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a458723 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | 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
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