File size: 3,662 Bytes
2fca478 | 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 | import OAI.Geometry.Perimeter.Regularity.NeighborhoodPlanes
import OAI.Geometry.Perimeter.Regularity.GraphDerivative
import OAI.Geometry.Perimeter.Regularity.UniformDerivativeContinuity
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal ContDiff InnerProductSpace
namespace RPISingularHarnack
variable {n : ℕ}
theorem c1_graph_from_contraction (hn : 1<n) {q M δ : ℝ}
(hq : 0<q) (hq1 : q<1) (hM : 0≤M) (hstep : SlabContractionProperty n q M δ)
{E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
{v : Euc n} (hv : ‖v‖=1) {a : ℝ} (ha : 0<a) (haδ : 4*a<δ)
(ha4 : a<1/4) (haM : 8*M*a≤1/2) (hachord : a*(16*M+1/q)≤1/2)
(hs : FullSlab E v 8 a) :
∃ f : normalPlane v → ℝ,
ContDiffOn ℝ 1 f (Metric.ball 0 (1/4)) ∧
(∀ y, ‖y‖<1/4 → |f y|≤a ∧ (y:Euc n)+f y•v∈perimeterSupport E ∩ Metric.ball 0 1) ∧
(∀ y z, ‖y‖<1/4 → ‖z‖<1/4 → |f y-f z|≤‖y-z‖) ∧
(∀ (y : normalPlane v) (t : ℝ), ‖y‖<1/4 → ‖(y:Euc n)+t•v‖<1 →
((y:Euc n)+t•v∈perimeterSupport E ↔ t=f y)) := by
have hchord : ∀ p∈perimeterSupport E ∩ Metric.ball 0 1,
∀ x∈perimeterSupport E ∩ Metric.ball 0 1,
|inner ℝ v (x-p)|≤(1/2)*‖x-p‖ := by
intro p hp x hx
exact (nearby_chord_bound hq hq1 hM hstep hmin hv ha haδ hs hp hx).trans
(mul_le_mul_of_nonneg_right hachord (norm_nonneg _))
obtain ⟨f,hf,hlip,hiff⟩ := single_graph_from_slab hn hmin hv ha.le ha4 hs hchord
have hwex (y : normalPlane v) : ∃ w : Euc n, ‖y‖<1/4 →
‖w-v‖≤8*M*a ∧ ∀ j x, x∈perimeterSupport E → ‖x-((y:Euc n)+f y•v)‖≤2*q^j →
|inner ℝ w (x-((y:Euc n)+f y•v))|≤4*a*(2*M+1/(4*q))*(1/2)^j*‖x-((y:Euc n)+f y•v)‖ := by
by_cases hy : ‖y‖<1/4
· obtain ⟨w,_,hwd,htan⟩ := nearby_full_support_planes hq hq1 hM hstep hmin hv ha haδ hs (hf y hy).2
exact ⟨w,fun _ ↦ ⟨hwd,htan⟩⟩
· exact ⟨v,fun h ↦ (hy h).elim⟩
choose w hw using hwex
let D : normalPlane v → normalPlane v →L[ℝ] ℝ := fun y ↦ graphTangentLinear v (w y)
let C : ℝ := 16*a*(2*M+1/(4*q))
have hrem : ∀ j y, y∈Metric.ball (0:normalPlane v) (1/4) →
∀ z, z∈Metric.ball (0:normalPlane v) (1/4) → ‖z-y‖≤q^j →
|f z-f y-D y (z-y)|≤C*(1/2)^j*‖z-y‖ := by
intro j y hy z hz hzy
have hy' : ‖y‖<1/4 := by simpa only [Metric.mem_ball,dist_zero_right] using hy
have hz' : ‖z‖<1/4 := by simpa only [Metric.mem_ball,dist_zero_right] using hz
have hh := graph_tangent_remainder hv ((hw y hy').1.trans haM) (hf z hz').2.1
(hlip z y hz' hy') hzy ((hw y hy').2 j) (by positivity : 0≤4*a*(2*M+1/(4*q)))
convert hh using 1
dsimp [C,D]
ring
have hD : ∀ y∈Metric.ball (0:normalPlane v) (1/4), HasFDerivAt f (D y) y := by
intro y hy
exact hasFDerivAt_of_geometric_remainder Metric.isOpen_ball hy hq (fun j z hz hzy ↦ hrem j y hy z hz hzy)
have hDC := continuous_derivative_of_geometric_remainder Metric.isOpen_ball hq
(fun x hx y hy ↦ hlip x y
(by simpa only [Metric.mem_ball,dist_zero_right] using hx)
(by simpa only [Metric.mem_ball,dist_zero_right] using hy)) hrem
refine ⟨f,?_,hf,hlip,hiff⟩
change ContDiffOn ℝ (0+1) f (Metric.ball 0 (1/4))
rw [contDiffOn_succ_iff_hasFDerivWithinAt_of_uniqueDiffOn Metric.isOpen_ball.uniqueDiffOn]
refine ⟨by simp, D,?_,fun y hy ↦ (hD y hy).hasFDerivWithinAt⟩
exact contDiffOn_zero.mpr (fun y hy ↦ (hDC y hy).continuousWithinAt)
end RPISingularHarnack
end
end OAI
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