File size: 2,830 Bytes
a458723 | 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 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory
open scoped ENNReal
namespace ScalarConductivity
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
def smoothJet (f : R3 → ℝ) (x : R3) : JetFiber :=
WithLp.toLp 2 (Fin.cases (f x) (fun i => gradient f x i))
def smoothJets : Set JetSpace :=
{ z | ∃ (f : R3 → ℝ) (_ : ContDiff ℝ (↑(⊤ : ℕ∞)) f)
(hf : MemLp (smoothJet f) 2 ballMeasure), z = hf.toLp (smoothJet f) }
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) }
def H1Space : Submodule ℝ JetSpace :=
(Submodule.span ℝ smoothJets).topologicalClosure
abbrev H1 := H1Space
def zeroTraceAmbient : Submodule ℝ JetSpace :=
(Submodule.span ℝ compactSmoothJets).topologicalClosure
def H10 : Submodule ℝ H1 := zeroTraceAmbient.comap H1Space.subtype
instance H10_isClosed : IsClosed (H10 : Set H1) := by
exact (Submodule.isClosed_topologicalClosure
(Submodule.span ℝ compactSmoothJets)).preimage continuous_subtype_val
abbrev TraceSpace := H1 ⧸ H10
abbrev TraceDual := TraceSpace →L[ℝ] ℝ
abbrev DNOperator := TraceSpace →L[ℝ] TraceDual
def trace (u : H1) : TraceSpace := H10.mkQ u
def weakGradient (u : H1) (x : R3) : R3 :=
WithLp.toLp 2 (fun i => (u.val x) i.succ)
def energy (γ : R3 → ℝ) (u v : H1) : ℝ :=
∫ x, γ x * inner ℝ (weakGradient u x) (weakGradient v x) ∂ballMeasure
def Harmonic (γ : R3 → ℝ) (u : H1) : Prop :=
∀ h : H1, h ∈ H10 → energy γ u h = 0
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
def EqualOneNearBoundary (γ : R3 → ℝ) : Prop :=
∃ r : ℝ, 0 < r ∧ r < 3 ∧ ∀ᵐ x ∂ballMeasure, r < ‖x‖ → γ x = 1
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
end ScalarConductivity
end
end OAI
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