| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Swept.Current |
| import OAI.Geometry.CAT0Fillings.Minimizers.CriticalEnergy |
| import OAI.Geometry.CAT0Fillings.Calculus.ClosedLevel |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter |
| open scoped Topology NNReal |
|
|
| namespace CAT0Fillings.ClosedCalculus |
|
|
| noncomputable def absApprox (ε t : ℝ) : ℝ := Real.sqrt (t^2+ε^2) |
| noncomputable def absApproxDeriv (ε t : ℝ) : ℝ := t / Real.sqrt (t^2+ε^2) |
|
|
| lemma absApprox_hasDerivAt {ε : ℝ} (hε : 0 < ε) (t : ℝ) : |
| HasDerivAt (absApprox ε) (absApproxDeriv ε t) t := by |
| have hh : 0 < t^2+ε^2 := add_pos_of_nonneg_of_pos (sq_nonneg _) (sq_pos_of_pos hε) |
| have hd := (((hasDerivAt_id t).pow 2).add_const (ε^2)).sqrt hh.ne' |
| convert hd using 1 |
| · rfl |
| · dsimp [absApproxDeriv] |
| norm_num |
| ring |
|
|
| lemma absApproxDeriv_bound {ε : ℝ} (hε : 0 < ε) (t : ℝ) : |absApproxDeriv ε t| ≤ 1 := by |
| have hh : 0 < t^2+ε^2 := add_pos_of_nonneg_of_pos (sq_nonneg _) (sq_pos_of_pos hε) |
| dsimp [absApproxDeriv] |
| rw [abs_div,abs_of_nonneg (Real.sqrt_nonneg _)] |
| apply (div_le_one (Real.sqrt_pos.mpr hh)).mpr |
| rw [←Real.sqrt_sq_eq_abs t] |
| exact Real.sqrt_le_sqrt (le_add_of_nonneg_right (sq_nonneg _)) |
|
|
| lemma absApprox_lipschitz {ε : ℝ} (hε : 0 < ε) : LipschitzWith 1 (absApprox ε) := by |
| apply lipschitzWith_of_nnnorm_deriv_le (fun t => (absApprox_hasDerivAt hε t).differentiableAt) |
| intro t |
| rw [←NNReal.coe_le_coe,coe_nnnorm,NNReal.coe_one,(absApprox_hasDerivAt hε t).deriv,Real.norm_eq_abs] |
| exact absApproxDeriv_bound hε t |
|
|
| lemma continuous_absApproxDeriv {ε : ℝ} (hε : 0 < ε) : Continuous (absApproxDeriv ε) := by |
| change Continuous (fun t : ℝ => t / Real.sqrt (t^2+ε^2)) |
| apply continuous_id.div (Real.continuous_sqrt.comp (continuous_id.pow 2 |>.add continuous_const)) |
| intro t |
| exact (Real.sqrt_pos.mpr (add_pos_of_nonneg_of_pos (sq_nonneg _) (sq_pos_of_pos hε))).ne' |
|
|
| lemma absApprox_error {ε : ℝ} (hε : 0 ≤ ε) (t : ℝ) : abs (absApprox ε t-|t|) ≤ ε := by |
| have hl : |t| ≤ absApprox ε t := by |
| rw [←Real.sqrt_sq_eq_abs t] |
| exact Real.sqrt_le_sqrt (le_add_of_nonneg_right (sq_nonneg _)) |
| rw [abs_of_nonneg (sub_nonneg.mpr hl)] |
| have hu : absApprox ε t ≤ |t|+ε := by |
| apply (Real.sqrt_le_left (by positivity)).mpr |
| rw [add_sq,sq_abs] |
| nlinarith [abs_nonneg t] |
| linarith |
|
|
| lemma absApproxDeriv_tendsto {e : ℕ → ℝ} (he : Tendsto e atTop (𝓝 0)) |
| (hepos : ∀ j, 0 < e j) (t : ℝ) : |
| Tendsto (fun j => absApproxDeriv (e j) t) atTop (𝓝 (t / |t|)) := by |
| by_cases ht : t = 0 |
| · subst t |
| rw [zero_div] |
| apply tendsto_const_nhds.congr |
| intro index |
| dsimp [absApproxDeriv] |
| rw [zero_pow (by decide : 2 ≠ 0),zero_add,Real.sqrt_sq (hepos index).le,zero_div] |
| have hab : 0 < |t| := abs_pos.mpr ht |
| have hh := (tendsto_const_nhds (x := t) (f := (atTop : Filter ℕ))).div ((tendsto_const_nhds.add (he.pow 2)).sqrt) |
| (show Real.sqrt (t^2+(0:ℝ)^2) ≠ 0 by simpa only [zero_pow (by decide : 2 ≠ 0),add_zero,Real.sqrt_sq_eq_abs] using hab.ne') |
| simp only [zero_pow (by decide : 2 ≠ 0),add_zero,Real.sqrt_sq_eq_abs] at hh |
| exact hh |
|
|
| end CAT0Fillings.ClosedCalculus |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal NNReal |
|
|
| namespace CAT0Fillings.ChartGeometry |
| open ClosedCalculus AnalyticMinimizer |
|
|
| 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_abs (P : q.Sobolev) : |
| ∃ Q : q.Sobolev, |
| (q.inclusion Q : X → ℝ) =ᵐ[MassMeasure.currentMassMeasure hT] |
| (fun x => |(q.inclusion P) x|) ∧ |
| ‖q.inclusion Q‖ = ‖q.inclusion P‖ ∧ |
| ‖q.closedGradient Q‖ ≤ ‖q.closedGradient P‖ := by |
| let e (j : ℕ) : ℝ := 1/((j:ℝ)+1) |
| have hepos (j : ℕ) : 0 < e j := by dsimp [e]; positivity |
| have he : Tendsto e atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat |
| have hb (t : ℝ) : abs (t/|t|) ≤ 1 := by |
| rw [abs_div,abs_abs] |
| by_cases ht : t = 0 |
| · simp [ht] |
| · rw [div_self (abs_pos.mpr ht).ne'] |
| obtain ⟨Q,hQ,hG⟩ := q.closed_chain_uniform (F := abs) (D := fun t => t/|t|) |
| (show LipschitzWith 1 abs from LipschitzWith.of_dist_le_mul (fun x y => by simpa only [Real.dist_eq,NNReal.coe_one,one_mul] using abs_abs_sub_abs_le_abs_sub x y)) (measurable_id.div measurable_id.abs) (by norm_num : (0:ℝ) ≤ 1) hb |
| (fun j => absApprox (e j)) (fun j => absApproxDeriv (e j)) (fun _ => 1) |
| (fun j => absApprox_lipschitz (hepos j)) (fun j => absApprox_hasDerivAt (hepos j)) |
| (fun j => continuous_absApproxDeriv (hepos j)) (fun j => absApproxDeriv_bound (hepos j)) |
| (absApproxDeriv_tendsto he hepos) e he (fun j => absApprox_error (hepos j).le) P |
| refine ⟨Q,hQ,?_,?_⟩ |
| · apply le_antisymm <;> apply Lp.norm_le_norm_of_ae_le |
| · filter_upwards [hQ] with x hx |
| simp only [hx,Real.norm_eq_abs,abs_abs,le_refl] |
| · filter_upwards [hQ] with x hx |
| simp only [hx,Real.norm_eq_abs,abs_abs,le_refl] |
| · apply Lp.norm_le_norm_of_ae_le |
| filter_upwards [hG] with w hw |
| rw [hw,norm_smul,Real.norm_eq_abs] |
| exact (mul_le_mul_of_nonneg_right (hb _) (norm_nonneg _)).trans_eq (one_mul _) |
|
|
| lemma nonnegative_minimizer {p A B : ℝ} (hA : 0 ≤ A) (u : q.Sobolev) |
| (hu : criticalNorm q.inclusion p u = 1) |
| (hmin : ∀ w, energy q.inclusion q.closedGradient A B u * (criticalNorm q.inclusion p w)^2 ≤ |
| energy q.inclusion q.closedGradient A B w) : |
| ∃ v : q.Sobolev, |
| (∀ᵐ x ∂MassMeasure.currentMassMeasure hT, 0 ≤ (q.inclusion v) x) ∧ |
| criticalNorm q.inclusion p v = 1 ∧ |
| energy q.inclusion q.closedGradient A B v = energy q.inclusion q.closedGradient A B u ∧ |
| ∀ w, energy q.inclusion q.closedGradient A B v * (criticalNorm q.inclusion p w)^2 ≤ |
| energy q.inclusion q.closedGradient A B w := by |
| obtain ⟨v,hv,hv2,hvg⟩ := q.closed_abs u |
| have hvp : criticalNorm q.inclusion p v = 1 := by |
| rw [criticalNorm,lpNorm_congr_ae hv _,lpNorm_fun_abs (Lp.aestronglyMeasurable _)] |
| exact hu |
| have hle : energy q.inclusion q.closedGradient A B v ≤ energy q.inclusion q.closedGradient A B u := by |
| dsimp [energy] |
| rw [hv2] |
| exact add_le_add (mul_le_mul_of_nonneg_left |
| ((sq_le_sq₀ (norm_nonneg _) (norm_nonneg _)).mpr hvg) hA) le_rfl |
| have hge := hmin v |
| rw [hvp,one_pow,mul_one] at hge |
| have heq := le_antisymm hle hge |
| refine ⟨v,?_,hvp,heq,?_⟩ |
| · filter_upwards [hv] with x hx |
| rw [hx] |
| exact abs_nonneg _ |
| · simpa only [heq] using hmin |
|
|
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| end OAI |
|
|
