| import OAI.Geometry.Perimeter.Regularity.ContinuousWithinAtSurfaceGradientOfLocalLevel |
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| namespace OAI |
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| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal ContDiff InnerProductSpace BoundedContinuousFunction |
| namespace RPISingularHarnack |
| variable {n : ℕ} |
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| theorem geometric_equicontinuous {X V : Type*} [PseudoMetricSpace X] [PseudoMetricSpace V] |
| {G : ℕ → X → V} {θ R C : ℝ} (hθ : 0<θ) (hR : 0<R) |
| (hmod : ∀ i j x y, dist x y≤θ^j*R → dist (G i x) (G i y)≤C*(1/2)^j) : |
| Equicontinuous G := by |
| intro x |
| rw [Metric.equicontinuousAt_iff] |
| intro ε hε |
| have ht : Tendsto (fun j : ℕ ↦ C*(1/2:ℝ)^j) atTop (𝓝 0) := by |
| simpa only [mul_zero] using |
| (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ)≤1/2) (by norm_num)).const_mul C |
| obtain ⟨j,hj⟩ := (ht.eventually (gt_mem_nhds hε)).exists |
| refine ⟨θ^j*R,by positivity,fun y hy i ↦ ?_⟩ |
| exact (hmod i j x y (by simpa only [dist_comm] using hy.le)).trans_lt hj |
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| theorem compact_field_subsequence {V : Type*} [NormedAddCommGroup V] [ProperSpace V] |
| {K : Set (Euc n)} (hK : IsCompact K) {G : ℕ → Euc n → V} |
| (hG : ∀ i, ContinuousOn (G i) K) {θ R C B : ℝ} (hθ : 0<θ) (hR : 0<R) |
| (hb : ∀ i x, x∈K → ‖G i x‖≤B) |
| (hmod : ∀ i j x, x∈K → ∀ y, y∈K → dist x y≤θ^j*R → ‖G i x-G i y‖≤C*(1/2)^j) : |
| ∃ W : Euc n → V, ContinuousOn W K ∧ ∃ σ : ℕ → ℕ, StrictMono σ ∧ |
| TendstoUniformlyOn (fun i ↦ G (σ i)) W atTop K := by |
| classical |
| let : CompactSpace K := isCompact_iff_compactSpace.mp hK |
| let F : ℕ → K →ᵇ V := fun i ↦ BoundedContinuousFunction.mkOfCompact |
| ⟨fun x ↦ G i x,continuousOn_iff_continuous_domRestrict.mp (hG i)⟩ |
| have hFeq (i : ℕ) (x : K) : F i x=G i x := rfl |
| have heq : Equicontinuous (fun i (x : K) ↦ F i x) := by |
| apply geometric_equicontinuous (C := C) hθ hR |
| intro i j x y hxy |
| simpa only [hFeq,dist_eq_norm] using hmod i j x x.property y y.property hxy |
| have hc : IsCompact (closure (range F)) := by |
| apply BoundedContinuousFunction.arzela_ascoli (Metric.closedBall (0:V) B) (isCompact_closedBall _ _) |
| · rintro g x ⟨i,rfl⟩ |
| exact Metric.mem_closedBall.mpr (by simpa only [dist_zero_right,hFeq] using hb i x x.property) |
| · intro x |
| rw [Metric.equicontinuousAt_iff] |
| intro ε hε |
| obtain ⟨δ,hδ,hh⟩ := Metric.equicontinuousAt_iff.mp (heq x) ε hε |
| refine ⟨δ,hδ,fun y hy g ↦ ?_⟩ |
| obtain ⟨i,hi⟩ := g.property |
| simpa only [←hi] using hh y hy i |
| obtain ⟨W,hW,σ,hσ,hlim⟩ := hc.tendsto_subseq (fun i ↦ subset_closure (mem_range_self i)) |
| let W' : Euc n → V := fun x ↦ if hx : x∈K then W ⟨x,hx⟩ else 0 |
| have hW'eq (x : K) : W' x=W x := by simp [W',x.property] |
| have hW' : ContinuousOn W' K := by |
| rw [continuousOn_iff_continuous_domRestrict] |
| exact W.continuous.congr (fun x ↦ (hW'eq x).symm) |
| refine ⟨W',hW',σ,hσ,?_⟩ |
| have hu := BoundedContinuousFunction.tendsto_iff_tendstoUniformly.mp hlim |
| rw [Metric.tendstoUniformlyOn_iff] |
| intro ε hε |
| filter_upwards [(Metric.tendstoUniformly_iff.mp hu) ε hε] with i hi x hx |
| have hh : dist (W ⟨x,hx⟩) (G (σ i) x)<ε := hi ⟨x,hx⟩ |
| simpa only [←hW'eq ⟨x,hx⟩] using hh |
| |
| end RPISingularHarnack |
| |
| end |
| |
| end OAI |
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