| 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 |
|
|
