| import Mathlib |
| import OAI.Analysis.Conductivity.Walls.CriticalWallTensor |
| import OAI.Analysis.Conductivity.Variational.ParametricSpatialDerivative |
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| namespace OAI |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set Matrix MeasureTheory Filter Topology |
| open scoped Matrix.Norms.Elementwise |
| variable {P : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] |
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| lemma gradientColumns_parametric_smoothOn {u : P×Coord3 → Fin 2 → ℝ} |
| {V : Set (P×Coord3)} (hV : IsOpen V) (hu : ContDiffOn ℝ (↑(⊤:ℕ∞)) u V) : |
| ContDiffOn ℝ (↑(⊤:ℕ∞)) (fun q => gradientColumns (fderiv ℝ (fun x => u (q.1,x)) q.2)) V := by |
| apply contDiffOn_pi.mpr |
| intro i |
| apply contDiffOn_pi.mpr |
| intro j |
| exact (contDiffOn_pi.mp ((spatialFDeriv_smoothOn hV hu).clm_apply contDiffOn_const)) j |
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| lemma coordinateDivergence_parametric_smoothOn {f : P×Coord3 → Coord3} |
| {V : Set (P×Coord3)} (hV : IsOpen V) (hf : ContDiffOn ℝ (↑(⊤:ℕ∞)) f V) : |
| ContDiffOn ℝ (↑(⊤:ℕ∞)) (fun q => coordinateDivergence (fun x => f (q.1,x)) q.2) V := by |
| apply ContDiffOn.sum |
| intro i _ |
| exact (spatialFDeriv_smoothOn hV (contDiffOn_pi.mp hf i)).clm_apply contDiffOn_const |
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| lemma symmetricSource_parametric_smoothOn {H : P×Coord3 → Mat3} {u : P×Coord3 → Fin 2 → ℝ} |
| {V : Set (P×Coord3)} (hV : IsOpen V) (hH : ContDiffOn ℝ (↑(⊤:ℕ∞)) H V) |
| (hu : ContDiffOn ℝ (↑(⊤:ℕ∞)) u V) (j : Fin 2) : |
| ContDiffOn ℝ (↑(⊤:ℕ∞)) (fun q => symmetricSource (fun x => H (q.1,x)) (fun x => u (q.1,x)) j q.2) V := by |
| apply coordinateDivergence_parametric_smoothOn hV |
| (f:=fun q : P×Coord3 => (H q*gradientColumns (fderiv ℝ (fun x => u (q.1,x)) q.2)).col j) |
| have hg := gradientColumns_parametric_smoothOn hV hu |
| apply contDiffOn_pi.mpr |
| intro i |
| change ContDiffOn ℝ (↑(⊤:ℕ∞)) (fun q => ∑ k, H q i k * |
| gradientColumns (fderiv ℝ (fun x => u (q.1,x)) q.2) k j) V |
| apply ContDiffOn.sum |
| intro k _ |
| exact (contDiffOn_pi.mp (contDiffOn_pi.mp hH i) k).mul |
| (contDiffOn_pi.mp (contDiffOn_pi.mp hg k) j) |
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| lemma wallCoordinatePair_parametric_smooth {v : P×Box3 → ℝ} |
| (hv : ContDiff ℝ (↑(⊤:ℕ∞)) v) : |
| ContDiff ℝ (↑(⊤:ℕ∞)) (fun q : P×Coord3 => wallCoordinatePair (fun y => v (q.1,y)) q.2) := by |
| apply contDiff_pi.mpr |
| intro j |
| fin_cases j |
| · exact (contDiff_apply ℝ ℝ (0:Fin 3)).comp contDiff_snd |
| · exact hv.comp (contDiff_fst.prodMk (boxCoordinates.contDiff.comp contDiff_snd)) |
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| lemma wallMatrix_parametric_smoothOn {A B : P×Box3 → ℝ} {V : Set P} |
| (hA : ContDiffOn ℝ (↑(⊤:ℕ∞)) A (V×ˢuniv)) |
| (hB : ContDiffOn ℝ (↑(⊤:ℕ∞)) B (V×ˢuniv)) : |
| ContDiffOn ℝ (↑(⊤:ℕ∞)) (fun q : P×Coord3 => |
| wallMatrix (fun y => A (q.1,y)) (fun y => B (q.1,y)) q.2) (V×ˢuniv) := by |
| have hm : ContDiff ℝ (↑(⊤:ℕ∞)) (fun q : P×Coord3 => (q.1,boxCoordinates q.2)) := |
| contDiff_fst.prodMk (boxCoordinates.contDiff.comp contDiff_snd) |
| have hmap : MapsTo (fun q : P×Coord3 => (q.1,boxCoordinates q.2)) (V×ˢuniv) (V×ˢuniv) := |
| fun q hq => ⟨hq.1,mem_univ _⟩ |
| apply contDiffOn_pi.mpr |
| intro i |
| apply contDiffOn_pi.mpr |
| intro j |
| fin_cases i <;> fin_cases j |
| all_goals first | exact contDiffOn_const | exact hA.comp hm.contDiffOn hmap | |
| exact hB.comp hm.contDiffOn hmap |
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| end ScalarConductivity |
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| end |
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|
| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology |
|
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| variable {P E F : Type*} [TopologicalSpace P] [TopologicalSpace E] |
| [NormedAddCommGroup F] |
|
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| lemma compact_parametric_norm_small {f : P×E → F} {p : P} {K : Set E} |
| (hK : IsCompact K) (hf : ∀ x∈K,ContinuousAt f (p,x)) |
| (hz : ∀ x∈K,f (p,x)=0) {ε : ℝ} (hε : 0<ε) : |
| ∀ᶠ q in 𝓝 p,∀ x∈K,‖f (q,x)‖≤ε := by |
| apply hK.eventually_forall_of_forall_eventually |
| intro x hx |
| have hn : ‖f (p,x)‖<ε := by rw [hz x hx,norm_zero]; exact hε |
| exact ((hf x hx).norm.eventually_lt_const hn).mono (fun _ h => h.le) |
|
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| end ScalarConductivity |
|
|
| namespace ScalarConductivity |
| open Set Filter Topology |
| variable {P : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] |
|
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| lemma compact_supported_family_C1_small {f : P×Coord3 → ℝ} {p : P} {V : Set P} |
| (hV : IsOpen V) (hp : p∈V) (hf : ContDiffOn ℝ (↑(⊤:ℕ∞)) f (V×ˢuniv)) |
| {K : Set Coord3} (hK : IsCompact K) |
| (hs : ∀ᶠ q in 𝓝 p,tsupport (fun x => f (q,x))⊆K) |
| (hz : ∀ x,f (p,x)=0) {ε : ℝ} (hε : 0<ε) : |
| ∀ᶠ q in 𝓝 p,UniformC1Bound (fun x => f (q,x)) ε := by |
| have hopen : IsOpen (V×ˢ(univ : Set Coord3)) := hV.prod isOpen_univ |
| have hc (x : Coord3) : ContDiffAt ℝ (↑(⊤:ℕ∞)) f (p,x) := |
| hf.contDiffAt (hopen.mem_nhds ⟨hp,mem_univ x⟩) |
| have hd := spatialFDeriv_smoothOn hopen hf |
| have he : (fun x => f (p,x))=fun _ => 0 := funext hz |
| have hzD (x : Coord3) : spatialFDeriv f (p,x)=0 := by |
| dsimp [spatialFDeriv]; rw [he]; simp |
| have hv := compact_parametric_norm_small hK (fun x _ => (hc x).continuousAt) |
| (fun x _ => hz x) hε |
| have hder := compact_parametric_norm_small hK |
| (fun x _ => (hd.contDiffAt (hopen.mem_nhds ⟨hp,mem_univ x⟩)).continuousAt) |
| (fun x _ => hzD x) hε |
| filter_upwards [hs,hv,hder] with q hq hv hd |
| intro x |
| by_cases hx : x∈K |
| · exact ⟨by simpa only [Real.norm_eq_abs] using hv x hx,hd x hx⟩ |
| · have hn : x∉tsupport (fun y => f (q,y)) := fun hh => hx (hq hh) |
| have hv : f (q,x)=0 := image_eq_zero_of_notMem_tsupport (f:=fun y => f (q,y)) hn |
| have hd := fderiv_of_notMem_tsupport ℝ hn |
| exact ⟨by change |f (q,x)|≤ε; rw [hv,abs_zero]; exact hε.le,by rw [hd,norm_zero]; exact hε.le⟩ |
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| end ScalarConductivity |
|
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| end |
|
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| end OAI |
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