| import Mathlib | |
| import OAI.Analysis.Conductivity.Branching.JoinHull | |
| import OAI.Analysis.Conductivity.Variational.ConjugateDiagonal | |
| import OAI.Analysis.Conductivity.Variational.CompactCompatibleReplacement | |
| 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_depth_reduction_two | |
| (μ : Measure Coord3) [μ.IsAddHaarMeasure] | |
| (u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) | |
| {U : Set Coord3} (hU : IsOpen U) (hUb : Bornology.IsBounded 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) | |
| {a b : ℝ} (ha : 0 < a) {T : Set DiagonalTriple} | |
| (hT : IsOpen T) (hsub : ∀ d ∈ T, IsFiniteLaminate a b d) | |
| (hperm : ∀ d ∈ T, ∀ e : Equiv.Perm (Fin 3), d ∘ e ∈ T) | |
| {p : Coord3} (hpU : p ∈ U) | |
| (hD : Function.Surjective (fderiv ℝ u p)) {n : ℕ} | |
| (hp : A p ∈ spectralExtension (depthClass T (n+1))) | |
| (hn : A p ∉ spectralExtension (depthClass T n)) : | |
| ∃ 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, | |
| (∀ x ∈ O, R.tensor x ∈ matrixFiniteLaminate a b) ∧ | |
| μ {x | x ∈ O ∧ R.tensor x ∉ spectralExtension (depthClass T n)} ≤ ENNReal.ofReal ε := by | |
| obtain ⟨Q, hQ, d, hd, hrep⟩ := hp | |
| have hdn : d ∉ depthClass T n := fun h => hn ⟨Q, hQ, d, h, hrep⟩ | |
| obtain ⟨i, θ, α, β, hθ, hθ1, hα, hβ, he, _⟩ := depthClass_plan hsub hd hdn | |
| have hre : conjugateDiagonal Q (coordinateJoin i θ α β) = A p := by | |
| apply Subtype.ext | |
| simpa only [conjugateDiagonal, he] using hrep.symm | |
| obtain ⟨v, L, B, η, c, m, C, hLv, hB, hBn, hη, hη1, hm, hpath, hC, hCs, | |
| hdet, hconst, hCG, hCa, hCb⟩ := spectral_join_plan ha | |
| (depthClass_subset_laminate hsub n) i hα hβ hθ hθ1 hQ | |
| rw [hre] at hconst | |
| have hTo : IsOpen (spectralExtension (depthClass T n)) := | |
| isOpen_spectralExtension (isOpen_depthClass ha hT hsub n) (depthClass_permute hperm n) | |
| let G := endpointGraph (-η) (1-η) (matrixFiniteLaminate a b) | |
| (spectralExtension (depthClass T n)) (spectralExtension (depthClass T n)) | |
| have hGo : IsOpen G := isOpen_endpointGraph _ _ (isOpen_matrixFiniteLaminate ha) hTo hTo | |
| have hGC : ∀ z ∈ Icc (-η) (1-η), (z, C z) ∈ G := | |
| path_mem_endpointGraph _ _ C hCG hCa hCb | |
| obtain ⟨W, hW, hpW, hWU, hWc, hop⟩ := exists_locally_smooth_compact_split μ u A | |
| hU hUb hpU hu hA hdiv hD v L hLv B hB hBn hη hη1 hm hpath C hC hCs hdet hconst hGo hGC | |
| refine ⟨W, hW, hpW, hWU, hWc, ?_⟩ | |
| intro O hO hOW ε hε | |
| obtain ⟨R, q, _, _, hqG, hqm⟩ := hop O hO hOW ε hε | |
| refine ⟨R, fun x hx => (hqG x hx).1, le_trans (measure_mono ?_) hqm⟩ | |
| intro x hx | |
| refine ⟨hx.1, ?_, ?_⟩ | |
| · exact (hqG x hx.1).2.1.resolve_right hx.2 | |
| · exact (hqG x hx.1).2.2.resolve_right hx.2 | |
| end ScalarConductivity | |
| namespace ScalarConductivity | |
| open Matrix Set MeasureTheory Filter Topology | |
| open scoped Matrix.Norms.Elementwise | |
| theorem exists_compact_depth_reduction_selected | |
| (J : Set (Fin 2)) | |
| (μ : Measure Coord3) [μ.IsAddHaarMeasure] | |
| (u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) | |
| {U : Set Coord3} (hU : IsOpen U) (hUb : Bornology.IsBounded U) | |
| (hu : ContDiffOn ℝ (↑(⊤ : ℕ∞)) u U) | |
| (hA : ContDiffOn ℝ (↑(⊤ : ℕ∞)) (fun x => (A x).val) U) | |
| (hdiv : ∀ j ∈ 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) | |
| {a b : ℝ} (ha : 0 < a) {T : Set DiagonalTriple} | |
| (hT : IsOpen T) (hsub : ∀ d ∈ T, IsFiniteLaminate a b d) | |
| (hperm : ∀ d ∈ T, ∀ e : Equiv.Perm (Fin 3), d ∘ e ∈ T) | |
| {p : Coord3} (hpU : p ∈ U) | |
| (hD : Function.Surjective (fderiv ℝ u p)) {n : ℕ} | |
| (hp : A p ∈ spectralExtension (depthClass T (n+1))) | |
| (hn : A p ∉ spectralExtension (depthClass T n)) : | |
| ∃ 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 : CompactSelectedReplacement J μ O u A, | |
| (∀ x ∈ O, R.tensor x ∈ matrixFiniteLaminate a b) ∧ | |
| μ {x | x ∈ O ∧ R.tensor x ∉ spectralExtension (depthClass T n)} ≤ ENNReal.ofReal ε := by | |
| obtain ⟨Q, hQ, d, hd, hrep⟩ := hp | |
| have hdn : d ∉ depthClass T n := fun h => hn ⟨Q, hQ, d, h, hrep⟩ | |
| obtain ⟨i, θ, α, β, hθ, hθ1, hα, hβ, he, _⟩ := depthClass_plan hsub hd hdn | |
| have hre : conjugateDiagonal Q (coordinateJoin i θ α β) = A p := by | |
| apply Subtype.ext | |
| simpa only [conjugateDiagonal, he] using hrep.symm | |
| obtain ⟨v, L, B, η, c, m, C, hLv, hB, hBn, hη, hη1, hm, hpath, hC, hCs, | |
| hdet, hconst, hCG, hCa, hCb⟩ := spectral_join_plan ha | |
| (depthClass_subset_laminate hsub n) i hα hβ hθ hθ1 hQ | |
| rw [hre] at hconst | |
| have hTo : IsOpen (spectralExtension (depthClass T n)) := | |
| isOpen_spectralExtension (isOpen_depthClass ha hT hsub n) (depthClass_permute hperm n) | |
| let G := endpointGraph (-η) (1-η) (matrixFiniteLaminate a b) | |
| (spectralExtension (depthClass T n)) (spectralExtension (depthClass T n)) | |
| have hGo : IsOpen G := isOpen_endpointGraph _ _ (isOpen_matrixFiniteLaminate ha) hTo hTo | |
| have hGC : ∀ z ∈ Icc (-η) (1-η), (z, C z) ∈ G := | |
| path_mem_endpointGraph _ _ C hCG hCa hCb | |
| obtain ⟨W, hW, hpW, hWU, hWc, hop⟩ := exists_locally_smooth_selected_split J μ u A | |
| hU hUb hpU hu hA hdiv hD v L hLv B hB hBn hη hη1 hm hpath C hC hCs hdet hconst hGo hGC | |
| refine ⟨W, hW, hpW, hWU, hWc, ?_⟩ | |
| intro O hO hOW ε hε | |
| obtain ⟨R, q, _, _, hqG, hqm⟩ := hop O hO hOW ε hε | |
| refine ⟨R, fun x hx => (hqG x hx).1, le_trans (measure_mono ?_) hqm⟩ | |
| intro x hx | |
| refine ⟨hx.1, ?_, ?_⟩ | |
| · exact (hqG x hx.1).2.1.resolve_right hx.2 | |
| · exact (hqG x hx.1).2.2.resolve_right hx.2 | |
| end ScalarConductivity | |
| namespace ScalarConductivity | |
| open Matrix Set MeasureTheory Filter Topology | |
| open scoped Matrix.Norms.Elementwise | |
| lemma gradientColumns_surjective {m : Type*} [Fintype m] [DecidableEq m] | |
| (D : Coord3 →L[ℝ] (m → ℝ)) | |
| (hD : LinearIndependent ℝ (gradientColumns D).col) : Function.Surjective D := by | |
| let E := gradientColumns D | |
| have hh : Eᵀ * (((Eᵀ * E)⁻¹ * Eᵀ)ᵀ) = 1 := by | |
| rw [← Matrix.transpose_mul, gram_leftInverse E hD, Matrix.transpose_one] | |
| have hs := Matrix.mulVec_surjective_iff_exists_right_inverse.mpr ⟨_, hh⟩ | |
| intro y | |
| obtain ⟨x, hx⟩ := hs y | |
| exact ⟨x, (gradientColumns_apply D x).symm.trans hx⟩ | |
| lemma exists_independent_pairedPotential_at {u : Coord3 → ℝ} {p : Coord3} | |
| (hu : DifferentiableAt ℝ u p) (hp : fderiv ℝ u p ≠ 0) : | |
| ∃ j : Fin 3, Function.Surjective (fderiv ℝ (pairedPotential u j) p) := by | |
| obtain ⟨i, hi⟩ := exists_coordinate_nonzero hp | |
| obtain ⟨j, hji⟩ := exists_ne i | |
| refine ⟨j, ?_⟩ | |
| rw [(pairedPotential_hasFDerivAt hu.hasFDerivAt j).fderiv] | |
| exact pairedFunctional_surjective hi hji.symm | |
| lemma pairedPotential_contDiffOn {u : Coord3 → ℝ} {O : Set Coord3} | |
| (hu : ContDiffOn ℝ (↑(⊤ : ℕ∞)) u O) (j : Fin 3) : | |
| ContDiffOn ℝ (↑(⊤ : ℕ∞)) (pairedPotential u j) O := by | |
| apply contDiffOn_pi.mpr | |
| intro i | |
| fin_cases i | |
| · exact hu | |
| · exact (contDiff_apply ℝ ℝ j).contDiffOn | |
| lemma column_derivative_ne_zero_of_independent {O : Set Coord3} (hO : IsOpen O) | |
| {v : Coord3 → Fin 2 → ℝ} (hv : DifferentiableOn ℝ v O) | |
| {x : Coord3} (hx : x ∈ O) | |
| (hr : LinearIndependent ℝ (gradientColumns (fderiv ℝ v x)).col) (j : Fin 2) : | |
| fderiv ℝ (fun y => v y j) x ≠ 0 := by | |
| intro hz | |
| have hh := ((ContinuousLinearMap.proj j).hasFDerivAt.comp x | |
| ((hv x hx).differentiableAt (hO.mem_nhds hx)).hasFDerivAt).fderiv | |
| have hzero : (gradientColumns (fderiv ℝ v x)).col j = 0 := by | |
| ext i | |
| have h := congrArg (fun L : Coord3 →L[ℝ] ℝ => L (Pi.single i 1)) (hh.symm.trans hz) | |
| exact h | |
| exact (hr.ne_zero j) hzero | |
| lemma conductivityFlux_affine_column {O : Set Coord3} (hO : IsOpen O) | |
| {u v : Coord3 → Fin 2 → ℝ} {l κ : Fin 2 → ℝ} (A : Coord3 → Symmetric3) | |
| (he : ∀ x ∈ O, ∀ j, u x j = l j * v x 0 + κ j) | |
| (hv : DifferentiableOn ℝ v O) {x : Coord3} (hx : x ∈ O) (j : Fin 2) : | |
| (conductivityFlux u A x).col j = l j • (conductivityFlux v A x).col 0 := by | |
| ext i | |
| change (∑ k, (A x).val i k * gradientColumns (fderiv ℝ u x) k j) = | |
| l j * ∑ k, (A x).val i k * gradientColumns (fderiv ℝ v x) k 0 | |
| simp_rw [gradientColumns_affine_column hO he hv hx] | |
| rw [Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro k _ | |
| ring | |
| lemma active_flux_harmonic_of_affine | |
| (μ : Measure Coord3) {O : Set Coord3} (hO : IsOpen O) | |
| {u v : Coord3 → Fin 2 → ℝ} {l κ : Fin 2 → ℝ} (A : Coord3 → Symmetric3) | |
| (he : ∀ x ∈ O, ∀ j, u x j = l j * v x 0 + κ j) | |
| (hv : DifferentiableOn ℝ v O) {j : Fin 2} (hj : l j ≠ 0) | |
| (hdiv : ∀ (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → HasCompactSupport ψ → | |
| tsupport ψ ⊆ O → (∫ x, fderiv ℝ ψ x ((conductivityFlux u A x).col j) ∂μ) = 0) : | |
| ∀ (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → HasCompactSupport ψ → | |
| tsupport ψ ⊆ O → (∫ x, fderiv ℝ ψ x ((conductivityFlux v A x).col 0) ∂μ) = 0 := by | |
| intro ψ hψ hc hs | |
| have heq : (∫ x, fderiv ℝ ψ x ((conductivityFlux u A x).col j) ∂μ) = | |
| l j * ∫ x, fderiv ℝ ψ x ((conductivityFlux v A x).col 0) ∂μ := by | |
| rw [derivative_test_integral_congr μ ψ _ | |
| (fun x => l j • (conductivityFlux v A x).col 0) | |
| (fun x hx => conductivityFlux_affine_column hO A he hv (hs hx) j)] | |
| simp_rw [map_smul, smul_eq_mul] | |
| exact integral_const_mul _ _ | |
| exact (mul_eq_zero.mp (heq.symm.trans (hdiv ψ hψ hc hs))).resolve_left hj | |
| def TwoFieldRankRegular (u : Coord3 → Fin 2 → ℝ) (O : Set Coord3) : Prop := | |
| (∀ x ∈ O, LinearIndependent ℝ (gradientColumns (fderiv ℝ u x)).col) ∨ | |
| (∃ (v : Coord3 → ℝ) (l κ : Fin 2 → ℝ), | |
| ContDiffOn ℝ (↑(⊤ : ℕ∞)) v O ∧ (∃ j, l j ≠ 0) ∧ | |
| (∀ x ∈ O, fderiv ℝ v x ≠ 0) ∧ | |
| (∀ x ∈ O, ∀ j, u x j = l j * v x + κ j)) ∨ | |
| ∃ κ : Fin 2 → ℝ, ∀ x ∈ O, u x = κ | |
| lemma TwoFieldRankRegular.mono {u : Coord3 → Fin 2 → ℝ} {O W : Set Coord3} | |
| (h : TwoFieldRankRegular u O) (hW : W ⊆ O) : TwoFieldRankRegular u W := by | |
| rcases h with h | h | ⟨κ, hκ⟩ | |
| · exact Or.inl (fun x hx => h x (hW hx)) | |
| · obtain ⟨v, l, κ, hv, hl, hd, he⟩ := h | |
| exact Or.inr (Or.inl ⟨v, l, κ, hv.mono hW, hl, fun x hx => hd x (hW hx), | |
| fun x hx => he x (hW hx)⟩) | |
| · exact Or.inr (Or.inr ⟨κ, fun x hx => hκ x (hW hx)⟩) | |
| end ScalarConductivity | |
| end | |
| end OAI | |