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a458723 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | import Mathlib
import OAI.Analysis.Conductivity.Branching.AttachedEndGradientL2
import OAI.Analysis.Conductivity.Sobolev.CutSlopeH1
namespace OAI
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 _
end ScalarConductivity
end
end OAI
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