| 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 | |