| import Mathlib |
| import OAI.Analysis.Conductivity.Sobolev.LocalPiolaFluxSub |
| import OAI.Analysis.Conductivity.Geometry.InverseChartPiolaOperator |
| import OAI.Analysis.Conductivity.Sobolev.LocalPullback |
|
|
| 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 Set Filter Topology MeasureTheory |
|
|
| theorem exists_chart_two_flux_localized |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| (hdim : Module.finrank ℝ E = 3) (μ : Measure E) [μ.IsAddHaarMeasure] |
| (X : OpenPartialHomeomorph E E) (hX : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X X.source) |
| (hXi : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X.symm X.target) |
| (D : E →L[ℝ] (Fin 2 → ℝ)) (hD : Function.Surjective D) |
| (u : E → Fin 2 → ℝ) (hu : ∀ y ∈ X.source, D (X y) = u y) |
| (χ : SmoothScalar E) (hχ : HasCompactSupport χ.val) (hχs : tsupport χ.val ⊆ X.target) |
| (L : E →L[ℝ] ℝ) (hL : L ≠ 0) |
| (h : SmoothScalar ℝ) (hp : Function.Periodic h.val 1) |
| (hm : ∫ t in (0 : ℝ)..1, h.val t = 0) |
| (R : Fin 2 → E) (hRs : D (R 0) 1 = D (R 1) 0) (hRn : ∀ j, L (R j) = 0) : |
| ∃ F : ℝ → Fin 2 → E → E, |
| (∀ k j, ContDiff ℝ (↑(⊤ : ℕ∞)) (F k j)) ∧ |
| (∀ k j, HasCompactSupport (F k j) ∧ tsupport (F k j) ⊆ X.source ∧ |
| tsupport (F k j) ⊆ X.symm '' tsupport χ.val) ∧ |
| (∀ k y, fderiv ℝ u y (F k 0 y) 1 = fderiv ℝ u y (F k 1 y) 0) ∧ |
| (∀ k j (ψ : E → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → |
| (∫ y, fderiv ℝ ψ y (F k j y) ∂μ) = 0) ∧ |
| ∀ j, TendstoUniformly |
| (fun k y => F k j y - localPiolaFlux X.symm |
| (fun x => (χ.val x * h.val (k * L x)) • R j) y) (fun _ => 0) atTop := by |
| classical |
| obtain ⟨G, hg, hs, hsym, hdiv, he⟩ := exists_physical_two_flux hdim μ D hD |
| χ hχ L hL h hp hm R hRs hRn |
| have hc (k : ℝ) (j : Fin 2) : HasCompactSupport (G k j) := |
| hχ.of_isClosed_subset (isClosed_tsupport _) (hs k j) |
| have hgs (k : ℝ) (j : Fin 2) : tsupport (G k j) ⊆ X.target := (hs k j).trans hχs |
| refine ⟨fun k j => localPiolaFlux X.symm (G k j), ?_, ?_, ?_, ?_, ?_⟩ |
| · intro k j |
| exact localPiolaFlux_smooth X.symm hXi hX (G k j) (hg k j) (hc k j) (hgs k j) |
| · intro k j |
| refine ⟨localPiolaFlux_hasCompactSupport X.symm _ (hc k j) (hgs k j), |
| localPiolaFlux_tsupport X.symm _ (hc k j) (hgs k j), ?_⟩ |
| apply closure_minimal |
| ((localPiolaFlux_support X.symm (G k j)).trans |
| (image_mono (fun _ hx => hs k j (subset_closure hx.1)))) |
| ((hχ.image_of_continuousOn (X.symm.continuousOn.mono hχs)).isClosed) |
| · intro k y |
| by_cases hy : y ∈ X.source |
| · rw [localPiolaFlux_potential_pairing X (hX.differentiableOn (by simp)) |
| (hXi.differentiableOn (by simp)) D u hu (G k 0) hy, |
| localPiolaFlux_potential_pairing X (hX.differentiableOn (by simp)) |
| (hXi.differentiableOn (by simp)) D u hu (G k 1) hy] |
| simp only [Pi.smul_apply, smul_eq_mul, hsym] |
| · simp [localPiolaFlux, hy] |
| · intro k j ψ hψ |
| exact localPiolaFlux_zero_pairing μ X.symm hXi hX _ (hc k j) (hgs k j) (hdiv k j) ψ hψ |
| · intro j |
| apply localPiolaFlux_uniform_error X.symm hXi hX hχ hχs |
| (fun k => G k j) (fun k x => (χ.val x * h.val (k * L x)) • R j) _ (he j) |
| intro k x hx |
| by_contra hn |
| have hz : χ.val x = 0 := image_eq_zero_of_notMem_tsupport hn |
| have hgz : G k j x = 0 := image_eq_zero_of_notMem_tsupport (fun hx => hn (hs k j hx)) |
| apply hx |
| simp [hz, hgz] |
|
|
| theorem exists_frozen_chart_two_flux |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| (hdim : Module.finrank ℝ E = 3) (μ : Measure E) [μ.IsAddHaarMeasure] |
| (X : OpenPartialHomeomorph E E) (hX : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X X.source) |
| (hXi : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X.symm X.target) |
| (D : E →L[ℝ] (Fin 2 → ℝ)) (hD : Function.Surjective D) |
| (u : E → Fin 2 → ℝ) (hu : ∀ y ∈ X.source, D (X y) = u y) |
| {a : E} (ha : a ∈ X.source) (hDa : fderiv ℝ X a = ContinuousLinearMap.id ℝ E) |
| (L : E →L[ℝ] ℝ) (hL : L ≠ 0) |
| (R : Fin 2 → E) (hRs : D (R 0) 1 = D (R 1) 0) (hRn : ∀ j, L (R j) = 0) |
| {ε : ℝ} (hε : 0 < ε) : |
| ∃ V : Set E, IsOpen V ∧ a ∈ V ∧ V ⊆ X.source ∧ |
| ∀ h : SmoothScalar ℝ, Function.Periodic h.val 1 → |
| (∫ t in (0 : ℝ)..1, h.val t = 0) → (∀ t, |h.val t| ≤ 1) → |
| ∀ χ : SmoothScalar E, HasCompactSupport χ.val → tsupport χ.val ⊆ X '' V → |
| (∀ x, 0 ≤ χ.val x ∧ χ.val x ≤ 1) → |
| ∃ F : ℝ → Fin 2 → E → E, |
| (∀ k j, ContDiff ℝ (↑(⊤ : ℕ∞)) (F k j)) ∧ |
| (∀ k j, HasCompactSupport (F k j) ∧ tsupport (F k j) ⊆ V ∧ |
| tsupport (F k j) ⊆ X.symm '' tsupport χ.val) ∧ |
| (∀ k y, fderiv ℝ u y (F k 0 y) 1 = fderiv ℝ u y (F k 1 y) 0) ∧ |
| (∀ k j (ψ : E → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → |
| (∫ y, fderiv ℝ ψ y (F k j y) ∂μ) = 0) ∧ |
| ∀ᶠ k : ℝ in atTop, ∀ j y, |
| ‖F k j y - (localPullback X χ.val y * h.val (k * L (X y))) • R j‖ < ε := by |
| let B := ‖R 0‖ + ‖R 1‖ + 1 |
| have hB : 0 < B := by dsimp [B]; positivity |
| have hR : ∀ j, ‖R j‖ ≤ B := by |
| intro j |
| fin_cases j <;> dsimp [B] <;> linarith only [norm_nonneg (R 0), norm_nonneg (R 1)] |
| obtain ⟨V, hV, haV, hVs, hfreeze⟩ := inverseChartPiola_freezing X hX hXi ha hDa |
| (div_pos (half_pos hε) hB) |
| refine ⟨V, hV, haV, hVs, ?_⟩ |
| intro h hp hm hb χ hχ hχV hχb |
| have hχs : tsupport χ.val ⊆ X.target := by |
| rintro z hz |
| obtain ⟨y, hy, rfl⟩ := hχV hz |
| exact X.map_source (hVs hy) |
| obtain ⟨F, hF, hFs, hsym, hdiv, he⟩ := exists_chart_two_flux_localized hdim μ X hX hXi |
| D hD u hu χ hχ hχs L hL h hp hm R hRs hRn |
| have hFi : ∀ k j, tsupport (F k j) ⊆ V := by |
| intro k j y hy |
| obtain ⟨z, hz, rfl⟩ := (hFs k j).2.2 hy |
| obtain ⟨w, hw, rfl⟩ := hχV hz |
| simpa only [X.left_inv (hVs hw)] using hw |
| have hχi : tsupport (localPullback X χ.val) ⊆ V := by |
| intro y hy |
| obtain ⟨z, hz, rfl⟩ := (localPullback_tsupport X χ.val hχ hχs).2.1 hy |
| obtain ⟨w, hw, rfl⟩ := hχV hz |
| simpa only [X.left_inv (hVs hw)] using hw |
| refine ⟨F, hF, fun k j => ⟨(hFs k j).1, hFi k j, (hFs k j).2.2⟩, hsym, hdiv, ?_⟩ |
| have herr : ∀ᶠ k : ℝ in atTop, ∀ j y, |
| ‖F k j y - localPiolaFlux X.symm |
| (fun z => (χ.val z * h.val (k * L z)) • R j) y‖ < ε / 2 := by |
| apply Filter.eventually_all.mpr |
| intro j |
| simpa only [dist_zero_left] using (Metric.tendstoUniformly_iff.mp (he j)) _ (half_pos hε) |
| filter_upwards [herr] with k hk j y |
| by_cases hy : y ∈ V |
| · have hfr := hfreeze (localPullback X χ.val) (fun z => h.val (k * L z)) (R j) y hy |
| have hpre : localPiolaFlux X.symm |
| (fun z => (localPullback X χ.val (X.symm z) * h.val (k * L z)) • R j) y = |
| localPiolaFlux X.symm (fun z => (χ.val z * h.val (k * L z)) • R j) y := by |
| rw [localPiolaFlux_inverse_eq X _ (hVs hy), localPiolaFlux_inverse_eq X _ (hVs hy), |
| localPullback_comp X _ (X.map_source (hVs hy))] |
| rw [hpre] at hfr |
| have hcb := localPullback_bounds X χ.val hχb y |
| have habs : |localPullback X χ.val y * h.val (k * L (X y))| ≤ 1 := by |
| rw [abs_mul, abs_of_nonneg hcb.1] |
| exact (mul_le_mul_of_nonneg_left (hb _) hcb.1).trans (by simpa using hcb.2) |
| have hnorm : |localPullback X χ.val y * h.val (k * L (X y))| * ‖R j‖ ≤ B := |
| (mul_le_mul_of_nonneg_right habs (norm_nonneg _)).trans (by simpa using hR j) |
| have hsmall : ‖localPiolaFlux X.symm |
| (fun z => (χ.val z * h.val (k * L z)) • R j) y - |
| (localPullback X χ.val y * h.val (k * L (X y))) • R j‖ ≤ ε / 2 := |
| hfr.trans ((mul_le_mul_of_nonneg_left hnorm (div_pos (half_pos hε) hB).le).trans |
| (le_of_eq (div_mul_cancel₀ _ hB.ne'))) |
| exact (norm_sub_le_norm_sub_add_norm_sub (F k j y) _ _).trans_lt (by linarith only [hk j y, hsmall]) |
| · have hFz : F k j y = 0 := image_eq_zero_of_notMem_tsupport (fun hz => hy (hFi k j hz)) |
| have hχz : localPullback X χ.val y = 0 := |
| image_eq_zero_of_notMem_tsupport (fun hz => hy (hχi hz)) |
| simpa only [hFz, hχz, zero_mul, zero_smul, sub_zero, norm_zero] using hε |
| |
| end ScalarConductivity |
| |
| end |
| |
| end OAI |
| |
