| import Mathlib |
| import OAI.Analysis.Conductivity.Geometry.LocalSplitSubpatchRetained |
| import OAI.Analysis.Conductivity.Flux.PiolaFluxAdd |
|
|
| 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 |
|
|
| theorem exists_compact_split_on_subpatch |
| (μ : Measure Coord3) [μ.IsAddHaarMeasure] |
| (u : Coord3 → Fin 2 → ℝ) (hu : ContDiff ℝ (↑(⊤ : ℕ∞)) u) |
| (A : Coord3 → Symmetric3) {p : Coord3} (hAp : ContinuousAt A p) |
| {U : Set Coord3} (hU : IsOpen U) (hUb : Bornology.IsBounded U) (hpU : p ∈ 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 ⟨X, W, hW, hpW, hWU, hWc, hWX, hX, hXi, hop⟩ := |
| exists_local_split_on_subpatch_retained μ u hu A hAp hD d L hLd B hB hBn |
| hθ hθ1 hm hpath C hC hdet hconstit hG hCG hU hpU |
| refine ⟨W, hW, hpW, hWU, hWc, ?_⟩ |
| intro O hO hOW ε hε |
| have hOU : O ⊆ U := hOW.trans (subset_closure.trans hWU) |
| have hOX : O ⊆ X.source := hOW.trans hWX |
| obtain ⟨χ, H, F, hχc, hχO, hχb, hHr, hF, hFs, hFdiv, hn⟩ := hop O hO hOW ε hε |
| obtain ⟨n, hn⟩ := hn.exists |
| obtain ⟨Y, hY, hYi, hoffO, _, hform, hpure, hstate⟩ := hn |
| have hχt : tsupport χ.val ⊆ X.target := hχO.trans (by |
| rintro y ⟨x, hx, rfl⟩; exact X.map_source (hOX hx)) |
| let K := X.symm '' tsupport χ.val |
| have hK : IsCompact K := hχc.image_of_continuousOn (X.symm.continuousOn.mono hχt) |
| have hKO : K ⊆ O := by |
| rintro x ⟨y, hy, rfl⟩ |
| obtain ⟨t, ht, rfl⟩ := hχO hy |
| simpa only [X.left_inv (hOX ht)] using ht |
| have hχp := localPullback_tsupport X χ.val hχc hχt |
| have hoff : ∀ x ∉ K, Y x = x := by |
| intro x hx |
| have hz := image_eq_zero_of_notMem_tsupport (fun h => hx (hχp.2.1 h)) |
| rw [hform x, hz, mul_zero, zero_mul, zero_smul, add_zero] |
| let z : Coord3 → ℝ := fun x => localPullback X χ.val x * |
| (smoothDirection 1 H).val ((n+1 : ℝ) * L (X x)) |
| have hz : ContDiff ℝ (↑(⊤ : ℕ∞)) z := |
| (localPullback_contDiff X hX.contDiffOn χ.val (smoothScalar_contDiff χ) hχc hχt).mul |
| ((smoothScalar_contDiff (smoothDirection 1 H)).comp |
| (contDiff_const.mul (L.contDiff.comp hX))) |
| have hzr : ∀ x, z x ∈ Icc (-θ) (1-θ) := fun x => |
| scalar_cutoff_segment hθ.le (by linarith) (localPullback_bounds X χ.val hχb x) (hHr _) |
| obtain ⟨R, hRt⟩ := synchronized_compact_replacement μ u hu A hU hUb hOU hK hKO hA hdiv |
| Y hY hYi hoff z hz C (fun x _ => hCs _ (hzr x)) (F (n+1 : ℝ)) |
| (hF _) (fun j => (hFs _ j).2.2) (hFdiv _) (fun x hx => |
| ⟨(hstate x hx).1, (hstate x hx).2.1, (hstate x hx).2.2.2⟩) |
| let q : Coord3 → ℝ := z ∘ Y.symm |
| have him : MapsTo Y.symm O O := (equiv_mapsTo_of_fixed_compl Y O hoffO).2 |
| refine ⟨R, q, hz.comp hYi, fun x => hzr _, ?_, hpure⟩ |
| intro x hx |
| have ht := hRt (Y.symm x) (him hx) |
| rw [Y.apply_symm_apply] at ht |
| rw [ht] |
| exact (hstate (Y.symm x) (him hx)).2.2.1 |
|
|
| end ScalarConductivity |
|
|
| end |
|
|
| end OAI |
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|
