| import OAI.Analysis.BernoulliCone.BoundedCalibration |
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| namespace OAI |
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| section |
| noncomputable section |
| open Set Filter InnerProductSpace |
| open scoped Topology ContDiff |
| namespace Bernoulli.DeSilvaJerison |
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|
| lemma upperFamily_eq_cap {x:Space 7} {l:ℝ} (hl:0<l) |
| (hp:0<upperFamily l x) (hq:Cylindrical.radiusSq axis (l⁻¹ • x)≤1) : |
| upperFamily l x=κ*l*smoothCap (l⁻¹ • x) := by |
| have hh:0 < max (upperSigned (l⁻¹ • x)) 0:= |
| pos_of_mul_pos_right hp (mul_nonneg κ_pos.le hl.le) |
| have hs:0<upperSigned (l⁻¹ • x):=by simpa using hh |
| rw [upperFamily,max_eq_left hs.le,upperSigned_cap hq] |
|
|
| /-- Smooth signed extensions of the inverse cap parameter through the actual |
| cap/exterior interface, obtained by IFT with local uniqueness. -/ |
| lemma capParameter_local_extension {p:Space 7×ℝ} (hp:U p.1<p.2) |
| (hq:Cylindrical.radiusSq axis ((upperParameter p.1 p.2)⁻¹ • p.1)≤1) : |
| ∃ψ:Space 7×ℝ→ℝ,ContDiffAt ℝ ∞ ψ p ∧ ψ p=upperParameter p.1 p.2 ∧ |
| ∀ᶠ z in 𝓝 p, U z.1<z.2 → |
| Cylindrical.radiusSq axis ((upperParameter z.1 z.2)⁻¹ • z.1)≤1 → |
| ψ z=upperParameter z.1 z.2 := by |
| let l:=upperParameter p.1 p.2 |
| have hl:0<l:=(upperParameter_spec hp).1 |
| have he:upperFamily l p.1=p.2:=(upperParameter_spec hp).2 |
| have hpos:0<upperFamily l p.1:=he.symm ▸ (U_nonneg p.1).trans_lt hp |
| have hc:ContDiffAt ℝ ∞ smoothCap (l⁻¹ • p.1):=smoothCap_contDiffAt hq |
| have hpc:0<smoothCap (l⁻¹ • p.1):=by |
| rw [←upperSigned_cap hq] |
| exact upperParameter_signed_pos hp |
| let F:(Space 7×ℝ)×ℝ→ℝ:=fun z=>κ*z.2*smoothCap (z.2⁻¹ • z.1.1)-z.1.2 |
| have hi:ContDiffAt ℝ ∞ (fun z:(Space 7×ℝ)×ℝ=>z.2⁻¹ • z.1.1) (p,l):= |
| (contDiffAt_snd.inv hl.ne').smul (contDiffAt_fst.fst) |
| have hF:ContDiffAt ℝ ∞ F (p,l):= |
| ((contDiffAt_const.mul contDiffAt_snd).mul (hc.comp (p,l) hi)).sub contDiffAt_fst.snd |
| let ρ:=κ*(smoothCap (l⁻¹ • p.1)-inner ℝ (l⁻¹ • p.1) (gradient smoothCap (l⁻¹ • p.1))) |
| have hρ:0<ρ:=mul_pos κ_pos (smoothCap_radial_defect_pos hq hpc) |
| have hd:HasDerivAt (fun s:ℝ=>F (p,s)) ρ l := by |
| have hh:=((Foliation.hasDerivAt_dilate p.1 hl.ne' |
| (hc.differentiableAt (by simp))).const_mul κ).sub_const p.2 |
| simpa only [F,mul_assoc] using hh |
| obtain ⟨ψ,hψ,hψ0,hψeq,huniq⟩:=Foliation.scalar_implicit_unique hF hd hρ.ne' |
| refine ⟨ψ,hψ,hψ0,?_⟩ |
| have hF0:F (p,l)=0:=by |
| dsimp [F] |
| rw [←upperFamily_eq_cap hl hpos hq,he,sub_self] |
| have ht:Tendsto (fun z:Space 7×ℝ=>(z,upperParameter z.1 z.2)) (𝓝 p) (𝓝 (p,l)):= |
| tendsto_id.prodMk_nhds (upperParameter_continuousAt hp) |
| filter_upwards [ht.eventually huniq] with z hz hzp hzq |
| apply hz.mp |
| rw [hF0] |
| dsimp [F] |
| have hzpos:0<upperFamily (upperParameter z.1 z.2) z.1:= |
| (upperParameter_spec hzp).2.symm ▸ (U_nonneg z.1).trans_lt hzp |
| rw [←upperFamily_eq_cap (upperParameter_spec hzp).1 hzpos hzq, |
| (upperParameter_spec hzp).2,sub_self] |
|
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| lemma gradient_smoothCap_contDiffAt {x:Space 7} (hq:Cylindrical.radiusSq axis x≤1) : |
| ContDiffAt ℝ ∞ (gradient smoothCap) x := by |
| exact (toDual ℝ (Space 7)).symm.contDiff.contDiffAt.comp x |
| ((smoothCap_contDiffAt hq).fderiv_right (by simp)) |
|
|
| /-- The actual one-sided cap flux admits a smooth germ even at the seam. |
| No regularity of the discontinuous full flux is asserted. -/ |
| theorem upperFlux_cap_local_extension {p:Space 7×ℝ} (hp:U p.1<p.2) |
| (hq:Cylindrical.radiusSq axis ((upperParameter p.1 p.2)⁻¹ • p.1)≤1) : |
| ∃B:Space 7×ℝ→Space 7,ContDiffAt ℝ ∞ B p ∧ B p=upperFlux p ∧ |
| ∀ᶠ z in 𝓝 p,U z.1<z.2 → |
| Cylindrical.radiusSq axis ((upperParameter z.1 z.2)⁻¹ • z.1)≤1 → B z=upperFlux z := by |
| obtain ⟨ψ,hψ,hψ0,he⟩:=capParameter_local_extension hp hq |
| let B:Space 7×ℝ→Space 7:=fun z=>κ • gradient smoothCap ((ψ z)⁻¹ • z.1) |
| have hψp:ψ p≠0:=by rw [hψ0]; exact (upperParameter_spec hp).1.ne' |
| have hn:ContDiffAt ℝ ∞ (fun z:Space 7×ℝ=>(ψ z)⁻¹ • z.1) p:= |
| (hψ.inv hψp).smul contDiffAt_fst |
| have hg:ContDiffAt ℝ ∞ (gradient smoothCap) ((ψ p)⁻¹ • p.1):=by |
| rw [hψ0]; exact gradient_smoothCap_contDiffAt hq |
| refine ⟨B,(hg.comp p hn).const_smul κ,?_,?_⟩ |
| · simp only [B,hψ0,upperFlux,ite_eq_left hp,ite_eq_left hq] |
| · filter_upwards [he] with z hz hzp hzq |
| simp only [B,hz hzp hzq,upperFlux,ite_eq_left hzp,ite_eq_left hzq] |
| end Bernoulli.DeSilvaJerison |
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| end |
| end |
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| end OAI |
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