import Mathlib import OAI.Analysis.Conductivity.Variational.PairedPotential import OAI.Analysis.Conductivity.Sobolev.LocalPullbackTsupportInside 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 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'] end ScalarConductivity end end OAI