| import OAI.Analysis.BernoulliCone.SmoothLocalizedExtension |
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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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| 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 |
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| /-- 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 |
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| end |
| end |
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| end OAI |
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