File size: 3,622 Bytes
9655094 | 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.Analysis.CircleDomains.Sobolev.SobolevBarrierCollapse
import OAI.Analysis.CircleDomains.Topology.PhysicalFrontierCrossings
namespace OAI
/-! A compact set trapped inside bounded regions whose boundary barriers
have vanishing energy has zero projection measure in both directions. -/
open Set Function Filter MeasureTheory Metric
open scoped Topology ENNReal
noncomputable section
namespace CircleDomainRigidity
theorem abs_planeTransverse_le_norm (i : Fin 2) (z : EuclideanPlane) :
|planeTransverse i z| ≤ ‖z‖ := by
unfold planeTransverse
split <;> simpa only [Real.norm_eq_abs] using PiLp.norm_apply_le z _
theorem projection_measure_zero_of_compact_barriers
{K : Set EuclideanPlane} (hK : IsCompact K)
{U : ℕ → Set EuclideanPlane} (hU : ∀ n, IsOpen (U n))
(hUc : ∀ n, IsCompact (closure (U n)))
{v : ℕ → EuclideanPlane → ℝ} (W : ∀ n, DeGiorgi.MemW1pWitness 2 (v n) univ)
(hv : ∀ n, Continuous (v n)) {L : ℝ} (hL : 0 < L)
(hs : ∀ n, tsupport (v n) ⊆ ball 0 L)
(hE : Tendsto (fun n => ∫ x, ‖(W n).weakGrad x‖^2) atTop (𝓝 0))
(htrap : ∀ᶠ n in atTop, K ⊆ closure (U n) ∧ ∀ x ∈ frontier (U n), v n x = 1)
(i : Fin 2) : volume (planeTransverse i '' K) = 0 := by
have hcross : ∀ᶠ n in atTop, ∀ y ∈ planeTransverse i '' K,
∃ c ∈ Icc (-L) L, v n (physicalCoordinateLine i y c) = 1 := by
filter_upwards [htrap] with n hn
rintro y ⟨z,hz,rfl⟩
obtain ⟨c,hc⟩ := exists_coordinate_frontier (hU n) (hUc n) i ⟨z,hn.1 hz,rfl⟩
have hone := hn.2 _ hc
have hsup : physicalCoordinateLine i (planeTransverse i z) c ∈ tsupport (v n) := by
apply subset_tsupport
change v n _ ≠ 0
rw [hone]
norm_num
have hnorm := mem_ball_zero_iff.mp (hs n hsup)
have hcoord := PiLp.norm_apply_le (physicalCoordinateLine i (planeTransverse i z) c) i
rw [physicalCoordinateLine_coordinate,Real.norm_eq_abs] at hcoord
exact ⟨c,abs_le.mp (hcoord.trans hnorm.le),hone⟩
have hJL : planeTransverse i '' K ⊆ Icc (-L) L := by
obtain ⟨n,hn⟩ := hcross.exists
intro y hy
obtain ⟨c,hc,hvone⟩ := hn y hy
have hsup : physicalCoordinateLine i y c ∈ tsupport (v n) := by
apply subset_tsupport
change v n _ ≠ 0
rw [hvone]
norm_num
have hnorm := mem_ball_zero_iff.mp (hs n hsup)
have hcoord := abs_planeTransverse_le_norm i (physicalCoordinateLine i y c)
rw [physicalCoordinateLine_transverse] at hcoord
exact abs_le.mp (hcoord.trans hnorm.le)
exact measure_zero_of_vanishing_sobolev_barriers W hv hL hs hE i
(hK.image (continuous_planeTransverse i)).measurableSet hJL hcross
theorem physical_subsingleton_of_zero_projection_measure
{K : Set EuclideanPlane} (hK : IsCompact K) (hconn : IsConnected K)
(hz : ∀ i : Fin 2, volume (planeTransverse i '' K) = 0) : K.Subsingleton := by
have hone (i : Fin 2) : (planeTransverse i '' K).Subsingleton := by
have hc := hconn.image _ (continuous_planeTransverse i).continuousOn
have he := eq_Icc_of_connected_compact hc (hK.image (continuous_planeTransverse i))
have hle := nonempty_Icc.mp (he ▸ hc.nonempty)
have hm := hz i
rw [he,Real.volume_Icc,ENNReal.ofReal_eq_zero] at hm
intro a ha b hb
rw [he] at ha hb
linarith [ha.1,ha.2,hb.1,hb.2]
intro x hx y hy
ext i
fin_cases i
· simpa [planeTransverse] using hone 1 (mem_image_of_mem _ hx) (mem_image_of_mem _ hy)
· simpa [planeTransverse] using hone 0 (mem_image_of_mem _ hx) (mem_image_of_mem _ hy)
end CircleDomainRigidity
end
end OAI
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