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110f4f3 | 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 | import OAI.Analysis.BernoulliCone.SmoothLocalizedExtension
namespace OAI
section
noncomputable section
open Set Filter InnerProductSpace
open scoped Topology ContDiff
namespace Bernoulli.DeSilvaJerison
def exteriorFlux (p:Space 7×ℝ):Space 7:=κ • gradient Zsigned p.1+
((p.2-κ*Zsigned p.1)/KSup p.1) • gradient KSup p.1
lemma exteriorFlux_contDiffAt {p:Space 7×ℝ} (hq:0<Cylindrical.radiusSq axis p.1) :
ContDiffAt ℝ ∞ exteriorFlux p := by
obtain ⟨hx,ha⟩:=radiusSq_pos_data hq
have hZ:ContDiffAt ℝ ∞ Zsigned p.1:=signed_contDiffAt hx ha
have hK:ContDiffAt ℝ ∞ KSup p.1:=KSup_contDiffAt hx ha
have hgZ:ContDiffAt ℝ ∞ (gradient Zsigned) p.1:=
(toDual ℝ (Space 7)).symm.contDiff.contDiffAt.comp p.1 (hZ.fderiv_right (by simp))
have hgK:ContDiffAt ℝ ∞ (gradient KSup) p.1:=
(toDual ℝ (Space 7)).symm.contDiff.contDiffAt.comp p.1 (hK.fderiv_right (by simp))
exact ((hgZ.comp p contDiffAt_fst).const_smul κ).add
(((contDiffAt_snd.sub (contDiffAt_const.mul (hZ.comp p contDiffAt_fst))).div
(hK.comp p contDiffAt_fst) (KSup_pos hx ha).ne').smul (hgK.comp p contDiffAt_fst))
lemma upperFlux_eq_exterior {p:Space 7×ℝ} (hp:U p.1<p.2)
(hq:0<Cylindrical.radiusSq axis p.1) (ht:p.2<seamHeight p.1) :
upperFlux p=exteriorFlux p := by
have hqn:¬Cylindrical.radiusSq axis ((upperParameter p.1 p.2)⁻¹ • p.1)≤1:=
fun h=>not_lt.mpr ((upperParameter_seam_iff hp hq).mp h) ht
have hqs:(upperParameter p.1 p.2)^2<Cylindrical.radiusSq axis p.1:=by
have hl:0<upperParameter p.1 p.2:=(upperParameter_spec hp).1
have hh:1<Cylindrical.radiusSq axis ((upperParameter p.1 p.2)⁻¹ • p.1):=not_le.mp hqn
rw [Cylindrical.radiusSq_smul,inv_pow,←div_eq_inv_mul,lt_div_iff₀ (sq_pos_of_pos hl),one_mul] at hh
exact hh
rw [upperFlux_eq_calibration hp (ne_of_gt (not_le.mp hqn)),upperCalibration_outer_affine hp hqs]
rfl
/-- The actual two branch fields have global C1 representatives on any compact
positive-height tube away from the axis. Only the cap-side field is extended
from a one-sided region; the full discontinuous flux is never declared smooth. -/
theorem compact_branch_extensions {K:Set (Space 7×ℝ)} (hK:IsCompact K)
(hp:∀p∈K,U p.1<p.2) (hq:∀p∈K,0<Cylindrical.radiusSq axis p.1) :
∃A B:Space 7×ℝ→Space 7,ContDiff ℝ 1 A ∧ ContDiff ℝ 1 B ∧
EqOn A exteriorFlux K ∧
∀p∈K,seamHeight p.1≤p.2 → B p=upperFlux p := by
let D:Set (Space 7×ℝ):={p|0<Cylindrical.radiusSq axis p.1}
have hD:IsOpen D:=isOpen_lt continuous_const
((Cylindrical.radiusSq_contDiff axis (n:=1)).continuous.comp continuous_fst)
have hKD:K⊆D:=hq
obtain ⟨A,hA,hAe⟩:=LocalExtension.exists_contDiff_extension (n:=1) hK hD hKD
(fun p hpd=>(exteriorFlux_contDiffAt hpd).of_le (by simp) |>.contDiffWithinAt)
let S:Set (Space 7×ℝ):={p∈K|seamHeight p.1≤p.2}
have hσ:ContinuousOn (fun p:Space 7×ℝ=>seamHeight p.1) K:=by
intro p hpk
exact ((seamHeight_contDiffAt (hq p hpk)).continuousAt.comp continuous_fst.continuousAt).continuousWithinAt
have hS:IsClosed S:=hK.isClosed.isClosed_le hσ continuousOn_snd
obtain ⟨B,hB,hBe⟩:=LocalExtension.exists_contDiff_one_of_closed_germs hS (F:=upperFlux) (by
intro p hps
have hqp:=((upperParameter_seam_iff (hp p hps.1) (hq p hps.1)).mpr hps.2)
obtain ⟨C,hC,hCp,hCe⟩:=upperFlux_cap_local_extension (hp p hps.1) hqp
refine ⟨C,hC.of_le (by simp),?_⟩
filter_upwards [hCe] with y hy hys
exact hy (hp y hys.1) ((upperParameter_seam_iff (hp y hys.1) (hq y hys.1)).mpr hys.2))
refine ⟨A,B,hA,hB,?_,?_⟩
· intro p hpk
exact (hAe.filter_mono (nhds_le_nhdsSet hpk)).self_of_nhds
· intro p hpk hpt
exact hBe ⟨hpk,hpt⟩
end Bernoulli.DeSilvaJerison
end
end
end OAI
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