File size: 3,466 Bytes
dae740b | 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 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory Set Filter
open scoped ENNReal Topology ContDiff
namespace Bernoulli
open scoped _root_.Bernoulli
abbrev Space (d : ℕ) := EuclideanSpace ℝ (Fin d)
/-- Compactly supported smooth real tests in the open set. -/
def Test {d : ℕ} (D : Set (Space d)) (φ : Space d → ℝ) : Prop :=
ContDiff ℝ ∞ φ ∧ HasCompactSupport φ ∧ tsupport φ ⊆ D
/-- Distributional gradient, tested in every constant direction. -/
def WeakGradient {d : ℕ} (D : Set (Space d))
(u : Space d → ℝ) (G : Space d → Space d) : Prop :=
∀ φ, Test D φ → ∀ h : Space d,
(∫ x in D, u x * (fderiv ℝ φ x) h) =
-(∫ x in D, inner ℝ (G x) h * φ x)
/-- H¹(D), with its weak gradient. -/
def H1 {d : ℕ} (D : Set (Space d))
(u : Space d → ℝ) (G : Space d → Space d) : Prop :=
MemLp u 2 (volume.restrict D) ∧
MemLp G 2 (volume.restrict D) ∧ WeakGradient D u G
/-- H¹₀(D) is H¹-closure of C∞ compactly supported functions.
Both L² errors tend to zero. This does not restrict competitors to smooth ones. -/
def H10 {d : ℕ} (D : Set (Space d))
(u : Space d → ℝ) (G : Space d → Space d) : Prop :=
H1 D u G ∧ ∀ ε : ℝ, 0 < ε → ∃ φ : Space d → ℝ,
Test D φ ∧
eLpNorm (fun x => u x - φ x) 2 (volume.restrict D) < ENNReal.ofReal ε ∧
eLpNorm (fun x => G x - gradient φ x) 2 (volume.restrict D) < ENNReal.ofReal ε
/-- Ballwise H¹_loc, on an open set (balls have compact closure inside D). -/
def H1Loc {d : ℕ} (D : Set (Space d))
(u : Space d → ℝ) (G : Space d → Space d) : Prop :=
∀ c : Space d, ∀ r : ℝ, 0 < r → Metric.closedBall c r ⊆ D →
H1 (Metric.ball c r) u G
/-- The exact volume-one Bernoulli energy, not just Dirichlet energy. -/
def energy {d : ℕ} (D : Set (Space d))
(u : Space d → ℝ) (G : Space d → Space d) : ℝ :=
∫ x in D, ‖G x‖ ^ 2 + if 0 < u x then (1 : ℝ) else 0
/-- Full ballwise local comparison. Competitor and gradient are arbitrary
Sobolev representatives on the ball, with zero-trace difference. -/
def LocalMinimizer {d : ℕ} (D : Set (Space d)) (u : Space d → ℝ) : Prop :=
IsOpen D ∧ ∃ G : Space d → Space d,
H1Loc D u G ∧ (∀ᵐ x ∂volume.restrict D, 0 ≤ u x) ∧
∀ c : Space d, ∀ r : ℝ, 0 < r → Metric.closedBall c r ⊆ D →
∀ (v : Space d → ℝ) (V : Space d → Space d),
H1 (Metric.ball c r) v V →
(∀ᵐ x ∂volume.restrict (Metric.ball c r), 0 ≤ v x) →
H10 (Metric.ball c r) (fun x => v x - u x) (fun x => V x - G x) →
energy (Metric.ball c r) u G ≤ energy (Metric.ball c r) v V
/-- Nonnegative global minimizer, with all centers and all positive radii. -/
def GlobalMinimizer {d : ℕ} (u : Space d → ℝ) : Prop :=
LocalMinimizer Set.univ u
def OneHomogeneous {d : ℕ} (u : Space d → ℝ) : Prop :=
∀ r : ℝ, 0 < r → ∀ᵐ x ∂volume, u (r • x) = r * u x
def Nonzero {d : ℕ} (u : Space d → ℝ) : Prop :=
¬ (u =ᵐ[volume] fun _ => 0)
/-- Flatness is a.e. equality of H¹ representatives. -/
def Flat {d : ℕ} (u : Space d → ℝ) : Prop :=
∃ e : Space d, ‖e‖ = 1 ∧ ∀ᵐ x ∂volume, u x = max (inner ℝ x e) 0
theorem nonflat_in_seven : ∃ u : Space 7 → ℝ,
GlobalMinimizer u ∧ Nonzero u ∧ OneHomogeneous u ∧ ¬ Flat u := by
sorry
end Bernoulli
end
end OAI
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