| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Calculus.PositiveExtension |
| import OAI.Geometry.CAT0Fillings.Calculus.Truncation |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
|
| namespace CAT0Fillings.ChartGeometry |
| open ClosedCalculus |
|
|
| lemma clamp_gradient_scalar {E : Type*} [Zero E] {t ε N : ℝ} (hεN : ε < N) (v : E) : |
| (if ε < min t N then (if t < N then v else 0) else 0) = |
| (if ε < t ∧ t < N then v else 0) := by |
| by_cases he : ε < t <;> by_cases ht : t < N <;> simp [lt_min_iff,he,ht,hεN] |
| 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_clamp (hz : IsCycle T) (P : q.Sobolev) {ε N : ℝ} (hεN : ε < N) : |
| ∃ Q : q.Sobolev, |
| (q.inclusion Q : X → ℝ) =ᵐ[MassMeasure.currentMassMeasure hT] |
| (fun x => max (min (q.inclusion P x) N) ε) ∧ |
| (q.closedGradient Q : _ → _) =ᵐ[q.atlasMeasure] |
| (fun w => if ε < q.inclusion P (q.atlasParam w) ∧ q.inclusion P (q.atlasParam w) < N |
| then q.closedGradient P w else 0) := by |
| obtain ⟨U,hU,hUG⟩ := q.closed_min_const hz P N |
| obtain ⟨V,hV,hVG⟩ := q.closed_max_const hz U ε |
| refine ⟨V,?_,?_⟩ |
| · filter_upwards [hU,hV] with x hu hv |
| rw [hv,hu] |
| · have hau := q.atlas_preserving.quasiMeasurePreserving.ae |
| (p := fun x : X => q.inclusion U x = min (q.inclusion P x) N) hU |
| filter_upwards [hau,hUG,hVG] with w hu hug hvg |
| rw [hvg,hu,hug] |
| exact clamp_gradient_scalar hεN _ |
|
|
| lemma closed_positive_bivariate {r : X → ℝ} {K : ℝ≥0} (hr : LipschitzWith K r) |
| {R : ℝ} (hrange : ∀ x, 0 ≤ r x ∧ r x ≤ R) |
| {F : ℝ × ℝ → ℝ} (hF : ContDiffOn ℝ 1 F {z | 0 < z.2}) |
| (P : q.Sobolev) {ε N : ℝ} (he : 0 < ε) (hN : ε ≤ N) |
| (hb : ∀ᵐ x ∂MassMeasure.currentMassMeasure hT, q.inclusion P x ∈ Icc ε N) : |
| ∃ 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 ⟨G,hG,heq⟩ := positive_strip_extension hF he hN |
| have hbound : ∀ᵐ x ∂MassMeasure.currentMassMeasure hT, ‖(r x,q.inclusion P x)‖ ≤ max R N := by |
| filter_upwards [hb] with x hx |
| rw [Prod.norm_def,Real.norm_eq_abs,Real.norm_eq_abs, |
| abs_of_nonneg (hrange x).1,abs_of_nonneg (he.le.trans hx.1)] |
| exact max_le_max (hrange x).2 hx.2 |
| obtain ⟨Q,hQ,hQG⟩ := q.closed_bounded_bivariate hr hG P hbound |
| refine ⟨Q,?_,?_⟩ |
| · filter_upwards [hQ,hb] with x hx hb |
| rw [hx,(heq _ hb).eq_of_nhds] |
| · have ha := q.atlas_preserving.quasiMeasurePreserving.ae |
| (p := fun x : X => q.inclusion P x ∈ Icc ε N) hb |
| filter_upwards [hQG,ha] with w hw ha |
| rw [hw,(heq _ ha).fderiv_eq] |
|
|
| lemma closed_positive_clamped_bivariate (hz : IsCycle T) |
| {r : X → ℝ} {K : ℝ≥0} (hr : LipschitzWith K r) |
| {R : ℝ} (hrange : ∀ x, 0 ≤ r x ∧ r x ≤ R) |
| {F : ℝ × ℝ → ℝ} (hF : ContDiffOn ℝ 1 F {z | 0 < z.2}) |
| (P : q.Sobolev) {ε N : ℝ} (he : 0 < ε) (hN : ε < N) : |
| ∃ Q : q.Sobolev, |
| (q.inclusion Q : X → ℝ) =ᵐ[MassMeasure.currentMassMeasure hT] |
| (fun x => F (r x,max (min (q.inclusion P x) N) ε)) ∧ |
| (q.closedGradient Q : _ → _) =ᵐ[q.atlasMeasure] |
| (fun w => fderiv ℝ F (r (q.atlasParam w),max (min (q.inclusion P (q.atlasParam w)) N) ε) (1,0) • q.gradient hr w + |
| fderiv ℝ F (r (q.atlasParam w),max (min (q.inclusion P (q.atlasParam w)) N) ε) (0,1) • |
| (if ε < q.inclusion P (q.atlasParam w) ∧ q.inclusion P (q.atlasParam w) < N |
| then q.closedGradient P w else 0)) := by |
| obtain ⟨U,hU,hUG⟩ := q.closed_clamp hz P hN |
| have hbound : ∀ᵐ x ∂MassMeasure.currentMassMeasure hT, q.inclusion U x ∈ Icc ε N := by |
| filter_upwards [hU] with x hx |
| rw [hx] |
| exact ⟨le_max_right _ _,max_le (min_le_right _ _) hN.le⟩ |
| obtain ⟨Q,hQ,hG⟩ := q.closed_positive_bivariate hr hrange hF U he hN.le hbound |
| refine ⟨Q,?_,?_⟩ |
| · filter_upwards [hQ,hU] with x hq hu |
| rw [hq,hu] |
| · have hau := q.atlas_preserving.quasiMeasurePreserving.ae |
| (p := fun x : X => q.inclusion U x = max (min (q.inclusion P x) N) ε) hU |
| filter_upwards [hG,hau,hUG] with w hq hu hug |
| simp only [hq,hu,hug] |
| end CAT0Fillings.ChartGeometry |
| end |
|
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| end OAI |
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