File size: 4,118 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 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | 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
|