| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open Set MeasureTheory |
| open scoped ContDiff |
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| namespace StrictHotSpots |
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| abbrev Plane := EuclideanSpace ℝ (Fin 2) |
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| def SmoothBoundary (Ω : Set Plane) : Prop := |
| ∀ p ∈ frontier Ω, ∃ (U : Set Plane) (ρ : Plane → ℝ), |
| IsOpen U ∧ p ∈ U ∧ ContDiffOn ℝ ∞ ρ U ∧ |
| fderiv ℝ ρ p ≠ 0 ∧ ρ p = 0 ∧ |
| Ω ∩ U = {x | ρ x < 0} ∩ U |
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| def AdmissibleDomain (Ω : Set Plane) : Prop := |
| Ω.Nonempty ∧ IsOpen Ω ∧ Bornology.IsBounded Ω ∧ |
| IsSimplyConnected Ω ∧ SmoothBoundary Ω |
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| def HasH1Gradient (Ω : Set Plane) (v : Plane → ℝ) (g : Plane → Plane) : Prop := |
| MemLp v 2 (volume.restrict Ω) ∧ MemLp g 2 (volume.restrict Ω) ∧ |
| ∀ φ : Plane → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ → tsupport φ ⊆ Ω → |
| ∀ e : Plane, |
| (∫ x in Ω, v x * (fderiv ℝ φ x) e) = |
| -(∫ x in Ω, (inner ℝ (g x) e) * φ x) |
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| def rayleighValues (Ω : Set Plane) : Set ℝ := |
| {r | ∃ (v : Plane → ℝ) (g : Plane → Plane), |
| HasH1Gradient Ω v g ∧ (∫ x in Ω, v x) = 0 ∧ |
| 0 < (∫ x in Ω, (v x) ^ 2) ∧ |
| r = (∫ x in Ω, ‖g x‖ ^ 2) / (∫ x in Ω, (v x) ^ 2)} |
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| def firstPositiveNeumannValue (Ω : Set Plane) : ℝ := sInf (rayleighValues Ω) |
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| def InFirstNeumannEigenspace (Ω : Set Plane) (u : Plane → ℝ) : Prop := |
| ContDiffOn ℝ ∞ u (closure Ω) ∧ HasH1Gradient Ω u (gradient u) ∧ |
| (∫ x in Ω, u x) = 0 ∧ |
| ∀ (v : Plane → ℝ) (g : Plane → Plane), HasH1Gradient Ω v g → |
| (∫ x in Ω, inner ℝ (gradient u x) (g x)) = |
| firstPositiveNeumannValue Ω * (∫ x in Ω, u x * v x) |
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| def MainConclusion (Ω : Set Plane) (u : Plane → ℝ) : Prop := |
| (∀ x ∈ Ω, gradient u x ≠ 0) ∧ |
| ∀ x ∈ Ω, sInf (u '' frontier Ω) < u x ∧ u x < sSup (u '' frontier Ω) |
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| end StrictHotSpots |
| end |
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| end OAI |
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