| 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 |
|
|
