| 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 |
|
|
