| import Mathlib |
| import OAI.Analysis.Conductivity.Model |
|
|
| namespace OAI |
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|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set Matrix Filter Topology |
| open scoped Matrix.Norms.Elementwise |
| |
| variable {ι P : Type*} [Fintype ι] [DecidableEq ι] [TopologicalSpace P] |
|
|
| omit [DecidableEq ι] in |
| lemma matrix_pi_mulVec_bound (A : Matrix ι ι ℝ) (b : ι → ℝ) (i : ι) : |
| ‖(A*ᵥb) i‖≤(Fintype.card ι:ℝ)*‖A‖*‖b‖ := by |
| change ‖∑ j,A i j*b j‖≤_ |
| calc |
| _≤∑ j,‖A i j*b j‖ := norm_sum_le _ _ |
| _≤∑ j,‖A‖*‖b‖ := by |
| apply Finset.sum_le_sum |
| intro j _ |
| rw [norm_mul] |
| exact mul_le_mul ((norm_le_pi_norm (A i) j).trans (norm_le_pi_norm A i)) |
| (norm_le_pi_norm b j) (norm_nonneg _) (norm_nonneg _) |
| _=_ := by simp [mul_assoc] |
| |
| theorem matrix_inverse_locally_bounded {M : P → Matrix ι ι ℝ} {p₀ : P} |
| (hM : ContinuousAt M p₀) (h₀ : IsUnit (M p₀)) : |
| ∃ C : ℝ,0<C ∧ ∀ᶠ p in 𝓝 p₀, |
| IsUnit (M p) ∧ (∀ b : ι → ℝ, |
| M p*ᵥ((M p)⁻¹*ᵥb)=b ∧ ∀ i,‖((M p)⁻¹*ᵥb) i‖≤C*‖b‖) := by |
| have hd₀ : (M p₀).det≠0 := isUnit_iff_ne_zero.mp ((isUnit_iff_isUnit_det _).mp h₀) |
| have hdi : ContinuousAt Ring.inverse (M p₀).det := by |
| simpa only [Ring.inverse_eq_inv'] using (continuousAt_inv₀ hd₀) |
| have hi : ContinuousAt (fun p => (M p)⁻¹) p₀ := |
| (continuousAt_matrix_inv _ hdi).comp hM |
| have hn : ∀ᶠ p in 𝓝 p₀,‖(M p)⁻¹‖<‖(M p₀)⁻¹‖+1 := |
| hi.norm.eventually (gt_mem_nhds (by linarith)) |
| have hd : ∀ᶠ p in 𝓝 p₀,(M p).det≠0 := |
| ((continuous_id.matrix_det).continuousAt.comp hM).tendsto.eventually (isOpen_ne.mem_nhds hd₀) |
| let C := ((Fintype.card ι:ℝ)+1)*(‖(M p₀)⁻¹‖+1) |
| have hC : 0<C := by dsimp [C]; positivity |
| refine ⟨C,hC,?_⟩ |
| filter_upwards [hn,hd] with p hp hpd |
| have hu : IsUnit (M p).det := isUnit_iff_ne_zero.mpr hpd |
| refine ⟨(isUnit_iff_isUnit_det _).mpr hu,?_⟩ |
| intro b |
| refine ⟨?_,?_⟩ |
| · rw [Matrix.mulVec_mulVec,Matrix.mul_nonsing_inv _ hu,Matrix.one_mulVec] |
| · intro i |
| apply (matrix_pi_mulVec_bound _ b i).trans |
| apply mul_le_mul_of_nonneg_right _ (norm_nonneg b) |
| have hn₀ : 0≤‖(M p₀)⁻¹‖+1 := by positivity |
| dsimp only [C] |
| have hh := mul_le_mul_of_nonneg_left hp.le (show (0:ℝ)≤Fintype.card ι by positivity) |
| nlinarith |
| |
| end ScalarConductivity |
| |
| end |
| |
| end OAI |
| |
