File size: 3,110 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 | import Mathlib
import OAI.Analysis.Conductivity.Sobolev.VoltageH1Completion
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open MeasureTheory Set Filter Topology TopologicalSpace Matrix
open scoped ENNReal Matrix.Norms.Elementwise
lemma uniform_matrix_laminate_coverage {c C : ℝ} (hc : 0 < c) (hcC : c ≤ C)
(A : Symmetric3)
(hA : ∀ v : Coord3, c * (v ⬝ᵥ v) ≤ v ⬝ᵥ (A.val *ᵥ v) ∧
v ⬝ᵥ (A.val *ᵥ v) ≤ C * (v ⬝ᵥ v)) :
A ∈ matrixFiniteLaminate (laminateLower c C) (laminateUpper C) := by
obtain ⟨Q,hQ,α,he⟩ := exists_orthogonal_diagonalization A.property
refine ⟨Q,hQ,α,(uniform_laminate_coverage hc hcC).2.2 α ?_,he⟩
intro i
let v : Coord3 := Pi.single i 1
have hnorm : (Q *ᵥ v) ⬝ᵥ (Q *ᵥ v) = 1 := by
rw [orthogonal_dotProduct hQ]
simp [v,dotProduct,Pi.single_apply]
have hback : Qᵀ *ᵥ (Q *ᵥ v) = v := by
rw [mulVec_mulVec,hQ.1,one_mulVec]
have hval : (Q *ᵥ v) ⬝ᵥ (A.val *ᵥ (Q *ᵥ v)) = α i := by
rw [he,conjugateDiagonal_quadratic,hback]
simp [v,Pi.single_apply]
simpa only [hnorm,hval,mul_one] using hA (Q *ᵥ v)
theorem finite_field_scalarization
(μ : Measure Coord3) [μ.IsAddHaarMeasure] [Measure.InnerRegularCompactLTTop μ]
{c C : ℝ} (hc : 0 < c) (hcC : c ≤ C)
{U : Set Coord3} (hUb : Bornology.IsBounded U) (hUm : MeasurableSet U)
[IsFiniteMeasure (μ.restrict U)] [(μ.restrict U).WeaklyRegular]
[SeparableSpace (Lp FieldVector 2 (μ.restrict U))]
(u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) (hAm : Measurable A)
(hu : MemLp u 2 (μ.restrict U))
(hE : MemLp (voltageGradient u) 2 (μ.restrict U))
(hF : MemLp (voltageFlux u A) 2 (μ.restrict U))
(huH1 : (hu.toLp _,hE.toLp _) ∈ voltageH1Jets μ U)
(hint : SmoothFluxIntegrable μ U (conductivityFlux u A))
(hdiv : ∀ j (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → HasCompactSupport ψ →
tsupport ψ ⊆ U → (∫ x, fderiv ℝ ψ x ((conductivityFlux u A x).col j) ∂μ) = 0)
(hreg : μ (U \ regularRegion u A U) = 0)
(hbound : ∀ᵐ x ∂μ, x ∈ U → ∀ v : Coord3,
c * (v ⬝ᵥ v) ≤ v ⬝ᵥ ((A x).val *ᵥ v) ∧
v ⬝ᵥ ((A x).val *ᵥ v) ≤ C * (v ⬝ᵥ v)) :
∃ (z : voltageH1Jets μ U) (F : Lp FieldVector 2 (μ.restrict U)) (s : Coord3 → ℝ),
Measurable s ∧ MemLp s ∞ (μ.restrict U) ∧
z.val-(hu.toLp _,hE.toLp _) ∈ zeroVoltageJets μ U ∧
(∀ᵐ x ∂μ.restrict U, s x ∈ Icc (laminateLower c C) (laminateUpper C) ∧
F x = s x • (z.val.2 x)) ∧
(∀ W : voltageH1Jets μ U, inner ℝ W.val.2 F = inner ℝ W.val.2 (hF.toLp _)) ∧
(∀ W : zeroVoltageJets μ U, inner ℝ W.val.2 F = 0) := by
apply exact_scalarization_H1 μ (uniform_laminate_coverage hc hcC).1
(uniform_laminate_coverage hc hcC).2.1 hUb hUm u A hAm hu hE hF huH1 hint hdiv hreg
filter_upwards [hbound] with x hx hxU
exact uniform_matrix_laminate_coverage hc hcC (A x) (hx hxU)
end ScalarConductivity
end
end
end OAI
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