File size: 3,746 Bytes
9eeb9c7 | 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 | import OAI.Analysis.HotSpots.TraceRank
namespace OAI
section ActualNoncritical
noncomputable section
open Set MeasureTheory Filter Metric Function Module
open scoped ENNReal NNReal Topology InnerProductSpace
namespace StrictHotSpots.Conformal.ClosedDiskChart
open PlaneGreen DiskH10 HilbertMultipliers
variable {Ω : Set Plane} (c : ClosedDiskChart Ω)
theorem noncritical_at_disk (hΩ : AdmissibleDomain Ω)
{u : Plane → ℝ} (hu : InFirstNeumannEigenspace Ω u) (hne : ∃ x ∈ Ω, u x ≠ 0)
(p : disk) : gradient u (c.F p) ≠ 0 := by
intro hz
let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1
have hμ := firstPositiveNeumannValue_pos_of_eigenfunction hΩ hu hne
obtain ⟨d,hd,hd1,hsub⟩ := c.subcritical hΩ.2.2.1 hΩ.1 hμ
obtain ⟨H,hN,hI,hC,hS,b,B,hinj,hb,hid⟩ := full_boundary_hilbert_embedding
c.densityBound_ne_top (c.density_bound hμ.le) hd.le hd1 hsub p
let := hN
let := hI
let := hC
let := hS
obtain ⟨I,e,_he⟩ := exists_hilbertBasis ℝ H
let : Countable I := orthonormal_countable e.orthonormal
have hnull := c.coordinate_null_trace hΩ hu hne p hz e hb hid
have hY : ContinuousOn (closedGradient Ω u) (closure Ω) :=
closedGradient_continuous hΩ.2.1 hΩ.2.2.2.2 hu.1
have he : EqOn (gradient u) (closedGradient Ω u) Ω := fun _ hx => (closedGradient_eq hΩ.2.1 hx).symm
let J : closedEigenSpace Ω hΩ.2.1 hΩ.2.2.1 :=
⟨closedEigenJet Ω u (closedGradient Ω u),u,closedGradient Ω u,hu.interior,hu.1.continuousOn,hY,he,rfl⟩
let T := closedTraceSpace c hΩ
let g : T := ⟨closedGradientTrace c hΩ J,J,rfl⟩
have hgv (s : Circle) : g.val s = closedGradient Ω u (c.F (circleBoundary s)) := rfl
obtain ⟨hf,hr⟩ := closedTraceSpace_rank c hΩ
let : Module.Finite ℝ T := hf
have hmem (i : I) : (fun s => inner ℝ (e i) (b (circleBoundary s)) • g.val s) ∈ T := by
obtain ⟨v,Y,hv,hvc,hY,hvY,ht⟩ := hnull i
let K : closedEigenSpace Ω hΩ.2.1 hΩ.2.2.1 :=
⟨closedEigenJet Ω v Y,v,Y,hv,hvc,hY,hvY,rfl⟩
refine ⟨K,?_⟩
funext s
exact ht (circleBoundary s)
have h0 := multiplier_contradiction e (hb.continuous.comp circleBoundary.continuous)
(hinj.comp circleBoundary.injective) T hr (fun U hU hne => closedTraceSpace_local_rank c hΩ hU hne)
g ((c.closedGradient_trace_continuous hΩ hu).comp circleBoundary.continuous) hmem
apply c.boundary_gradient_not_zero hΩ hu.interior hu.1.continuousOn hY he hne
intro x hx
have hh := congrArg (fun f : T => f.val (c.circleBoundaryHomeomorph.symm ⟨x,hx⟩)) h0
change closedGradient Ω u
(c.circleBoundaryHomeomorph (c.circleBoundaryHomeomorph.symm ⟨x,hx⟩)) = 0 at hh
simpa only [Homeomorph.apply_symm_apply] using hh
include c
theorem noncritical (hΩ : AdmissibleDomain Ω)
{u : Plane → ℝ} (hu : InFirstNeumannEigenspace Ω u) (hne : ∃ x ∈ Ω, u x ≠ 0) :
∀ x ∈ Ω, gradient u x ≠ 0 := by
intro x hx
have hp : c.G x ∈ disk := by
have hh := c.diskMap.map_target hx
change c.diskMap.symm x ∈ disk at hh
rwa [c.diskMap_inverse_agrees hx] at hh
have hh := c.noncritical_at_disk hΩ hu hne ⟨c.G x,hp⟩
simpa only [c.right_inverse x (subset_closure hx)] using hh
end StrictHotSpots.Conformal.ClosedDiskChart
namespace StrictHotSpots
theorem main_theorem (Ω : Set Plane) (hΩ : AdmissibleDomain Ω)
(u : Plane → ℝ) (hu : InFirstNeumannEigenspace Ω u)
(hne : ∃ x ∈ Ω, u x ≠ 0) : MainConclusion Ω u := by
obtain ⟨c⟩ := Conformal.exists_closed_disk_chart hΩ
exact strict_boundary_extrema_of_noncritical hΩ.1 hΩ.2.1 hΩ.2.2.1 hu.1.continuousOn
(c.noncritical hΩ hu hne)
end StrictHotSpots
end
end ActualNoncritical
end OAI
|