| import Mathlib |
| import OAI.Analysis.Conductivity.Flux.TwoFluxColumns |
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| namespace OAI |
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| noncomputable section |
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| namespace ScalarConductivity |
|
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| open MeasureTheory |
| open scoped ENNReal |
| open Matrix Filter Topology |
| section |
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| lemma fderiv_equiv_symm_comp {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (X : E ≃ E) (hX : Differentiable ℝ X) (hXi : Differentiable ℝ X.symm) (x : E) : |
| (fderiv ℝ X.symm (X x)).comp (fderiv ℝ X x) = ContinuousLinearMap.id ℝ E := by |
| rw [← fderiv_comp x (hXi _) (hX _)] |
| have hx : (X.symm ∘ X : E → E) = id := funext X.symm_apply_apply |
| rw [hx, fderiv_id] |
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| lemma fderiv_equiv_comp_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (X : E ≃ E) (hX : Differentiable ℝ X) (hXi : Differentiable ℝ X.symm) (x : E) : |
| (fderiv ℝ X x).comp (fderiv ℝ X.symm (X x)) = ContinuousLinearMap.id ℝ E := by |
| simpa only [Equiv.symm_symm, Equiv.symm_apply_apply] using |
| fderiv_equiv_symm_comp X.symm hXi hX (X x) |
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| def derivativeEquiv {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (X : E ≃ E) (hX : Differentiable ℝ X) (hXi : Differentiable ℝ X.symm) (x : E) : E ≃L[ℝ] E := |
| ContinuousLinearEquiv.equivOfInverse' (fderiv ℝ X x) (fderiv ℝ X.symm (X x)) |
| (fderiv_equiv_comp_symm X hX hXi x) (fderiv_equiv_symm_comp X hX hXi x) |
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| lemma fderiv_equiv_det_ne_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (X : E ≃ E) (hX : Differentiable ℝ X) (hXi : Differentiable ℝ X.symm) (x : E) : |
| (fderiv ℝ X x).det ≠ 0 := by |
| have hc := congrArg (fun L : E →L[ℝ] E => L.det) (fderiv_equiv_symm_comp X hX hXi x) |
| simp only [ContinuousLinearMap.det, ContinuousLinearMap.toLinearMap_comp, |
| ContinuousLinearMap.coe_id, LinearMap.det_comp, LinearMap.det_id] at hc |
| exact right_ne_zero_of_mul_eq_one hc |
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| lemma piolaFlux_derivative_pairing {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (X : E ≃ E) (hX : Differentiable ℝ X) (hXi : Differentiable ℝ X.symm) |
| (u : E → ℝ) (hu : Differentiable ℝ u) (F : E → E) (y : E) : |
| fderiv ℝ (u ∘ X.symm) y (piolaFlux X F y) = |
| |(fderiv ℝ X (X.symm y)).det|⁻¹ * fderiv ℝ u (X.symm y) (F (X.symm y)) := by |
| rw [fderiv_comp y (hu _) (hXi _)] |
| simp only [piolaFlux, ContinuousLinearMap.comp_apply, map_smul, smul_eq_mul] |
| congr 1 |
| have hc := fderiv_equiv_symm_comp X hX hXi (X.symm y) |
| rw [X.apply_symm_apply] at hc |
| exact congrArg (fun L : E →L[ℝ] E => fderiv ℝ u (X.symm y) (L (F (X.symm y)))) hc |
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| def smoothEquivHomeomorph {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (X : E ≃ E) (hX : Differentiable ℝ X) (hXi : Differentiable ℝ X.symm) : E ≃ₜ E := |
| { X with continuous_toFun := hX.continuous, continuous_invFun := hXi.continuous } |
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| lemma piolaFlux_hasCompactSupport {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (X : E ≃ E) (hX : Differentiable ℝ X) (hXi : Differentiable ℝ X.symm) |
| (F : E → E) (hF : HasCompactSupport F) : HasCompactSupport (piolaFlux X F) := by |
| apply (hF.comp_homeomorph (smoothEquivHomeomorph X hX hXi).symm).mono |
| intro y hy |
| contrapose! hy |
| simp only [Function.mem_support, not_not] at hy ⊢ |
| change F (X.symm y) = 0 at hy |
| simp only [piolaFlux, hy, map_zero, smul_zero] |
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| lemma piolaFlux_zero_pairing {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| (μ : Measure E) [μ.IsAddHaarMeasure] |
| (X : E ≃ E) (hX : ContDiff ℝ (↑(⊤ : ℕ∞)) X) |
| (hXi : ContDiff ℝ (↑(⊤ : ℕ∞)) X.symm) (F : E → E) |
| (hF : ∀ ψ : E → ℝ, ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → |
| (∫ x, fderiv ℝ ψ x (F x) ∂μ) = 0) |
| (ψ : E → ℝ) (hψ : ContDiff ℝ (↑(⊤ : ℕ∞)) ψ) : |
| (∫ y, fderiv ℝ ψ y (piolaFlux X F y) ∂μ) = 0 := by |
| rw [piolaFlux_pairing μ X (hX.differentiable (by simp)) |
| (fderiv_equiv_det_ne_zero X (hX.differentiable (by simp)) (hXi.differentiable (by simp))) |
| F ψ (hψ.differentiable (by simp))] |
| exact hF _ (hψ.comp hX) |
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| lemma equiv_mapsTo_of_fixed_compl {E : Type*} (X : E ≃ E) (Q : Set E) |
| (hX : ∀ x ∉ Q, X x = x) : Set.MapsTo X Q Q ∧ Set.MapsTo X.symm Q Q := by |
| have hi : ∀ x ∉ Q, X.symm x = x := by |
| intro x hx |
| exact (X.symm_apply_eq).2 (hX x hx).symm |
| constructor |
| · intro x hx |
| by_contra hn |
| have he : X (X x) = X x := hX (X x) hn |
| exact hn ((X.injective he).symm ▸ hx) |
| · intro x hx |
| by_contra hn |
| have he : X.symm (X.symm x) = X.symm x := hi (X.symm x) hn |
| exact hn ((X.symm.injective he).symm ▸ hx) |
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| end |
|
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| open Set MeasureTheory Filter Topology |
| open scoped BigOperators |
|
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| def flatPulse (θ δ t : ℝ) : ℝ := |
| Real.smoothTransition (t / δ) - Real.smoothTransition ((t - θ) / δ) |
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| lemma flatPulse_smooth (θ δ : ℝ) : ContDiff ℝ (↑(⊤ : ℕ∞)) (flatPulse θ δ) := by |
| exact (Real.smoothTransition.contDiff.comp (contDiff_id.div_const δ)).sub |
| (Real.smoothTransition.contDiff.comp ((contDiff_id.sub contDiff_const).div_const δ)) |
|
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| lemma flatPulse_zero_left {θ δ t : ℝ} (hθ : 0 ≤ θ) (hδ : 0 < δ) (ht : t ≤ 0) : |
| flatPulse θ δ t = 0 := by |
| rw [flatPulse, Real.smoothTransition.zero_of_nonpos (div_nonpos_of_nonpos_of_nonneg ht hδ.le), |
| Real.smoothTransition.zero_of_nonpos |
| (div_nonpos_of_nonpos_of_nonneg (by linarith) hδ.le), sub_self] |
|
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| lemma flatPulse_zero_right {θ δ t : ℝ} (hθ : 0 ≤ θ) (hδ : 0 < δ) (ht : θ + δ ≤ t) : |
| flatPulse θ δ t = 0 := by |
| rw [flatPulse, Real.smoothTransition.one_of_one_le ((le_div_iff₀ hδ).2 (by linarith)), |
| Real.smoothTransition.one_of_one_le ((le_div_iff₀ hδ).2 (by linarith)), sub_self] |
|
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| lemma flatPulse_one {θ δ t : ℝ} (hδ : 0 < δ) (ht : δ ≤ t) (htθ : t ≤ θ) : |
| flatPulse θ δ t = 1 := by |
| rw [flatPulse, Real.smoothTransition.one_of_one_le ((le_div_iff₀ hδ).2 (by linarith)), |
| Real.smoothTransition.zero_of_nonpos |
| (div_nonpos_of_nonpos_of_nonneg (sub_nonpos.mpr htθ) hδ.le), sub_zero] |
|
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| lemma flatPulse_bounds {θ δ : ℝ} (hδ : 0 < δ) (hδθ : δ ≤ θ) (t : ℝ) : |
| 0 ≤ flatPulse θ δ t ∧ flatPulse θ δ t ≤ 1 := by |
| by_cases ht : t ≤ θ |
| · rw [flatPulse, Real.smoothTransition.zero_of_nonpos |
| (div_nonpos_of_nonpos_of_nonneg (sub_nonpos.mpr ht) hδ.le), sub_zero] |
| exact ⟨Real.smoothTransition.nonneg _, Real.smoothTransition.le_one _⟩ |
| · rw [flatPulse, Real.smoothTransition.one_of_one_le |
| ((le_div_iff₀ hδ).2 (by linarith))] |
| exact ⟨sub_nonneg.2 (Real.smoothTransition.le_one _), |
| sub_le_self _ (Real.smoothTransition.nonneg _)⟩ |
|
|
| lemma flatPulse_support {θ δ : ℝ} (hθ : 0 ≤ θ) (hδ : 0 < δ) : |
| Function.support (flatPulse θ δ) ⊆ Ioo 0 (θ + δ) := by |
| intro t ht |
| constructor |
| · by_contra hn |
| exact ht (flatPulse_zero_left hθ hδ (not_lt.mp hn)) |
| · by_contra hn |
| exact ht (flatPulse_zero_right hθ hδ (not_lt.mp hn)) |
|
|
| def periodicPulse (θ δ t : ℝ) : ℝ := ∑ᶠ n : ℤ, flatPulse θ δ (t - n) |
|
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| lemma pulse_translates_locallyFinite {θ δ : ℝ} (hθ : 0 ≤ θ) (hδ : 0 < δ) : |
| LocallyFinite (fun n : ℤ => Function.support (fun t : ℝ => flatPulse θ δ (t - n))) := by |
| have hl : Tendsto (fun n : ℤ => (n : ℝ)) atTop atTop := tendsto_intCast_atTop_atTop |
| have hu : Tendsto (fun n : ℤ => (n : ℝ) + (θ + δ)) atBot atBot := by |
| exact tendsto_atBot_add_const_right _ _ (tendsto_intCast_atBot_iff.2 tendsto_id) |
| apply (locallyFinite_Icc_of_tendsto hl hu).subset |
| intro n t ht |
| have ht' := flatPulse_support hθ hδ ht |
| exact ⟨by linarith [ht'.1], by linarith [ht'.2]⟩ |
|
|
| lemma periodicPulse_smooth {θ δ : ℝ} (hθ : 0 ≤ θ) (hδ : 0 < δ) : |
| ContDiff ℝ (↑(⊤ : ℕ∞)) (periodicPulse θ δ) := by |
| rw [← contMDiff_iff_contDiff] |
| exact contMDiff_finsum (fun n => |
| (contMDiff_iff_contDiff).2 ((flatPulse_smooth θ δ).comp (contDiff_id.sub contDiff_const))) |
| (pulse_translates_locallyFinite hθ hδ) |
|
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| lemma periodicPulse_periodic (θ δ : ℝ) : Function.Periodic (periodicPulse θ δ) 1 := by |
| intro t |
| unfold periodicPulse |
| calc |
| _ = ∑ᶠ n : ℤ, flatPulse θ δ (t + 1 - (n + 1 : ℤ)) := |
| (finsum_comp_equiv (Equiv.addRight (1 : ℤ))).symm |
| _ = _ := by congr 1; funext n; congr 1; push_cast; ring |
|
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| lemma periodicPulse_eq {θ δ t : ℝ} (hθ : 0 ≤ θ) (hδ : 0 < δ) |
| (hwidth : θ + δ ≤ 1) (ht : t ∈ Icc 0 1) : periodicPulse θ δ t = flatPulse θ δ t := by |
| unfold periodicPulse |
| have hz : ∀ n : ℤ, n ≠ 0 → flatPulse θ δ (t - n) = 0 := by |
| intro n hn |
| rcases lt_or_gt_of_ne hn with hn | hn |
| · apply flatPulse_zero_right hθ hδ |
| have hn' : (n : ℝ) ≤ -1 := by exact_mod_cast (show n ≤ -1 by omega) |
| linarith [ht.1] |
| · apply flatPulse_zero_left hθ hδ |
| have hn' : (1 : ℝ) ≤ n := by exact_mod_cast (show 1 ≤ n by omega) |
| linarith [ht.2] |
| simpa using finsum_eq_single _ 0 hz |
|
|
| lemma periodicPulse_bounds {θ δ : ℝ} (hδ : 0 < δ) (hδθ : δ ≤ θ) |
| (hwidth : θ + δ ≤ 1) (t : ℝ) : |
| 0 ≤ periodicPulse θ δ t ∧ periodicPulse θ δ t ≤ 1 := by |
| have ht : t - (⌊t⌋ : ℤ) ∈ Icc (0 : ℝ) 1 := by |
| constructor |
| · exact sub_nonneg.mpr (Int.floor_le t) |
| · linarith [Int.lt_floor_add_one t] |
| have he : periodicPulse θ δ (t - (⌊t⌋ : ℤ)) = periodicPulse θ δ t := by |
| simpa using (periodicPulse_periodic θ δ).sub_zsmul_eq (x := t) ⌊t⌋ |
| rw [← he, periodicPulse_eq (hδ.le.trans hδθ) hδ hwidth ht] |
| exact flatPulse_bounds hδ hδθ _ |
|
|
| lemma smoothStep_integral {δ a b : ℝ} (hδ : 0 < δ) (ha : a ≤ 0) (hb : δ ≤ b) : |
| (∫ t in a..b, Real.smoothTransition (t / δ)) = |
| (∫ t in (0 : ℝ)..δ, Real.smoothTransition (t / δ)) + (b - δ) := by |
| have hs : Continuous (fun t : ℝ => Real.smoothTransition (t / δ)) := by fun_prop |
| have hl : (∫ t in a..0, Real.smoothTransition (t / δ)) = 0 := by |
| calc |
| _ = ∫ _t in a..0, (0 : ℝ) := by |
| apply intervalIntegral.integral_congr |
| intro t ht |
| rw [uIcc_of_le ha] at ht |
| exact Real.smoothTransition.zero_of_nonpos |
| (div_nonpos_of_nonpos_of_nonneg ht.2 hδ.le) |
| _ = 0 := by simp |
| have hr : (∫ t in δ..b, Real.smoothTransition (t / δ)) = b - δ := by |
| calc |
| _ = ∫ _t in δ..b, (1 : ℝ) := by |
| apply intervalIntegral.integral_congr |
| intro t ht |
| apply Real.smoothTransition.one_of_one_le |
| apply (le_div_iff₀ hδ).2 |
| rw [uIcc_of_le hb] at ht |
| simpa only [one_mul] using ht.1 |
| _ = _ := by simp |
| rw [← intervalIntegral.integral_add_adjacent_intervals (hs.intervalIntegrable a 0) |
| (hs.intervalIntegrable 0 b), |
| ← intervalIntegral.integral_add_adjacent_intervals (hs.intervalIntegrable 0 δ) |
| (hs.intervalIntegrable δ b), hl, hr, zero_add] |
|
|
| lemma flatPulse_integral {θ δ : ℝ} (hθ : 0 ≤ θ) (hδ : 0 < δ) |
| (hwidth : θ + δ ≤ 1) : (∫ t in (0 : ℝ)..1, flatPulse θ δ t) = θ := by |
| have hs : Continuous (fun t : ℝ => Real.smoothTransition (t / δ)) := by fun_prop |
| have hs' : Continuous (fun t : ℝ => Real.smoothTransition ((t - θ) / δ)) := by fun_prop |
| simp only [flatPulse] |
| rw [intervalIntegral.integral_sub (hs.intervalIntegrable _ _) |
| (hs'.intervalIntegrable _ _)] |
| rw [intervalIntegral.integral_comp_sub_right (fun t : ℝ => Real.smoothTransition (t / δ)) θ] |
| rw [smoothStep_integral (δ := δ) (a := 0) (b := 1) hδ le_rfl (by linarith), |
| smoothStep_integral (δ := δ) (a := 0 - θ) (b := 1 - θ) hδ (by linarith) (by linarith)] |
| ring |
|
|
| lemma periodicPulse_integral {θ δ : ℝ} (hθ : 0 ≤ θ) (hδ : 0 < δ) |
| (hwidth : θ + δ ≤ 1) : (∫ t in (0 : ℝ)..1, periodicPulse θ δ t) = θ := by |
| convert flatPulse_integral hθ hδ hwidth using 1 |
| apply intervalIntegral.integral_congr |
| intro t ht |
| exact periodicPulse_eq hθ hδ hwidth (by simpa using ht) |
|
|
| theorem exists_two_state_wave {θ δ : ℝ} (hδ : 0 < δ) (hδθ : δ ≤ θ) |
| (hwidth : θ + δ ≤ 1) : |
| ∃ h : ℝ → ℝ, |
| ContDiff ℝ (↑(⊤ : ℕ∞)) h ∧ Function.Periodic h 1 ∧ |
| (∫ t in (0 : ℝ)..1, h t) = 0 ∧ |
| (∀ t, -θ ≤ h t ∧ h t ≤ 1 - θ) ∧ |
| (∀ t ∈ Icc 0 1, h t ≠ -θ → h t ≠ 1 - θ → |
| t ∈ Ioo 0 δ ∪ Ioo θ (θ + δ)) := by |
| have hθ : 0 ≤ θ := hδ.le.trans hδθ |
| refine ⟨fun t => periodicPulse θ δ t - θ, |
| (periodicPulse_smooth hθ hδ).sub contDiff_const, ?_, ?_, ?_, ?_⟩ |
| · intro t |
| change periodicPulse θ δ (t + 1) - θ = periodicPulse θ δ t - θ |
| rw [show periodicPulse θ δ (t + 1) = periodicPulse θ δ t from periodicPulse_periodic θ δ t] |
| · rw [intervalIntegral.integral_sub ((periodicPulse_smooth hθ hδ).continuous.intervalIntegrable _ _) |
| intervalIntegrable_const, periodicPulse_integral hθ hδ hwidth] |
| simp |
| · intro t |
| obtain ⟨hl, hu⟩ := periodicPulse_bounds hδ hδθ hwidth t |
| constructor <;> linarith |
| · intro t ht hl hu |
| change periodicPulse θ δ t - θ ≠ -θ at hl |
| change periodicPulse θ δ t - θ ≠ 1 - θ at hu |
| rw [periodicPulse_eq hθ hδ hwidth ht] at hl hu |
| by_cases htd : t < δ |
| · left |
| refine ⟨?_, htd⟩ |
| by_contra hn |
| exact hl (by rw [flatPulse_zero_left hθ hδ (not_lt.mp hn)]; ring) |
| · right |
| refine ⟨?_, ?_⟩ |
| · by_contra hn |
| exact hu (by rw [flatPulse_one hδ (not_lt.mp htd) (not_lt.mp hn)]) |
| · by_contra hn |
| exact hl (by rw [flatPulse_zero_right hθ hδ (not_lt.mp hn)]; ring) |
|
|
| end ScalarConductivity |
|
|
| end |
|
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| end OAI |
|
|
