| import Mathlib |
| import OAI.Analysis.Conductivity.Geometry.CompactSplitSubpatch |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory |
| open scoped ENNReal |
| open Matrix Filter Topology |
| open Set MeasureTheory Filter Topology |
| open scoped BigOperators |
| open Set MeasureTheory Filter Topology |
| open scoped Manifold |
| open Set Filter |
| open scoped Topology |
| open Set Filter MeasureTheory |
| open scoped Topology Manifold ENNReal |
| open Set |
| namespace ScalarConductivity |
| open Matrix Set MeasureTheory Filter Topology |
| open scoped Matrix.Norms.Elementwise |
|
|
| lemma fderiv_eq_of_eqOn_open {E V : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup V] [NormedSpace ℝ V] {O : Set E} (hO : IsOpen O) |
| {u v : E → V} (he : EqOn u v O) {x : E} (hx : x ∈ O) : |
| fderiv ℝ u x = fderiv ℝ v x := by |
| apply Filter.EventuallyEq.fderiv_eq |
| exact Filter.EventuallyEq.filter_mono (he.eventuallyEq_of_mem (hO.mem_nhds hx)) le_rfl |
|
|
| def CompactTwoFieldReplacement.changeOld |
| {μ : Measure Coord3} {O : Set Coord3} (hO : IsOpen O) |
| {u v : Coord3 → Fin 2 → ℝ} {A B : Coord3 → Symmetric3} |
| (R : CompactTwoFieldReplacement μ O u A) (he : EqOn u v O) (hAB : EqOn A B O) : |
| CompactTwoFieldReplacement μ O v B where |
| du := R.du |
| dF := R.dF |
| tensor := R.tensor |
| smooth_du := R.smooth_du |
| compact_du := R.compact_du |
| support_du := R.support_du |
| smooth_dF := R.smooth_dF |
| compact_dF := R.compact_dF |
| support_dF := R.support_dF |
| cauchy_dF := R.cauchy_dF |
| smooth_tensor := R.smooth_tensor |
| rank := by |
| intro x hx |
| rw [← fderiv_eq_of_eqOn_open hO (show EqOn (fun y => u y+R.du y) |
| (fun y => v y+R.du y) O from fun y hy => by dsimp only; rw [he hy]) hx] |
| exact R.rank x hx |
| constitutive := by |
| intro x hx |
| rw [← fderiv_eq_of_eqOn_open hO (show EqOn (fun y => u y+R.du y) |
| (fun y => v y+R.du y) O from fun y hy => by dsimp only; rw [he hy]) hx] |
| rw [R.constitutive x hx] |
| simp only [conductivityFlux, hAB hx, fderiv_eq_of_eqOn_open hO he hx] |
|
|
| theorem exists_locally_smooth_compact_split |
| (μ : Measure Coord3) [μ.IsAddHaarMeasure] |
| (u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) {p : Coord3} |
| {U : Set Coord3} (hU : IsOpen U) (hUb : Bornology.IsBounded U) (hpU : p ∈ U) |
| (hu : ContDiffOn ℝ (↑(⊤ : ℕ∞)) u U) |
| (hA : ContDiffOn ℝ (↑(⊤ : ℕ∞)) (fun x => (A x).val) U) |
| (hdiv : ∀ j (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → HasCompactSupport ψ → |
| tsupport ψ ⊆ U → (∫ x, fderiv ℝ ψ x ((conductivityFlux u A x).col j) ∂μ) = 0) |
| (hD : Function.Surjective (fderiv ℝ u p)) |
| (d : Coord3) (L : Coord3 →L[ℝ] ℝ) (hLd : L d = 1) |
| (B : Mat3) (hB : B.IsSymm) (hBn : ∀ v, L (B *ᵥ v) = 0) |
| {θ c m : ℝ} (hθ : 0 < θ) (hθ1 : θ < 1) |
| (hm : 0 < m) (hpath : ∀ z ∈ Icc (-θ) (1-θ), m ≤ 1+c*z) |
| (C : ℝ → Symmetric3) (hC : ∀ z ∈ Icc (-θ) (1-θ), ContinuousAt C z) |
| (hCs : ∀ z ∈ Icc (-θ) (1-θ), ContDiffAt ℝ (↑(⊤ : ℕ∞)) (fun t => (C t).val) z) |
| (hdet : ∀ z ∈ Icc (-θ) (1-θ), 0 < (splitJacobian d L c z).det) |
| (hconstit : ∀ z ∈ Icc (-θ) (1-θ), |
| (C z).val = (splitJacobian d L c z).det⁻¹ • |
| (splitJacobian d L c z * ((A p).val + z • B) * (splitJacobian d L c z)ᵀ)) |
| {G : Set (ℝ × Symmetric3)} (hG : IsOpen G) |
| (hCG : ∀ z ∈ Icc (-θ) (1-θ), (z, C z) ∈ G) : |
| ∃ W : Set Coord3, IsOpen W ∧ p ∈ W ∧ closure W ⊆ U ∧ IsCompact (closure W) ∧ |
| ∀ O : Set Coord3, IsOpen O → O ⊆ W → ∀ ε : ℝ, 0 < ε → |
| ∃ (R : CompactTwoFieldReplacement μ O u A) (q : Coord3 → ℝ), |
| ContDiff ℝ (↑(⊤ : ℕ∞)) q ∧ (∀ x, q x ∈ Icc (-θ) (1-θ)) ∧ |
| (∀ x ∈ O, (q x, R.tensor x) ∈ G) ∧ |
| μ {x | x ∈ O ∧ q x ≠ -θ ∧ q x ≠ 1-θ} ≤ ENNReal.ofReal ε := by |
| obtain ⟨v, V, hv, _, hV, hpV, hVU, he⟩ := exists_smooth_compact_extension |
| hU hUb isCompact_singleton (singleton_subset_iff.mpr hpU) u hu |
| have hep : fderiv ℝ v p = fderiv ℝ u p := fderiv_eq_of_eqOn_open hV he (hpV (mem_singleton p)) |
| have hAp : ContinuousAt A p := |
| Topology.IsInducing.subtypeVal.continuousAt_iff.mpr |
| (hA.continuousOn.continuousAt (hU.mem_nhds hpU)) |
| have hdivv : ∀ j (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → HasCompactSupport ψ → |
| tsupport ψ ⊆ V → (∫ x, fderiv ℝ ψ x ((conductivityFlux v A x).col j) ∂μ) = 0 := by |
| intro j ψ hψ hc hs |
| rw [derivative_test_integral_congr μ ψ _ (fun x => (conductivityFlux u A x).col j) (by |
| intro x hx |
| simp only [conductivityFlux, fderiv_eq_of_eqOn_open hV he (hs hx)])] |
| exact hdiv j ψ hψ hc (hs.trans hVU) |
| obtain ⟨W, hW, hpW, hWV, hWc, hw⟩ := exists_compact_split_on_subpatch μ v hv A hAp |
| hV (hUb.subset hVU) (hpV (mem_singleton p)) (hA.mono hVU) hdivv (hep ▸ hD) |
| d L hLd B hB hBn hθ hθ1 hm hpath C hC hCs hdet hconstit hG hCG |
| refine ⟨W, hW, hpW, hWV.trans hVU, hWc, ?_⟩ |
| intro O hO hOW ε hε |
| obtain ⟨R, q, hq, hqr, hqG, hqm⟩ := hw O hO hOW ε hε |
| exact ⟨R.changeOld hO (he.mono (hOW.trans (subset_closure.trans hWV))) (fun _ _ => rfl), |
| q, hq, hqr, hqG, hqm⟩ |
|
|
| end ScalarConductivity |
|
|
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology |
|
|
| theorem local_compact_piola_difference_regular |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| (Y : E ≃ E) (hY : ContDiff ℝ (↑(⊤ : ℕ∞)) Y) |
| (hi : ContDiff ℝ (↑(⊤ : ℕ∞)) Y.symm) |
| {K U : Set E} (hU : IsOpen U) (hUb : Bornology.IsBounded U) |
| (hK : IsCompact K) (hKU : K ⊆ U) (hoff : ∀ x ∉ K, Y x = x) |
| (F : E → E) (hF : ContDiffOn ℝ (↑(⊤ : ℕ∞)) F U) : |
| ContDiff ℝ (↑(⊤ : ℕ∞)) (fun x => piolaFlux Y F x-F x) ∧ |
| HasCompactSupport (fun x => piolaFlux Y F x-F x) ∧ |
| tsupport (fun x => piolaFlux Y F x-F x) ⊆ K := by |
| obtain ⟨G, W, hG, _, _, hKW, _, he⟩ := exists_smooth_compact_extension hU hUb hK hKU F hF |
| have hsame : (fun x => piolaFlux Y F x-F x) = fun x => piolaFlux Y G x-G x := by |
| funext x |
| by_cases hx : x ∈ K |
| · have hix := (equiv_mapsTo_of_fixed_compl Y K hoff).2 hx |
| dsimp only [piolaFlux] |
| rw [he (hKW hx), he (hKW hix)] |
| · rw [piolaFlux_eq_off Y hK.isClosed hoff F hx, |
| piolaFlux_eq_off Y hK.isClosed hoff G hx, sub_self, sub_self] |
| have hs : tsupport (fun x => piolaFlux Y F x-F x) ⊆ K := by |
| apply closure_minimal _ hK.isClosed |
| intro x hx |
| by_contra hn |
| exact hx (by dsimp only; rw [piolaFlux_eq_off Y hK.isClosed hoff F hn, sub_self]) |
| refine ⟨?_, hK.of_isClosed_subset isClosed_closure hs, hs⟩ |
| rw [hsame] |
| exact (piolaFlux_contDiff Y hY hi G hG).sub hG |
|
|
| theorem local_compact_updated_flux_regular |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| (Y : E ≃ E) (hY : ContDiff ℝ (↑(⊤ : ℕ∞)) Y) |
| (hi : ContDiff ℝ (↑(⊤ : ℕ∞)) Y.symm) |
| {K U : Set E} (hU : IsOpen U) (hUb : Bornology.IsBounded U) |
| (hK : IsCompact K) (hKU : K ⊆ U) (hoff : ∀ x ∉ K, Y x = x) |
| (F δ : E → E) (hF : ContDiffOn ℝ (↑(⊤ : ℕ∞)) F U) |
| (hδ : ContDiff ℝ (↑(⊤ : ℕ∞)) δ) (hsδ : tsupport δ ⊆ K) : |
| ContDiff ℝ (↑(⊤ : ℕ∞)) (fun x => piolaFlux Y (fun z => F z+δ z) x-F x) ∧ |
| HasCompactSupport (fun x => piolaFlux Y (fun z => F z+δ z) x-F x) ∧ |
| tsupport (fun x => piolaFlux Y (fun z => F z+δ z) x-F x) ⊆ K := by |
| obtain ⟨hs, _, hsk⟩ := local_compact_piola_difference_regular Y hY hi hU hUb hK hKU hoff F hF |
| have hp := piolaFlux_contDiff Y hY hi δ hδ |
| have hps := piolaFlux_tsupport_subset Y hK.isClosed hoff δ hsδ |
| have heq : (fun x => piolaFlux Y (fun z => F z+δ z) x-F x) = |
| fun x => (piolaFlux Y F x-F x)+piolaFlux Y δ x := by |
| rw [piolaFlux_add] |
| funext x; simp only [sub_eq_add_neg]; ac_rfl |
| have htotal : tsupport (fun x => piolaFlux Y (fun z => F z+δ z) x-F x) ⊆ K := by |
| rw [heq] |
| exact (tsupport_add _ _).trans (union_subset hsk hps) |
| refine ⟨?_, hK.of_isClosed_subset isClosed_closure htotal, htotal⟩ |
| rw [heq] |
| exact hs.add hp |
|
|
| end ScalarConductivity |
|
|
| end |
|
|
| end OAI |
|
|
