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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 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | 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
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