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159357f | 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 | import Mathlib
import OAI.Analysis.Conductivity.Flux.FluxPotentialSymbol
namespace OAI
noncomputable section
open MeasureTheory
open scoped ENNReal
namespace ScalarConductivity
open Matrix
def divSymbolTwo (ξ : Fin 3 → ℝ) : Matrix (Fin 2) (Fin 5) ℝ :=
!![ξ 0, ξ 1, 0, ξ 2, 0; 0, ξ 0, ξ 1, 0, ξ 2]
def divAdjointTwo (ξ : Fin 3 → ℝ) : Matrix (Fin 5) (Fin 2) ℝ :=
!![ξ 0, 0; ξ 1 / 2, ξ 0 / 2; 0, ξ 1; ξ 2, 0; 0, ξ 2]
theorem divSymbolTwo_gram_det (ξ : Fin 3 → ℝ) :
(divSymbolTwo ξ * divAdjointTwo ξ).det =
(ξ 0 ^ 2 + ξ 1 ^ 2 + ξ 2 ^ 2) *
((ξ 0 ^ 2 + ξ 1 ^ 2) / 2 + ξ 2 ^ 2) := by
simp [divSymbolTwo, divAdjointTwo, Matrix.det_fin_two]
ring
theorem divSymbolTwo_gram_det_pos {ξ : Fin 3 → ℝ} (hξ : ξ ≠ 0) :
0 < (divSymbolTwo ξ * divAdjointTwo ξ).det := by
have hp : 0 < ξ 0 ^ 2 + ξ 1 ^ 2 + ξ 2 ^ 2 := by
by_contra hn
have h0 : ξ 0 = 0 := by nlinarith [sq_nonneg (ξ 1), sq_nonneg (ξ 2)]
have h1 : ξ 1 = 0 := by nlinarith [sq_nonneg (ξ 0), sq_nonneg (ξ 2)]
have h2 : ξ 2 = 0 := by nlinarith [sq_nonneg (ξ 0), sq_nonneg (ξ 1)]
apply hξ
funext i
fin_cases i <;> assumption
rw [divSymbolTwo_gram_det]
exact mul_pos hp (by nlinarith [sq_nonneg (ξ 0), sq_nonneg (ξ 1), sq_nonneg (ξ 2)])
theorem divSymbolTwo_surjective {ξ : Fin 3 → ℝ} (hξ : ξ ≠ 0) :
Function.Surjective (divSymbolTwo ξ).mulVec := by
intro v
refine ⟨divAdjointTwo ξ *ᵥ ((divSymbolTwo ξ * divAdjointTwo ξ)⁻¹ *ᵥ v), ?_⟩
rw [Matrix.mulVec_mulVec, Matrix.mulVec_mulVec,
Matrix.mul_nonsing_inv _ (isUnit_iff_ne_zero.mpr (ne_of_gt (divSymbolTwo_gram_det_pos hξ))),
Matrix.one_mulVec]
theorem twoField_potential_range {ξ : Fin 3 → ℝ} (hξ : ξ ≠ 0) :
Set.range (fluxPotentialSymbol (divSymbolTwo ξ) (divAdjointTwo ξ)).mulVec =
{v | divSymbolTwo ξ *ᵥ v = 0} :=
fluxPotentialSymbol_range _ _ (ne_of_gt (divSymbolTwo_gram_det_pos hξ))
end ScalarConductivity
end
end OAI
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