| import OAI.Geometry.Perimeter.Regularity.HeightCompactness |
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| namespace OAI |
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| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal |
| namespace RPISingularHarnack |
| open RPI.Boundary |
| variable {n : ℕ} |
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| def compactLpExtension (K : Set (Euc n)) |
| (g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n))) : Euc n → ℝ := |
| Function.extend (Subtype.val : K → Euc n) g (fun _ ↦ 0) |
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| @[simp] theorem compactLpExtension_coe (K : Set (Euc n)) |
| (g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n))) (x : K) : |
| compactLpExtension K g x = g x := |
| Subtype.val_injective.extend_apply _ _ x |
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| theorem compactLpExtension_zero (K : Set (Euc n)) |
| (g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n))) {x : Euc n} (hx : x ∉ K) : |
| compactLpExtension K g x = 0 := by |
| apply Function.extend_apply' |
| rintro ⟨y,rfl⟩; exact hx y.property |
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| theorem compactLpExtension_integrable (K : Set (Euc n)) (hK : IsCompact K) |
| (g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n))) : |
| Integrable (compactLpExtension K g) volume := by |
| have hi : IntegrableOn (compactLpExtension K g) K volume := by |
| rw [integrableOn_iff_comap_subtypeVal hK.measurableSet] |
| simpa only [Function.comp_def,compactLpExtension_coe] using |
| (memLp_one_iff_integrable.mp (Lp.memLp g)) |
| exact hi.integrable_of_forall_notMem_eq_zero (fun _ hx ↦ compactLpExtension_zero K g hx) |
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| theorem compactLpExtension_error (K : Set (Euc n)) (hK : IsCompact K) |
| (g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n))) |
| (f : Euc n → ℝ) (hi : Integrable f volume) (hs : Function.support f ⊆ K) : |
| (∫⁻ x, ‖f x-compactLpExtension K g x‖ₑ) = edist (compactFunctionLp K hK f hi) g := by |
| rw [Lp.edist_def,eLpNorm_one_eq_lintegral_enorm |
| ((Lp.aestronglyMeasurable (compactFunctionLp K hK f hi)).sub (Lp.aestronglyMeasurable g))] |
| have hz : Function.support (fun x ↦ ‖f x-compactLpExtension K g x‖ₑ) ⊆ K := by |
| intro x hx |
| by_contra hnot |
| have hf : f x=0 := Function.notMem_support.mp (fun h ↦ hnot (hs h)) |
| simp [hf,compactLpExtension_zero K g hnot] at hx |
| rw [← setLIntegral_eq_of_support_subset hz,← lintegral_subtype_comap hK.measurableSet] |
| apply lintegral_congr_ae |
| filter_upwards [compactFunctionLp_ae K hK f hi] with x hx |
| simp only [compactLpExtension_coe,Pi.sub_apply,hx] |
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| theorem compactLpExtension_tendsto (K : Set (Euc n)) (hK : IsCompact K) |
| (f : ℕ → Euc n → ℝ) (hi : ∀ i, Integrable (f i) volume) |
| (hs : ∀ i, Function.support (f i) ⊆ K) |
| (g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n))) |
| (hlim : Tendsto (fun i ↦ compactFunctionLp K hK (f i) (hi i)) atTop (𝓝 g)) : |
| Tendsto (fun i ↦ ∫⁻ x, ‖f i x-compactLpExtension K g x‖ₑ) atTop (𝓝 0) := by |
| have he (i : ℕ) := compactLpExtension_error K hK g (f i) (hi i) (hs i) |
| simp_rw [he] |
| simpa only [edist_self] using hlim.edist (tendsto_const_nhds : Tendsto (fun _ : ℕ ↦ g) atTop (𝓝 g)) |
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| theorem compact_support_L1_subsequence |
| (K : Set (Euc n)) (hK : IsCompact K) (f : ℕ → Euc n → ℝ) |
| (hm : ∀ i, Measurable (f i)) (hi : ∀ i, Integrable (f i) volume) |
| (hs : ∀ i, Function.support (f i) ⊆ K) |
| (A : ℝ) (hA : 0 ≤ A) (hb : ∀ i, (∫ x, ‖f i x‖) ≤ A) |
| (D : ℝ≥0∞) (hD : D ≠ ⊤) |
| (ht : ∀ i w, (∫⁻ x, ‖f i (x+w)-f i x‖ₑ) ≤ ‖w‖ₑ * D) : |
| ∃ g : Euc n → ℝ, Integrable g volume ∧ Function.support g ⊆ K ∧ |
| ∃ σ : ℕ → ℕ, StrictMono σ ∧ |
| Tendsto (fun i ↦ ∫⁻ x, ‖f (σ i) x-g x‖ₑ) atTop (𝓝 0) := by |
| obtain ⟨g,σ,hσ,hg⟩ := compactFunctionLp_subsequence_of_translation K hK f hm hi A hA hb D hD ht |
| refine ⟨compactLpExtension K g,compactLpExtension_integrable K hK g,?_,σ,hσ,?_⟩ |
| · intro x hx |
| by_contra hnot |
| exact hx (compactLpExtension_zero K g hnot) |
| · exact compactLpExtension_tendsto K hK (fun i ↦ f (σ i)) (fun i ↦ hi (σ i)) |
| (fun i ↦ hs (σ i)) g hg |
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| end RPISingularHarnack |
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| end |
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| end OAI |
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