| import Mathlib |
| import OAI.Analysis.Conductivity.Variational.PairedPotential |
| import OAI.Analysis.Conductivity.Sobolev.LocalPullbackTsupportInside |
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| namespace OAI |
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| 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 ENNReal Matrix.Norms.Elementwise |
| theorem exists_local_split_on_subpatch_retained |
| (μ : Measure Coord3) [μ.IsAddHaarMeasure] |
| (u : Coord3 → Fin 2 → ℝ) (hu : ContDiff ℝ (↑(⊤ : ℕ∞)) u) |
| (A : Coord3 → Symmetric3) {p : Coord3} (hA : ContinuousAt A p) |
| (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) |
| (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) |
| {U : Set Coord3} (hU : IsOpen U) (hpU : p ∈ U) : |
| ∃ (X : OpenPartialHomeomorph Coord3 Coord3) (W : Set Coord3), |
| IsOpen W ∧ p ∈ W ∧ closure W ⊆ U ∧ IsCompact (closure W) ∧ W ⊆ X.source ∧ |
| ContDiff ℝ (↑(⊤ : ℕ∞)) X ∧ |
| ContDiffOn ℝ (↑(⊤ : ℕ∞)) X.symm X.target ∧ |
| ∀ O : Set Coord3, IsOpen O → O ⊆ W → |
| ∀ ε : ℝ, 0 < ε → |
| ∃ (χ : SmoothScalar Coord3) (H : SmoothScalar ℝ) (F : ℝ → Fin 2 → Coord3 → Coord3), |
| HasCompactSupport χ.val ∧ tsupport χ.val ⊆ X '' O ∧ |
| (∀ x, 0 ≤ χ.val x ∧ χ.val x ≤ 1) ∧ |
| (∀ t, -θ ≤ (smoothDirection 1 H).val t ∧ (smoothDirection 1 H).val t ≤ 1-θ) ∧ |
| (∀ k j, ContDiff ℝ (↑(⊤ : ℕ∞)) (F k j)) ∧ |
| (∀ k j, HasCompactSupport (F k j) ∧ tsupport (F k j) ⊆ O ∧ tsupport (F k j) ⊆ X.symm '' tsupport χ.val) ∧ |
| (∀ k j (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → |
| (∫ y, fderiv ℝ ψ y (F k j y) ∂μ) = 0) ∧ |
| ∀ᶠ n : ℕ in atTop, ∃ Y : Coord3 ≃ Coord3, |
| ContDiff ℝ (↑(⊤ : ℕ∞)) Y ∧ ContDiff ℝ (↑(⊤ : ℕ∞)) Y.symm ∧ |
| (∀ x ∉ O, Y x = x) ∧ Y '' O = O ∧ |
| (∀ x, Y x = x + (c * (n+1 : ℝ)⁻¹ * localPullback X χ.val x * |
| H.val ((n+1 : ℝ) * L (X x))) • d) ∧ |
| μ {x | x ∈ O ∧ |
| localPullback X χ.val (Y.symm x) * |
| (smoothDirection 1 H).val ((n+1 : ℝ) * L (X (Y.symm x))) ≠ -θ ∧ |
| localPullback X χ.val (Y.symm x) * |
| (smoothDirection 1 H).val ((n+1 : ℝ) * L (X (Y.symm x))) ≠ 1-θ} ≤ |
| ENNReal.ofReal ε ∧ |
| ∀ x ∈ O, |
| let z := localPullback X χ.val x * (smoothDirection 1 H).val ((n+1 : ℝ) * L (X x)) |
| let E := gradientColumns (fderiv ℝ u x) |
| let Ft := (A x).val * E + Matrix.of (fun i j => F (n+1 : ℝ) j x i) |
| let J := operatorMatrix (fderiv ℝ Y x) |
| let At := repairPushed (C z) J E Ft |
| 0 < J.det ∧ LinearIndependent ℝ E.col ∧ (z, At) ∈ G ∧ |
| At.val * gradientColumns (fderiv ℝ (u ∘ Y.symm) (Y x)) = pushFlux J Ft := by |
| have hb : 0 ≤ 1-θ := by linarith |
| have hb1 : 1-θ ≤ 1 := by linarith |
| obtain ⟨X, V, hV, hpV, hVU, hVX, hX, hXi, M, hM, hop⟩ := |
| exists_local_two_field_split μ u hu A hA hD d L hLd B hB hBn |
| hθ.le hb hθ1.le hb1 hm hpath C hC hdet hconstit hG hCG hU hpU |
| obtain ⟨W, hW, hpW, hWV, hWc, hXW, D, hDpos, hDb⟩ := |
| exists_bounded_chart_neighborhood X hX.continuous hXi hV hVX hpV |
| have hWV' : W ⊆ V := subset_closure.trans hWV |
| have hWX : W ⊆ X.source := hWV'.trans hVX |
| have hL : L ≠ 0 := by intro hz; simp only [hz, _root_.zero_apply] at hLd; norm_num at hLd |
| refine ⟨X, W, hW, hpW, hWV.trans hVU, hWc, hWX, hX, hXi, ?_⟩ |
| intro O hO hOW ε hε |
| have hOX : O ⊆ X.source := hOW.trans hWX |
| have hOV : O ⊆ V := hOW.trans hWV' |
| have hXO := hXW.subset (image_mono hOW) |
| obtain ⟨χ, H, hχc, hχW, hχb, hHp, hHm, hHb, hHr, hpure⟩ := |
| exists_chart_pure_profile μ X hO hOX (hXi.differentiableOn (by simp)) |
| hXO hXO.measure_lt_top.ne hDpos (fun x hx => hDb x (image_mono hOW hx)) L hL hθ hθ1 (div_pos hε hM) |
| obtain ⟨F, hF, hFs, hdiv, hlim⟩ := hop H hHp hHm hHb hHr χ hχc |
| (hχW.trans (image_mono hOV)) hχb |
| have hχt : tsupport χ.val ⊆ X.target := hχW.trans (by |
| rintro y ⟨x, hx, rfl⟩; exact X.map_source (hOX hx)) |
| have hχp : tsupport (localPullback X χ.val) ⊆ O := |
| localPullback_tsupport_inside X χ.val hχc hOX hχW |
| refine ⟨χ, H, F, hχc, hχW, hχb, hHr, hF, ?_, hdiv, ?_⟩ |
| · intro k j |
| refine ⟨(hFs k j).1, ?_, (hFs k j).2.2⟩ |
| intro x hx |
| obtain ⟨y, hy, rfl⟩ := (hFs k j).2.2 hx |
| obtain ⟨z, hz, rfl⟩ := hχW hy |
| simpa only [X.left_inv (hOX hz)] using hz |
| have hnat : Tendsto (fun n : ℕ => (n+1 : ℝ)) atTop atTop := |
| Filter.tendsto_atTop_add_const_right _ 1 tendsto_natCast_atTop_atTop |
| filter_upwards [hnat.eventually hlim, hpure] with n hn hnp |
| obtain ⟨Y, hY, hYi, _, _, hform, hYM, hstate⟩ := hn |
| have hoff : ∀ x ∉ O, Y x = x := by |
| intro x hx |
| have hz := image_eq_zero_of_notMem_tsupport (fun h => hx (hχp h)) |
| rw [hform x, hz, mul_zero, zero_mul, zero_smul, add_zero] |
| have himage := equiv_image_set_of_eq_off Y O hoff |
| refine ⟨Y, hY, hYi, hoff, himage, hform, ?_, fun x hx => hstate x (hOV hx)⟩ |
| let z : Coord3 → ℝ := fun x => localPullback X χ.val x * |
| (smoothDirection 1 H).val ((n+1 : ℝ) * L (X x)) |
| have hz : Continuous z := |
| (localPullback_contDiff X hX.contDiffOn χ.val (smoothScalar_contDiff χ) hχc hχt).continuous.mul |
| ((smoothScalar_contDiff (smoothDirection 1 H)).continuous.comp |
| (continuous_const.mul (L.continuous.comp hX.continuous))) |
| have he := pushed_exception_measure_le μ Y (hY.differentiable (by simp)) |
| hO.measurableSet himage z hz (-θ) (1-θ) hYM |
| calc |
| _ ≤ ENNReal.ofReal M * μ {x | x ∈ O ∧ z x ≠ -θ ∧ z x ≠ 1-θ} := he |
| _ ≤ ENNReal.ofReal M * ENNReal.ofReal (ε/M) := mul_le_mul_right hnp _ |
| _ = ENNReal.ofReal ε := by |
| rw [← ENNReal.ofReal_mul hM.le, mul_div_cancel₀ ε hM.ne'] |
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| end ScalarConductivity |
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| end |
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| end OAI |
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