File size: 1,914 Bytes
9eeb9c7 | 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 | import Mathlib
namespace OAI
noncomputable section
open Set MeasureTheory
open scoped ContDiff
namespace StrictHotSpots
abbrev Plane := EuclideanSpace ℝ (Fin 2)
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
def AdmissibleDomain (Ω : Set Plane) : Prop :=
Ω.Nonempty ∧ IsOpen Ω ∧ Bornology.IsBounded Ω ∧
IsSimplyConnected Ω ∧ SmoothBoundary Ω
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)
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)}
def firstPositiveNeumannValue (Ω : Set Plane) : ℝ := sInf (rayleighValues Ω)
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)
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)
end StrictHotSpots
end
end OAI
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