| import Mathlib |
| import OAI.Analysis.Conductivity.Branching.AttachedEndGradientL2 |
| import OAI.Analysis.Conductivity.Sobolev.CutSlopeH1 |
|
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| namespace OAI |
|
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| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology |
|
|
| lemma differentiable_cut_positive {X : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X] |
| {χ τ : X → ℝ} {x : X} (hχ : DifferentiableAt ℝ χ x) |
| (hτ : DifferentiableAt ℝ τ x) (hne : τ x≠0) (κ : ℝ) : |
| DifferentiableAt ℝ (fun x => κ*(χ x*max 0 (τ x))) x := by |
| rcases lt_or_gt_of_ne hne with hn | hp |
| · have he : (fun x => κ*(χ x*max 0 (τ x))) =ᶠ[𝓝 x] (fun _ => 0) := by |
| filter_upwards [hτ.continuousAt.eventually (gt_mem_nhds hn)] with y hy |
| simp [max_eq_left hy.le] |
| exact (differentiableAt_const (0:ℝ)).congr_of_eventuallyEq he |
| · have he : (fun x => κ*(χ x*max 0 (τ x))) =ᶠ[𝓝 x] (fun x => κ*(χ x*τ x)) := by |
| filter_upwards [hτ.continuousAt.eventually (lt_mem_nhds hp)] with y hy |
| rw [max_eq_right hy.le] |
| exact ((hχ.mul hτ).const_mul κ).congr_of_eventuallyEq he |
|
|
| lemma fderiv_cut_positive {X : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X] |
| {χ τ : X → ℝ} {x : X} (hχ : DifferentiableAt ℝ χ x) |
| (hτ : DifferentiableAt ℝ τ x) (hne : τ x≠0) (κ : ℝ) (v : X) : |
| fderiv ℝ (fun x => κ*(χ x*max 0 (τ x))) x v= |
| κ*(fderiv ℝ χ x v*max 0 (τ x)+χ x*(if 0<τ x then fderiv ℝ τ x v else 0)) := by |
| rcases lt_or_gt_of_ne hne with hn | hp |
| · have he : (fun x => κ*(χ x*max 0 (τ x))) =ᶠ[𝓝 x] (fun _ => 0) := by |
| filter_upwards [hτ.continuousAt.eventually (gt_mem_nhds hn)] with y hy |
| simp [max_eq_left hy.le] |
| rw [he.fderiv_eq,fderiv_const_apply] |
| simp [max_eq_left hn.le,not_lt_of_ge hn.le] |
| · have he : (fun x => κ*(χ x*max 0 (τ x))) =ᶠ[𝓝 x] (fun x => κ*(χ x*τ x)) := by |
| filter_upwards [hτ.continuousAt.eventually (lt_mem_nhds hp)] with y hy |
| rw [max_eq_right hy.le] |
| have hh := ((hχ.hasFDerivAt.mul hτ.hasFDerivAt).const_mul κ).fderiv |
| simp only [Pi.mul_apply] at hh |
| rw [he.fderiv_eq,hh] |
| simp [smul_apply,add_apply,smul_eq_mul,max_eq_right hp.le,hp] |
| ring |
|
|
| theorem compact_collar_slope_value_gradient {χ : (Fin 3 → ℝ) → ℝ} |
| (hχ : ContDiff ℝ (↑(⊤:ℕ∞)) χ) (hc : HasCompactSupport χ) (hχb : ∀ x,|χ x|≤1) |
| {a : ℝ} (ha : a≠0) {b : ℝ} (hb : b∈Icc (-(1:ℝ)/100) (1/100)) |
| (κ l r : ℝ) (hl : -(1:ℝ)/100≤l) (hr : r≤1/100) |
| (hχs : tsupport χ⊆sourceClosedCollarBand l r) : |
| ∃ w : H1,w∈H10 ∧ (∀ᵐ x∂ballMeasure, |
| weakValue w x=κ*(χ (WithLp.ofLp x)*max 0 (a*(sourceCollarTime (WithLp.ofLp x)-b))) ∧ |
| ∀ i,weakGradient w x i=κ*(fderiv ℝ χ (WithLp.ofLp x) (Pi.single i 1)*max 0 (a*(sourceCollarTime (WithLp.ofLp x)-b))+ |
| χ (WithLp.ofLp x)*(if 0<a*(sourceCollarTime (WithLp.ofLp x)-b) then |
| fderiv ℝ (fun y => a*(sourceCollarTime y-b)) (WithLp.ofLp x) (Pi.single i 1) else 0))) := by |
| obtain ⟨w,hw,hwe⟩ := compact_collar_slope_H1 hχ hc hχb a b κ l r hl hr hχs |
| refine ⟨w,hw,?_⟩ |
| obtain ⟨K,hK⟩ := attachedEndTime_lipschitz a b |
| have hd := ae_restrict_of_ae (s:=ball) ((PiLp.volume_preserving_ofLp (Fin 3)).quasiMeasurePreserving.ae |
| (hK.ae_differentiableAt (μ:=volume))) |
| have hn := ae_restrict_of_ae (s:=ball) ((PiLp.volume_preserving_ofLp (Fin 3)).quasiMeasurePreserving.ae |
| (sourceColevel_ae_ne hb)) |
| filter_upwards [hwe,hd,hn] with x hx hd hn |
| refine ⟨hx.1,fun i => ?_⟩ |
| have hne : a*(sourceCollarTime (WithLp.ofLp x)-b)≠0 := mul_ne_zero ha (sub_ne_zero.mpr hn) |
| have hχd := hχ.differentiable (by simp) (WithLp.ofLp x) |
| have hD := differentiable_cut_positive hχd hd hne κ |
| let E : R3 ≃L[ℝ] (Fin 3 → ℝ) := PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin 3 => ℝ) |
| have hE := hD.hasFDerivAt.comp x E.hasFDerivAt |
| simp only [Function.comp_def] at hE |
| rw [hx.2 i,hE.differentiableAt.lineDeriv_eq_fderiv,hE.fderiv,ContinuousLinearMap.comp_apply] |
| have hi : E.toContinuousLinearMap (EuclideanSpace.single i 1)=Pi.single i 1 := rfl |
| rw [hi] |
| exact fderiv_cut_positive hχd hd hne κ _ |
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| end ScalarConductivity |
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| end |
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| end OAI |
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