| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory |
| open scoped ENNReal |
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| namespace ScalarConductivity |
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| abbrev R3 := EuclideanSpace ℝ (Fin 3) |
| abbrev JetFiber := EuclideanSpace ℝ (Fin 4) |
| def ball : Set R3 := Metric.ball 0 3 |
| def ballMeasure : Measure R3 := volume.restrict ball |
| abbrev JetSpace := Lp JetFiber 2 ballMeasure |
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| def smoothJet (f : R3 → ℝ) (x : R3) : JetFiber := |
| WithLp.toLp 2 (Fin.cases (f x) (fun i => gradient f x i)) |
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| def smoothJets : Set JetSpace := |
| { z | ∃ (f : R3 → ℝ) (_ : ContDiff ℝ (↑(⊤ : ℕ∞)) f) |
| (hf : MemLp (smoothJet f) 2 ballMeasure), z = hf.toLp (smoothJet f) } |
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| def compactSmoothJets : Set JetSpace := |
| { z | ∃ (f : R3 → ℝ) (_ : ContDiff ℝ (↑(⊤ : ℕ∞)) f) |
| (_ : HasCompactSupport f) (_ : tsupport f ⊆ ball) |
| (hf : MemLp (smoothJet f) 2 ballMeasure), z = hf.toLp (smoothJet f) } |
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| def H1Space : Submodule ℝ JetSpace := |
| (Submodule.span ℝ smoothJets).topologicalClosure |
| abbrev H1 := H1Space |
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| def zeroTraceAmbient : Submodule ℝ JetSpace := |
| (Submodule.span ℝ compactSmoothJets).topologicalClosure |
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| def H10 : Submodule ℝ H1 := zeroTraceAmbient.comap H1Space.subtype |
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| instance H10_isClosed : IsClosed (H10 : Set H1) := by |
| exact (Submodule.isClosed_topologicalClosure |
| (Submodule.span ℝ compactSmoothJets)).preimage continuous_subtype_val |
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| abbrev TraceSpace := H1 ⧸ H10 |
| abbrev TraceDual := TraceSpace →L[ℝ] ℝ |
| abbrev DNOperator := TraceSpace →L[ℝ] TraceDual |
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| def trace (u : H1) : TraceSpace := H10.mkQ u |
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| def weakGradient (u : H1) (x : R3) : R3 := |
| WithLp.toLp 2 (fun i => (u.val x) i.succ) |
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| def energy (γ : R3 → ℝ) (u v : H1) : ℝ := |
| ∫ x, γ x * inner ℝ (weakGradient u x) (weakGradient v x) ∂ballMeasure |
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| def Harmonic (γ : R3 → ℝ) (u : H1) : Prop := |
| ∀ h : H1, h ∈ H10 → energy γ u h = 0 |
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| def IsDirichletToNeumann (γ : R3 → ℝ) (Λ : DNOperator) : Prop := |
| ∀ f : TraceSpace, ∃ u : H1, |
| trace u = f ∧ Harmonic γ u ∧ |
| (∀ u' : H1, trace u' = f → Harmonic γ u' → u' = u) ∧ |
| ∀ v : H1, Λ f (trace v) = energy γ u v |
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| def EqualOneNearBoundary (γ : R3 → ℝ) : Prop := |
| ∃ r : ℝ, 0 < r ∧ r < 3 ∧ ∀ᵐ x ∂ballMeasure, r < ‖x‖ → γ x = 1 |
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| def MainStatement : Prop := |
| ∃ (γ₀ γ₁ : R3 → ℝ) (c C : ℝ) (Λ : DNOperator), |
| 0 < c ∧ c < C ∧ |
| Measurable γ₀ ∧ Measurable γ₁ ∧ |
| MemLp γ₀ ∞ ballMeasure ∧ MemLp γ₁ ∞ ballMeasure ∧ |
| (∀ᵐ x ∂ballMeasure, c ≤ γ₀ x ∧ γ₀ x ≤ C) ∧ |
| (∀ᵐ x ∂ballMeasure, c ≤ γ₁ x ∧ γ₁ x ≤ C) ∧ |
| EqualOneNearBoundary γ₀ ∧ EqualOneNearBoundary γ₁ ∧ |
| 0 < ballMeasure {x | γ₀ x ≠ γ₁ x} ∧ |
| IsDirichletToNeumann γ₀ Λ ∧ IsDirichletToNeumann γ₁ Λ |
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| end ScalarConductivity |
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| end |
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| end OAI |
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