| import OAI.Analysis.CircleDomains.Modulus.CompactBarrierProjection |
| import OAI.Analysis.CircleDomains.Topology.AnchoredCurveImages |
| import OAI.Analysis.CircleDomains.Sobolev.PhysicalRoundIntegrals |
|
|
| namespace OAI |
|
|
| /-! Concrete compact global barriers surrounding a fixed compact set. -/ |
| open Set Function Filter MeasureTheory Metric |
| open scoped Topology ENNReal |
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| structure CompactBarrierData (K : Set EuclideanPlane) (L η : ℝ) where |
| region : Set EuclideanPlane |
| open_region : IsOpen region |
| compact_closure : IsCompact (closure region) |
| contains : K ⊆ closure region |
| value : EuclideanPlane → ℝ |
| witness : DeGiorgi.MemW1pWitness 2 value univ |
| continuous : Continuous value |
| support : tsupport value ⊆ ball 0 L |
| one : ∀ x ∈ frontier region, value x = 1 |
| energy : (∫ x, ‖witness.weakGrad x‖^2) < η^2 |
|
|
| theorem compact_barrier_data_subsingleton |
| {K : Set EuclideanPlane} (hK : IsCompact K) (hc : IsConnected K) |
| {L : ℝ} (hL : 0 < L) |
| (D : ∀ s : ℕ, CompactBarrierData K L (1/((s:ℝ)+1))) : K.Subsingleton := by |
| have hE : Tendsto (fun s => ∫ x, ‖(D s).witness.weakGrad x‖^2) atTop (𝓝 0) := by |
| have ht := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).pow 2 |
| simp only [zero_pow (by decide : 2 ≠ 0)] at ht |
| exact squeeze_zero (fun s => integral_nonneg (fun x => sq_nonneg _)) |
| (fun s => (D s).energy.le) ht |
| apply physical_subsingleton_of_zero_projection_measure hK hc |
| intro i |
| exact projection_measure_zero_of_compact_barriers hK |
| (fun s => (D s).open_region) (fun s => (D s).compact_closure) |
| (fun s => (D s).witness) (fun s => (D s).continuous) hL |
| (fun s => (D s).support) hE |
| (Eventually.of_forall (fun s => ⟨(D s).contains,(D s).one⟩)) i |
|
|
| theorem exists_compact_barrier_of_curve_images |
| {K : Set ℂ} (hK : IsCompact K) (J : ℂ ≃ₜ ℂ) (Jn : ℕ → ℂ ≃ₜ ℂ) |
| (γ : Circle → ℂ) (γn : ℕ → Circle → ℂ) |
| (hγ : Continuous γ) (hfront : range γ = frontier (jordanInterior J)) |
| (hfrontn : ∀ n, range (γn n) = frontier (jordanInterior (Jn n))) |
| (hlim : TendstoUniformly γn γ atTop) |
| {q : ℂ} {qn : ℕ → ℂ} (hq : q ∈ jordanInterior J) |
| (hqlim : Tendsto qn atTop (𝓝 q)) (hqn : ∀ n, qn n ∈ jordanInterior (Jn n)) |
| (hKJ : K ⊆ jordanInterior J) {L η : ℝ} |
| (hbarrier : ∀ᶠ n in atTop, |
| ∃ (v : EuclideanPlane → ℝ) (W : DeGiorgi.MemW1pWitness 2 v univ), |
| Continuous v ∧ tsupport v ⊆ ball 0 L ∧ |
| (∀ t, v (complexPlaneCoordinates.symm (γn n t)) = 1) ∧ |
| (∫ x, ‖W.weakGrad x‖^2) < η^2) : |
| Nonempty (CompactBarrierData (complexPlaneCoordinates.symm '' K) L η) := by |
| have hret := eventually_compact_subset_jordanInterior_of_anchor J Jn γ γn hγ hfront |
| (Eventually.of_forall hfrontn) hlim hq hqlim (Eventually.of_forall hqn) hK hKJ |
| obtain ⟨n,hn,hb⟩ := (hret.and hbarrier).exists |
| obtain ⟨v,W,hv,hs,hone,hE⟩ := hb |
| let U := complexPlaneCoordinates ⁻¹' jordanInterior (Jn n) |
| have hUc : IsCompact (closure U) := by |
| have he : closure U = complexPlaneCoordinates.symm '' closure (jordanInterior (Jn n)) := by |
| calc |
| closure U = complexPlaneCoordinates ⁻¹' closure (jordanInterior (Jn n)) := |
| (complexPlaneCoordinates.toHomeomorph.preimage_closure _).symm |
| _ = _ := by |
| ext x |
| constructor |
| · intro hx |
| exact ⟨complexPlaneCoordinates x,hx,complexPlaneCoordinates.symm_apply_apply x⟩ |
| · rintro ⟨z,hz,rfl⟩ |
| simpa only [mem_preimage,complexPlaneCoordinates.apply_symm_apply] using hz |
| rw [he] |
| exact (isCompact_closure_jordanInterior (Jn n)).image complexPlaneCoordinates.symm.continuous |
| refine ⟨⟨U,(jordanInterior_isOpen _).preimage complexPlaneCoordinates.continuous,hUc, |
| ?_,v,W,hv,hs,?_,hE⟩⟩ |
| · rintro x ⟨z,hz,rfl⟩ |
| apply subset_closure |
| change complexPlaneCoordinates (complexPlaneCoordinates.symm z) ∈ jordanInterior (Jn n) |
| rw [complexPlaneCoordinates.apply_symm_apply] |
| exact hn hz |
| · intro x hx |
| have hx' : complexPlaneCoordinates x ∈ frontier (jordanInterior (Jn n)) := by |
| change x ∈ complexPlaneCoordinates ⁻¹' frontier (jordanInterior (Jn n)) |
| have he : complexPlaneCoordinates ⁻¹' frontier (jordanInterior (Jn n)) = frontier U := |
| complexPlaneCoordinates.toHomeomorph.preimage_frontier _ |
| rwa [he] |
| obtain ⟨t,ht⟩ := (hfrontn n).symm ▸ hx' |
| have h := hone t |
| rwa [ht,complexPlaneCoordinates.symm_apply_apply] at h |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
