File size: 3,948 Bytes
f61e547 | 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 81 82 | import OAI.Probability.FirstPassage.SphereCharts
import OAI.Probability.FirstPassage.NormBoundary
namespace OAI
namespace PlanarFPP
open Set MeasureTheory
lemma HasC1CurveChart.preimage_isometry (S : Set Plane) (v : Plane)
(e : Plane ≃ₗᵢ[ℝ] Plane) (h : HasC1CurveChart S (e v)) :
HasC1CurveChart (e ⁻¹' S) v := by
obtain ⟨U,hU,hv,hvS,γ,θ,hγ,hθ,hmem,hleft,hright,hreg⟩ := h
refine ⟨e ⁻¹' U,hU.preimage e.continuous,hv,hvS,e.symm ∘ γ,θ ∘ e,?_,?_,?_,?_,?_,?_⟩
· exact e.symm.toContinuousLinearEquiv.contDiff.comp hγ
· exact hθ.comp e.continuous.continuousOn (fun _ hz => hz)
· intro t
simpa only [Function.comp_apply,mem_inter_iff,mem_preimage,e.apply_symm_apply] using hmem t
· intro t
simpa only [Function.comp_apply,e.apply_symm_apply] using hleft t
· intro z hz
change e.symm (γ (θ (e z)))=z
rw [hright (e z) hz,e.symm_apply_apply]
· intro t
obtain ⟨d,hd,hne⟩ := hreg t
refine ⟨e.symm d,e.symm.toContinuousLinearEquiv.hasFDerivAt.comp_hasDerivAt t hd,?_⟩
intro hz
exact hne (e.symm.injective (by simpa using hz))
variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {τ : Edge → Ω → ℝ}
theorem actual_sphere_curve_chart (henv : ExponentialEnvironment P τ) (v : Plane)
(hv : actualSourceTimeConstant v=1) : HasC1CurveChart {z | actualSourceTimeConstant z=1} v := by
let μ := actualSourceTimeConstant
have horizontal (z : Plane) (hz : μ z=1) (hx : z 0 ≠ 0) : HasC1CurveChart {w | μ w=1} z := by
rcases lt_or_gt_of_ne hx with hneg | hpos
· exact sphere_horizontal_chart μ actualSourceTimeConstant_norm (actual_profile_contDiff henv) (-1)
(by norm_num) z hz (by linarith)
· exact sphere_horizontal_chart μ actualSourceTimeConstant_norm (actual_profile_contDiff henv) 1
(by norm_num) z hz (by simpa using hpos)
by_cases hx : v 0 ≠ 0
· exact horizontal v hv hx
· have hvne : v ≠ 0 := by
intro he
have hz := (actualSourceTimeConstant_norm.zero_iff v).mpr he
linarith
have hy : v 1 ≠ 0 := by
intro hy
apply hvne
ext i
fin_cases i <;> simp_all
have hsv : μ (swapPlane v)=1 := by dsimp [μ]; rw [actualSourceTimeConstant_swap henv,hv]
have hs := horizontal (swapPlane v) hsv (by simpa using hy)
have hh := HasC1CurveChart.preimage_isometry {z | μ z=1} v swapPlane hs
have he : swapPlane ⁻¹' {z | μ z=1}={z | μ z=1} := by
ext z
simp only [mem_preimage,mem_ofPred_eq,μ,actualSourceTimeConstant_swap henv]
rwa [he] at hh
theorem actual_boundary_curve (henv : ExponentialEnvironment P τ) (v : Plane)
(hv : v ∈ frontier {z | actualSourceTimeConstant z ≤ 1}) :
HasC1CurveChart (frontier {z | actualSourceTimeConstant z ≤ 1}) v := by
rw [norm_ball_frontier _ actualSourceTimeConstant_norm] at hv ⊢
exact actual_sphere_curve_chart henv v hv
theorem manuscriptMain (Ω : Type) [MeasurableSpace Ω] (P : Measure Ω) (τ : Edge → Ω → ℝ)
(henv : ExponentialEnvironment P τ) :
∃ μ : Plane → ℝ, TimeConstantNorm μ ∧ IsTimeConstant P τ μ ∧
DifferentiableAwayFromOrigin μ ∧ UniqueNormalizedSupports μ ∧ C1UnitSphere μ ∧
(∀ v ∈ frontier {z | μ z ≤ 1}, ∃! L : Set Plane,
∃ g : Plane →L[ℝ] ℝ, g ≠ 0 ∧ (∀ z, μ z ≤ 1 → g z ≤ g v) ∧ L={z | g z=g v}) ∧
(∀ v ∈ frontier {z | μ z ≤ 1}, HasC1CurveChart (frontier {z | μ z ≤ 1}) v) := by
exact ⟨actualSourceTimeConstant,actualSourceTimeConstant_norm,actualSourceTimeConstant_isTimeConstant henv,
actual_differentiable_away henv,
unique_supports_of_differentiable _ actualSourceTimeConstant_norm (actual_differentiable_away henv),
actual_C1_unit_sphere henv,
norm_supporting_line_unique _ actualSourceTimeConstant_norm (actual_differentiable_away henv),
actual_boundary_curve henv⟩
end PlanarFPP
end OAI
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