File size: 3,751 Bytes
785c727 | 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 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory Set
open scoped ENNReal
namespace CubicTorus
abbrev Cover := EuclideanSpace ℝ (Fin 3)
abbrev Torus := PiLp 2 (fun _ : Fin 3 => AddCircle (1 : ℝ))
/-- The quotient R³ → R³/Z³ in orthonormal coordinates. -/
def quotientMap (x : Cover) : Torus :=
WithLp.toLp 2 (fun i => (x i : AddCircle (1 : ℝ)))
/-- One unit-volume fundamental domain; its boundary is immaterial to integration. -/
def fundamentalCube : Set Cover := {x | ∀ i, 0 ≤ x i ∧ x i < 1}
/-- The Riemannian volume measure of the unit flat torus. -/
def torusVolume : Measure Torus :=
Measure.map quotientMap (volume.restrict fundamentalCube)
/-- An integer lattice vector in the Euclidean cover. -/
def latticeVector (n : Fin 3 → ℤ) : Cover := WithLp.toLp 2 (fun i => (n i : ℝ))
/-- The smooth vector fields on the flat torus with pointwise length at most one,
expressed as Z³-periodic fields on its Euclidean cover. The explicit ℕ∞ index
is C^∞; top of WithTop ℕ∞ would instead mean analytic in this Mathlib. -/
def AdmissibleField (F : Cover → Cover) : Prop :=
ContDiff ℝ (⊤ : ℕ∞) F ∧
(∀ (n : Fin 3 → ℤ) (x : Cover), F (x + latticeVector n) = F x) ∧
∀ x, ‖F x‖ ≤ 1
/-- Euclidean divergence in the orthonormal coordinate frame. -/
def divergence (F : Cover → Cover) (x : Cover) : ℝ :=
∑ i : Fin 3, (fderiv ℝ F x (EuclideanSpace.single i 1)) i
/-- De Giorgi perimeter, allowed to be infinite; no regularity of E is imposed. -/
def perimeter (E : Set Torus) : ℝ≥0∞ :=
⨆ (F : Cover → Cover) (_ : AdmissibleField F),
ENNReal.ofReal (∫ x in fundamentalCube ∩ quotientMap ⁻¹' E, divergence F x)
/-- Finite-perimeter measurable regions, including all null-set modifications. -/
def FinitePerimeter (E : Set Torus) : Prop :=
NullMeasurableSet E torusVolume ∧ perimeter E < ⊤
/-- A global perimeter minimizer under the prescribed volume constraint. -/
def IsMinimizerAt (V : ℝ) (E : Set Torus) : Prop :=
FinitePerimeter E ∧ torusVolume E = ENNReal.ofReal V ∧
∀ F : Set Torus, FinitePerimeter F → torusVolume F = ENNReal.ofReal V →
perimeter E ≤ perimeter F
/-- The candidate unit-torus isoperimetric profile. -/
def candidateProfile (V : ℝ) : ℝ :=
let v := min V (1 - V)
min ((36 * Real.pi) ^ (1 / 3 : ℝ) * v ^ (2 / 3 : ℝ))
(min (2 * Real.sqrt (Real.pi * v)) 2)
/-- Canonical centered ball. Its radius on the listed branch is below 1/2. -/
def canonicalBall (V : ℝ) : Set Torus :=
Metric.ball 0 ((3 * V / (4 * Real.pi)) ^ (1 / 3 : ℝ))
/-- Canonical tube around the shortest closed geodesic in coordinate direction 0. -/
def canonicalTube (V : ℝ) : Set Torus :=
{x | ‖x 1‖ ^ 2 + ‖x 2‖ ^ 2 < (Real.sqrt (V / Real.pi)) ^ 2}
/-- Canonical slab of width V between two parallel coordinate tori. -/
def canonicalSlab (V : ℝ) : Set Torus := {x | ‖x 0‖ < V / 2}
/-- Equality of regions up to ambient flat isometry and a null set. -/
def CongruentAE (E F : Set Torus) : Prop :=
∃ φ : Torus ≃ᵢ Torus, E =ᵐ[torusVolume] (φ '' F)
/-- Exactly the closed phase ranges, so adjacent types are BOTH included at ties. -/
def SmallStandardRegion (V : ℝ) (E : Set Torus) : Prop :=
(V ≤ 4 * Real.pi / 81 ∧ CongruentAE E (canonicalBall V)) ∨
(4 * Real.pi / 81 ≤ V ∧ V ≤ 1 / Real.pi ∧ CongruentAE E (canonicalTube V)) ∨
(1 / Real.pi ≤ V ∧ CongruentAE E (canonicalSlab V))
/-- The equality class at every interior volume, including complementation. -/
def StandardRegionAt (V : ℝ) (E : Set Torus) : Prop :=
if V ≤ 1 / 2 then SmallStandardRegion V E
else SmallStandardRegion (1 - V) Eᶜ
end CubicTorus
end
end OAI
|