| import Mathlib |
| import OAI.Analysis.Conductivity.Variational.DivSymbolTwo |
| import OAI.Analysis.Conductivity.Scalarization.SmoothScalarAlgebra |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory |
| open scoped ENNReal |
| namespace ScalarConductivity |
| open Matrix |
|
|
| def airySymbol (ξ : Fin 3 → ℝ) : Fin 5 → ℝ := |
| ![ξ 1 ^ 2, -(ξ 0 * ξ 1), ξ 0 ^ 2, 0, 0] |
|
|
| def transverseSymbol (ξ : Fin 3 → ℝ) (p q r : ℝ) : Fin 5 → ℝ := |
| ![ξ 2 * p, ξ 2 * q, ξ 2 * r, |
| -(ξ 0 * p) - ξ 1 * q, -(ξ 0 * q) - ξ 1 * r] |
|
|
| theorem airy_transverse_span {ξ : Fin 3 → ℝ} (hξ : ξ ≠ 0) |
| {F : Fin 5 → ℝ} (hF : divSymbolTwo ξ *ᵥ F = 0) : |
| ∃ α p q r : ℝ, F = α • airySymbol ξ + transverseSymbol ξ p q r := by |
| have h0 := congr_fun hF 0 |
| have h1 := congr_fun hF 1 |
| simp [divSymbolTwo, Matrix.mulVec, dotProduct, Fin.sum_univ_succ] at h0 h1 |
| by_cases hz : ξ 2 = 0 |
| · by_cases hx : ξ 0 = 0 |
| · have hy : ξ 1 ≠ 0 := by |
| intro hy |
| apply hξ |
| funext i |
| fin_cases i <;> simp [hx, hy, hz] |
| have hb : F 1 = 0 := by simpa [hx, hz, hy] using h0 |
| have hc : F 2 = 0 := by simpa [hx, hz, hy] using h1 |
| refine ⟨F 0 / ξ 1 ^ 2, 0, -F 3 / ξ 1, -F 4 / ξ 1, ?_⟩ |
| funext i |
| fin_cases i <;> simp [airySymbol, transverseSymbol, hx, hz, hb, hc, hy] <;> field_simp |
| · have hb : ξ 0 * F 1 = -(ξ 1 * F 2) := by |
| simp only [hz, zero_mul, add_zero] at h1 |
| linarith |
| have ha : ξ 0 ^ 2 * F 0 = ξ 1 ^ 2 * F 2 := by |
| simp only [hz, zero_mul, add_zero] at h0 |
| nlinarith [congrArg (fun t => ξ 1 * t) hb, congrArg (fun t => ξ 0 * t) h0] |
| refine ⟨F 2 / ξ 0 ^ 2, -F 3 / ξ 0 + ξ 1 * F 4 / ξ 0 ^ 2, |
| -F 4 / ξ 0, 0, ?_⟩ |
| funext i |
| fin_cases i <;> simp [airySymbol, transverseSymbol, hz] <;> |
| field_simp [hx] <;> nlinarith [hb, ha] |
| · refine ⟨0, F 0 / ξ 2, F 1 / ξ 2, F 2 / ξ 2, ?_⟩ |
| funext i |
| fin_cases i <;> simp [airySymbol, transverseSymbol] <;> |
| field_simp <;> nlinarith [h0, h1] |
|
|
| theorem tsupport_smoothDirection_subset {E : Type*} [NormedAddCommGroup E] |
| [NormedSpace ℝ E] (v : E) (f : SmoothScalar E) : |
| tsupport (smoothDirection v f).val ⊆ tsupport f.val := by |
| exact (tsupport_comp_subset (g := fun L : E →L[ℝ] ℝ => L v) (by simp) |
| (fderiv ℝ f.val)).trans (tsupport_fderiv_subset ℝ) |
|
|
| theorem compactSupport_smoothDirection {E : Type*} [NormedAddCommGroup E] |
| [NormedSpace ℝ E] (v : E) {f : SmoothScalar E} (hf : HasCompactSupport f.val) : |
| HasCompactSupport (smoothDirection v f).val := |
| hf.of_isClosed_subset isClosed_closure (tsupport_smoothDirection_subset v f) |
|
|
| end ScalarConductivity |
|
|
| end |
|
|
| end OAI |
|
|
