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159357f | 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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | import Mathlib
import OAI.Analysis.Conductivity.Flux.C1WeakDivergence
import OAI.Analysis.Conductivity.Sobolev.AxialPairRank
namespace OAI
noncomputable section
namespace ScalarConductivity
open Real Set Filter Topology MeasureTheory Matrix
open scoped Matrix.Norms.Elementwise
lemma axialPair_mem_regularRegion_of_model
{f F : Coord3 → ℝ} {A B : Coord3 → Symmetric3} {x : Coord3}
(hF : ContDiff ℝ (↑(⊤:ℕ∞)) F)
(hB : ContDiff ℝ (↑(⊤:ℕ∞)) (fun y => (B y).val))
(hf : f =ᶠ[𝓝 x] F) (hA : A =ᶠ[𝓝 x] B)
{i : Fin 3} (hi : i≠0) (hd : direction (Pi.single i 1) F x≠0)
{U : Set Coord3} (hU : IsOpen U) (hx : x∈U) :
x∈regularRegion (axialPair f) A U := by
have hc : Continuous (direction (Pi.single i 1) F) :=
direction_continuous (hF.of_le (by simp)) _
have hn : ∀ᶠ y in 𝓝 x,direction (Pi.single i 1) F y≠0 :=
(isOpen_ne_fun hc continuous_const).mem_nhds hd
have hh : {y | y∈U ∧ f y=F y ∧ A y=B y ∧ direction (Pi.single i 1) F y≠0}∈𝓝 x := by
filter_upwards [hU.mem_nhds hx,hf,hA,hn] with y hy hfy hAy hny
exact ⟨hy,hfy,hAy,hny⟩
obtain ⟨O,hOsub,hO,hxO⟩ := mem_nhds_iff.mp hh
refine mem_regularRegion_iff.mpr ⟨O,fun y hy => (hOsub hy).1,?_,hxO⟩
apply (axialPair_regularPatch hF hB hi hO (fun y hy => (hOsub hy).2.2.2)).congr
· intro y hy
simp only [axialPair,(hOsub hy).2.1]
· intro y hy
exact (hOsub hy).2.2.1
lemma axialPair_mem_regularRegion_of_zero
{f : Coord3 → ℝ} {A B : Coord3 → Symmetric3} {x : Coord3}
(hB : ContDiff ℝ (↑(⊤:ℕ∞)) (fun y => (B y).val))
(hf : f =ᶠ[𝓝 x] fun _ => 0) (hA : A =ᶠ[𝓝 x] B)
{U : Set Coord3} (hU : IsOpen U) (hx : x∈U) :
x∈regularRegion (axialPair f) A U := by
have hh : {y | y∈U ∧ f y=0 ∧ A y=B y}∈𝓝 x := by
filter_upwards [hU.mem_nhds hx,hf,hA] with y hy hfy hAy
exact ⟨hy,hfy,hAy⟩
obtain ⟨O,hOsub,hO,hxO⟩ := mem_nhds_iff.mp hh
have hp : RegularPatch (axialPair (fun _ => 0)) B O :=
⟨hO,(axialPair_smooth contDiff_const).contDiffOn,hB.contDiffOn,axialPair_zero_rank O⟩
refine mem_regularRegion_iff.mpr ⟨O,fun y hy => (hOsub hy).1,?_,hxO⟩
apply hp.congr
· intro y hy
simp only [axialPair,(hOsub hy).2.1]
· intro y hy
exact (hOsub hy).2.2
lemma pureModeValue_smooth {f : ℝ → ℝ} (hf : ContDiff ℝ (↑(⊤:ℕ∞)) f) (a : Fin 3) :
ContDiff ℝ (↑(⊤:ℕ∞)) (pureModeValue f a) := by
unfold pureModeValue
fun_prop
lemma pureModeTensor_smooth {f : Coord3 → ℝ} (hf : ContDiff ℝ (↑(⊤:ℕ∞)) f) (a : Fin 3) :
ContDiff ℝ (↑(⊤:ℕ∞)) (fun x => pureModeTensor (f x) a) := by
apply contDiff_pi.mpr; intro i
apply contDiff_pi.mpr; intro j
simp only [pureModeTensor,Matrix.diagonal_apply]
split_ifs <;> first | exact hf | exact contDiff_const
lemma crossingTensor_smooth (L : ℝ) (a b : Fin 3) :
ContDiff ℝ (↑(⊤:ℕ∞)) (crossingTensor L a b) := by
have hf := crossingResidualFirst_smooth L
have hg := crossingResidualSecond_smooth L
have hff := crossingFirst_smooth L
have hgg := crossingSecond_smooth L
have hxx : ContDiff ℝ (↑(⊤:ℕ∞)) (fun x : Coord3 => crossingXX L (x 0) (x a) (x b)) := by
unfold crossingXX crossingHxx
fun_prop (disch := intro x; positivity)
have hyy : ContDiff ℝ (↑(⊤:ℕ∞)) (fun x : Coord3 => crossingYY L (x 0) (x a) (x b)) := by
unfold crossingYY crossingHxx
fun_prop (disch := intro x; positivity)
have hxy : ContDiff ℝ (↑(⊤:ℕ∞)) (fun x : Coord3 => crossingXY L (x 0) (x a) (x b)) := by
unfold crossingXY crossingHxy
fun_prop (disch := intro x; positivity)
apply contDiff_pi.mpr; intro i
apply contDiff_pi.mpr; intro j
unfold crossingTensor
split_ifs <;> fun_prop
end ScalarConductivity
end
end OAI
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