| theorem finite_compatibility |
| (B : CompatibleLevelBarrier d π previous flags EStar regions f W₁) |
| (n : ℕ) (labels : Fin n → ℕ) : |
| (volume.prod (Measure.pi (fun _ : Fin n => (volume : Measure UnitAddCircle)))) |
| (freeComponentFailure (d.dust \ B.exclusions) |
| (circleBandGraph d.quotientDomain B.coordinate.quotient) |
| (fun i => previous.graph i) labels) = 0 := by |
| cases n with |
| | zero => simp only [freeComponentFailure_empty_labels, measure_empty] |
| | succ n => exact B.compatible n labels |
|
|
| theorem controlledSize_le_of_partial_bounds |
| (B : CompatibleLevelBarrier d π previous flags EStar regions f W₁) |
| {A : ℝ≥0∞} (hA : A ≠ ∞) |
| (hbound : ∀i : Fin 2, eLpNorm (fun x => (fderiv ℝ B.coordinate.physical x) |
| (if i = 0 then (1, 0) else (0, 1))) 2 |
| (volume.restrict (FiniteTransfer.realChartDomain d.physicalDomain \ |
| (fun x : ℝ × ℝ => π (Complex.equivRealProdCLM.symm x : Sphere)) ⁻¹' B.penalty)) ≤ A) : |
| B.controlledSize ≤ 2 * A + countingSquareNorm B.weight := by |
| refine add_le_add ?_ le_rfl |
| have hb := physical_coordinate_gradient_size_le d π B.coordinate |
| (fun _ hx => hx.1) hA hbound |
| have h := ENNReal.ofReal_le_ofReal hb |
| simpa only [ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 2), |
| ENNReal.ofReal_ofNat, ENNReal.ofReal_toReal hA] using h |
|
|
| end CompatibleLevelBarrier |
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
