| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Calculus.ClosedBivariate |
|
|
| namespace OAI |
|
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| section |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
|
| namespace CAT0Fillings.ChartGeometry |
| open ClosedCalculus |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X (k+1)} |
| {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
|
|
| lemma closed_compact_bivariate {r : X → ℝ} {K : ℝ≥0} (hr : LipschitzWith K r) |
| {F : ℝ × ℝ → ℝ} (hF : ContDiff ℝ 1 F) (hs : HasCompactSupport F) (P : q.Sobolev) : |
| ∃ Q : q.Sobolev, |
| (q.inclusion Q : X → ℝ) =ᵐ[MassMeasure.currentMassMeasure hT] |
| (fun x => F (r x,q.inclusion P x)) ∧ |
| (q.closedGradient Q : _ → _) =ᵐ[q.atlasMeasure] |
| (fun w => fderiv ℝ F (r (q.atlasParam w),q.inclusion P (q.atlasParam w)) (1,0) • q.gradient hr w + |
| fderiv ℝ F (r (q.atlasParam w),q.inclusion P (q.atlasParam w)) (0,1) • q.closedGradient P w) := by |
| obtain ⟨B,hB⟩ := ContDiff.lipschitzWith_of_hasCompactSupport hs hF one_ne_zero |
| exact q.closed_bivariate hr hB (hasFDerivAt_binary (hF.differentiable one_ne_zero)) |
| ((hF.continuous_fderiv one_ne_zero).clm_apply continuous_const) |
| ((hF.continuous_fderiv one_ne_zero).clm_apply continuous_const) |
| B.coe_nonneg (fun t => (binary_coeff_bound hB t).1) (fun t => (binary_coeff_bound hB t).2) P |
|
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| lemma closed_bounded_bivariate {r : X → ℝ} {K : ℝ≥0} (hr : LipschitzWith K r) |
| {F : ℝ × ℝ → ℝ} (hF : ContDiff ℝ 1 F) (P : q.Sobolev) {M : ℝ} |
| (hb : ∀ᵐ x ∂MassMeasure.currentMassMeasure hT, ‖(r x,q.inclusion P x)‖ ≤ M) : |
| ∃ Q : q.Sobolev, |
| (q.inclusion Q : X → ℝ) =ᵐ[MassMeasure.currentMassMeasure hT] |
| (fun x => F (r x,q.inclusion P x)) ∧ |
| (q.closedGradient Q : _ → _) =ᵐ[q.atlasMeasure] |
| (fun w => fderiv ℝ F (r (q.atlasParam w),q.inclusion P (q.atlasParam w)) (1,0) • q.gradient hr w + |
| fderiv ℝ F (r (q.atlasParam w),q.inclusion P (q.atlasParam w)) (0,1) • q.closedGradient P w) := by |
| let η : ContDiffBump (0 : ℝ × ℝ) := |
| { rIn := max M 0+1, rOut := max M 0+2, |
| rIn_pos := by positivity, rIn_lt_rOut := by linarith } |
| let G (t : ℝ × ℝ) := η t * F t |
| have hG : ContDiff ℝ 1 G := η.contDiff.mul hF |
| have he (t : ℝ × ℝ) (ht : ‖t‖ ≤ M) : G =ᶠ[𝓝 t] F := by |
| have hball : t ∈ Metric.ball 0 η.rIn := by |
| simp only [Metric.mem_ball,dist_zero_right] |
| dsimp [η] |
| exact lt_of_le_of_lt (ht.trans (le_max_left _ _)) (lt_add_one _) |
| filter_upwards [η.eventuallyEq_one_of_mem_ball hball] with z hz |
| dsimp [G] |
| rw [hz,Pi.one_apply,one_mul] |
| obtain ⟨Q,hQ,hQG⟩ := q.closed_compact_bivariate hr hG (η.hasCompactSupport.mul_right) P |
| refine ⟨Q,?_,?_⟩ |
| · filter_upwards [hQ,hb] with x hx hb |
| rw [hx,(he _ hb).eq_of_nhds] |
| · have ha : ∀ᵐ w ∂q.atlasMeasure, ‖(r (q.atlasParam w),q.inclusion P (q.atlasParam w))‖ ≤ M := |
| q.atlas_preserving.quasiMeasurePreserving.ae (p := fun x : X => ‖(r x,q.inclusion P x)‖ ≤ M) hb |
| filter_upwards [hQG,ha] with w hw ha |
| rw [hw,(he _ ha).fderiv_eq] |
|
|
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| section |
|
|
| open Set Filter |
| open scoped Topology |
|
|
| namespace CAT0Fillings.ClosedCalculus |
|
|
| lemma positive_strip_extension {F : ℝ × ℝ → ℝ} |
| (hF : ContDiffOn ℝ 1 F {z | 0 < z.2}) {ε N : ℝ} |
| (he : 0 < ε) (hN : ε ≤ N) : |
| ∃ G : ℝ × ℝ → ℝ, ContDiff ℝ 1 G ∧ |
| ∀ z : ℝ × ℝ, z.2 ∈ Icc ε N → G =ᶠ[𝓝 z] F := by |
| let c : ℝ := (N+ε)/2 |
| let η : ContDiffBump c := |
| { rIn := (N-ε)/2+ε/4, rOut := (N-ε)/2+ε/2, |
| rIn_pos := by nlinarith, rIn_lt_rOut := by nlinarith } |
| let G (z : ℝ × ℝ) := η z.2 * F z |
| have hs (z : ℝ × ℝ) (hz : z.2 ≤ 0) : G =ᶠ[𝓝 z] 0 := by |
| have ho : z.2 ∉ Metric.closedBall c η.rOut := by |
| rw [Metric.mem_closedBall,Real.dist_eq] |
| dsimp [c,η] |
| have hneg : z.2-(N+ε)/2 < 0 := by nlinarith |
| rw [abs_of_neg hneg] |
| nlinarith |
| have ht : z.2 ∉ tsupport η := by simpa only [η.tsupport_eq] using ho |
| have hzero := notMem_tsupport_iff_eventuallyEq.mp ht |
| filter_upwards [hzero.comp_tendsto continuous_snd.continuousAt] with w hw |
| change η w.2 * F w = 0 |
| change η w.2 = 0 at hw |
| rw [hw,zero_mul] |
| have hG : ContDiff ℝ 1 G := by |
| rw [contDiff_iff_contDiffAt] |
| intro z |
| by_cases hz : 0 < z.2 |
| · exact ((η.contDiff.comp contDiff_snd).contDiffAt).mul |
| (hF.contDiffAt ((isOpen_lt continuous_const continuous_snd).mem_nhds hz)) |
| · exact contDiffAt_const.congr_of_eventuallyEq (hs z (le_of_not_gt hz)) |
| refine ⟨G,hG,?_⟩ |
| intro z hz |
| have hb : z.2 ∈ Metric.ball c η.rIn := by |
| rw [Metric.mem_ball,Real.dist_eq,abs_lt] |
| dsimp [c,η] |
| constructor <;> nlinarith [hz.1,hz.2] |
| have hone := η.eventuallyEq_one_of_mem_ball hb |
| filter_upwards [hone.comp_tendsto continuous_snd.continuousAt] with w hw |
| dsimp [G] |
| change η w.2 = 1 at hw |
| rw [hw,one_mul] |
| end CAT0Fillings.ClosedCalculus |
| end |
|
|
| end OAI |
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