| import Mathlib | |
| import OAI.Analysis.Conductivity.Variational.LocalSymmetricPiola | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace ScalarConductivity | |
| open Set Matrix MeasureTheory Filter Topology | |
| open scoped Matrix.Norms.Elementwise | |
| lemma matrix_three_product_norm_bound (A B : Mat3) : ‖A*B‖≤3*‖A‖*‖B‖ := by | |
| apply Matrix.norm_le_iff (by positivity) |>.mpr | |
| intro i j | |
| rw [Matrix.mul_apply] | |
| calc | |
| _ ≤ ∑ k : Fin 3,‖A i k*B k j‖ := norm_sum_le _ _ | |
| _ ≤ ∑ _k : Fin 3,‖A‖*‖B‖ := by | |
| apply Finset.sum_le_sum | |
| intro k hk | |
| rw [norm_mul] | |
| exact mul_le_mul (Matrix.norm_entry_le_entrywise_sup_norm A) | |
| (Matrix.norm_entry_le_entrywise_sup_norm B) (norm_nonneg _) (norm_nonneg _) | |
| _ = _ := by simp; ring | |
| theorem localPiolaTensor_linear_bound | |
| (X : OpenPartialHomeomorph Coord3 Coord3) | |
| (hX : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X X.source) | |
| (hXi : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X.symm X.target) | |
| {K : Set Coord3} (hK : IsCompact K) (hKs : K⊆X.source) : | |
| ∃ C : ℝ,0<C ∧ ∀ (B : Coord3 → Mat3),Function.support B⊆K → ∀ y, | |
| ‖localPiolaTensor X B y‖≤C*‖B (X.symm y)‖ := by | |
| classical | |
| let A : Coord3 → Mat3 := fun x => operatorMatrix (fderiv ℝ X x) | |
| let g : Coord3 → ℝ := fun x => 9*|(fderiv ℝ X x).det|⁻¹*‖A x‖*‖A x‖ | |
| have hg : ContinuousOn g X.source := by | |
| intro x hx | |
| have hd : ContDiffAt ℝ (↑(⊤ : ℕ∞)) (fderiv ℝ X) x := | |
| (hX.contDiffAt (X.open_source.mem_nhds hx)).fderiv_right (by simp) | |
| have hdM := (operatorMatrix_contDiff (m := Fin 3) (n := Fin 3)).contDiffAt.comp x hd | |
| have hd0 := local_fderiv_det_ne_zero X (hX.differentiableOn (by simp)) | |
| (hXi.differentiableOn (by simp)) hx | |
| have hdet := ((contDiff_clm_det (E := Coord3) (↑(⊤ : ℕ∞))).contDiffAt.comp x hd).abs hd0 | |
| exact (((continuousAt_const.mul ((hdet.inv (abs_ne_zero.mpr hd0)).continuousAt)).mul | |
| hdM.continuousAt.norm).mul hdM.continuousAt.norm).continuousWithinAt | |
| obtain ⟨D,hD⟩ := (hK.image_of_continuousOn (hg.mono hKs)).isBounded.exists_norm_le | |
| refine ⟨max D 1,lt_of_lt_of_le zero_lt_one (le_max_right _ _),?_⟩ | |
| intro B hsB y | |
| by_cases hy : y∈X.target | |
| · by_cases hxy : X.symm y∈K | |
| · have hbound : g (X.symm y)≤ max D 1 := | |
| (le_abs_self _).trans ((hD _ (mem_image_of_mem _ hxy)).trans (le_max_left _ _)) | |
| simp only [localPiolaTensor,ite_eq_left hy,norm_smul,Real.norm_eq_abs,abs_inv,abs_abs] | |
| calc | |
| _ ≤ |(fderiv ℝ X (X.symm y)).det|⁻¹ * (3*‖A (X.symm y)*B (X.symm y)‖*‖(A (X.symm y))ᵀ‖) := | |
| mul_le_mul_of_nonneg_left (matrix_three_product_norm_bound _ _) (inv_nonneg.mpr (abs_nonneg _)) | |
| _ ≤ |(fderiv ℝ X (X.symm y)).det|⁻¹ * (3*(3*‖A (X.symm y)‖*‖B (X.symm y)‖)*‖A (X.symm y)‖) := by | |
| rw [Matrix.norm_transpose] | |
| gcongr | |
| exact matrix_three_product_norm_bound _ _ | |
| _ = g (X.symm y)*‖B (X.symm y)‖ := by dsimp [g]; ring | |
| _ ≤ _ := mul_le_mul_of_nonneg_right hbound (norm_nonneg _) | |
| · have hz : B (X.symm y)=0 := by | |
| by_contra hn | |
| exact hxy (hsB hn) | |
| simp [localPiolaTensor,hy,hz] | |
| · simp only [localPiolaTensor,ite_eq_right hy,norm_zero] | |
| exact mul_nonneg ((le_max_right D 1).trans' zero_le_one) (norm_nonneg _) | |
| end ScalarConductivity | |
| end | |
| end | |
| end OAI | |