| import Mathlib |
| import OAI.Analysis.Conductivity.Variational.SynchronizedChartWaves |
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| namespace OAI |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Matrix |
| open scoped Matrix.Norms.Elementwise |
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| lemma potential_component_fderiv {u : Coord3 → Fin 2 → ℝ} |
| (hu : Differentiable ℝ u) (j : Fin 2) (x v : Coord3) : |
| fderiv ℝ (fun y => u y j) x v=fderiv ℝ u x v j := by |
| rw [(hasFDerivAt_pi'.mp (hu x).hasFDerivAt j).fderiv] |
| rfl |
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| lemma symmetric_potential_pairing (D : Coord3 →L[ℝ] (Fin 2 → ℝ)) |
| (B : Mat3) (hB : B.IsSymm) : |
| D ((B*gradientColumns D).col 0) 1=D ((B*gradientColumns D).col 1) 0 := by |
| rw [gradientColumns_pairing,gradientColumns_pairing] |
| have hs : ((gradientColumns D)ᵀ*(B*gradientColumns D)).IsSymm := by |
| unfold Matrix.IsSymm |
| rw [transpose_mul,transpose_mul,transpose_transpose,hB] |
| exact Matrix.mul_assoc _ _ _ |
| exact congrArg (fun M : Matrix (Fin 2) (Fin 2) ℝ => M 0 1) hs |
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| theorem compact_symmetric_source_moments |
| (u : Coord3 → Fin 2 → ℝ) (hu : ContDiff ℝ (↑(⊤ : ℕ∞)) u) |
| (H : Coord3 → Mat3) (hH : ∀ x,(H x).IsSymm) |
| (r : Fin 2 → Coord3 → ℝ) |
| (hr : ∀ j,ContDiff ℝ (↑(⊤ : ℕ∞)) (r j)) |
| (hsr : ∀ j,HasCompactSupport (r j)) |
| (hw : ∀ j (ψ : Coord3 → ℝ),ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → |
| (∫ x,fderiv ℝ ψ x ((H x*gradientColumns (fderiv ℝ u x)).col j))= |
| -(∫ x,ψ x*r j x)) : |
| (∫ x,r 0 x)=0 ∧ (∫ x,r 1 x)=0 ∧ |
| (∫ x,u x 1*r 0 x-u x 0*r 1 x)=0 := by |
| have hz (j : Fin 2) : (∫ x,r j x)=0 := by |
| have hh := hw j (fun _ => 1) contDiff_const |
| simpa using hh |
| have huj (j : Fin 2) : ContDiff ℝ (↑(⊤ : ℕ∞)) (fun x => u x j) := |
| (contDiff_pi.mp hu) j |
| have he : (∫ x,u x 1*r 0 x)=(∫ x,u x 0*r 1 x) := by |
| have hw₁ := hw 0 (fun x => u x 1) (huj 1) |
| have hw₂ := hw 1 (fun x => u x 0) (huj 0) |
| have hp : (fun x => fderiv ℝ (fun y => u y 1) x ((H x*gradientColumns (fderiv ℝ u x)).col 0))= |
| (fun x => fderiv ℝ (fun y => u y 0) x ((H x*gradientColumns (fderiv ℝ u x)).col 1)) := by |
| funext x |
| rw [potential_component_fderiv (hu.differentiable (by simp)), |
| potential_component_fderiv (hu.differentiable (by simp))] |
| exact symmetric_potential_pairing _ _ (hH x) |
| rw [hp] at hw₁ |
| linarith |
| refine ⟨hz 0,hz 1,?_⟩ |
| have hi₁ : Integrable (fun x => u x 1*r 0 x) := |
| ((huj 1).continuous.mul (hr 0).continuous).integrable_of_hasCompactSupport (hsr 0).mul_left |
| have hi₂ : Integrable (fun x => u x 0*r 1 x) := |
| ((huj 0).continuous.mul (hr 1).continuous).integrable_of_hasCompactSupport (hsr 1).mul_left |
| rw [integral_sub hi₁ hi₂,he,sub_self] |
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| end ScalarConductivity |
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| end |
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| end OAI |
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