| 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 |
|
|
